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The regularity of weak solutions for certain n-dimensional strongly coupled parabolic systems

  • Qi-Jian Tan EMAIL logo
Published/Copyright: July 27, 2022
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Advanced Nonlinear Studies
From the journal Advanced Nonlinear Studies Volume 22 Issue 1

Abstract

This paper is concerned with the n-dimensional strongly coupled parabolic systems with triangular form in the cylinder Ω × ( 0 , T ] . We investigate L 2 and Hölder regularity of the derivatives of weak solutions ( u 1 , u 2 ) for the systems in the following two cases: one is that the boundedness of u 1 and u 2 has not been shown in existence result of solutions; the other is that the boundedness of u 1 or u 2 has been shown in existence result of solutions. By using difference ratios and Steklov averages methods and various estimates, we prove that if ( u 1 , u 2 ) is a weak solution of the system, then for any Ω ′ ⊂ ⊂ Ω and t ′ ∈ ( 0 , T ) , u 1 , u 2 belong to C α ′ , α ′ / 2 ( Ω ¯ ′ × [ t ′ , T ] ) and W 2 2 , 1 ( Ω ′ × ( t ′ , T ] ) under certain conditions, and u 1 , u 2 belong to C 2 + α ′ , 1 + α ′ / 2 ( Ω ¯ ′ × [ t ′ , T ] ) under stronger assumptions. Applications of these results are given to two ecological models with cross-diffusion.

Keywords: strongly coupled parabolic systems; regularity; weak solutions; triangular systems
MSC 2010: 35K55; 35B65; 35K57

1 Introduction

Strongly coupled elliptic and parabolic systems often appear in different fields of physics, chemistry, biology, ecology, and engineering sciences (see [5,8,11,21,25,26, 27,29] and the references therein). In the past three decades, they have been given considerable attention in the literature in both theory and applications. The two important issues are the existence and regularity of solutions. The papers in [1,2,3, 5,6,7, 8,10,15,18,19,20,25,28,30] are concerned with the existence of weak and classical solutions, and those in [14,15,16, 17,22,23] are for the regularity of weak solutions.

Local existence (in time) of solutions to the strongly coupled parabolic systems with boundary conditions and initial conditions was established by Amann in a series of important papers [1,2,3] under the conditions that the boundary of the domain and the functions making up the system are smooth. There are extensive literature using the results of [1,2,3]. Hence, the solutions obtained in them have good regularity (see [6, 19, 23, 28] and the references therein). In other literature, since the functions making up the strongly coupled parabolic systems do not satisfy the conditions in [1,2,3], the results of [1,2,3] cannot be used, and the solutions obtained in these studies are weak solutions (see [6,8,10,11,26]). Then it is meaningful to investigate the regularity of the weak solutions. In previous studies [22,23], applying Gagliardo-Nirenberg type inequalities, the regularity of weak solutions for the one-dimensional cross-diffusion systems in population dynamics was studied by Shim, and the uniform W 2 1 -bound of the solutions was obtained. In a series of papers [14,15, 16,17], assuming different additional structure conditions, the regularity of weak solutions for the n-dimensional strongly coupled parabolic systems was investigated by Le. The Hölder regularity of bounded solutions was obtained in [14] by the perturbation method, and imbedding theorems of Campanato-Morrey spaces, the Hölder regularity of BMO weak solutions was proved in [16] by using the nonlinear heat approximation and BMO preserving homotopy, and the partial regularity results for bounded weak solutions were established in [17] by the method of heat approximation. Besides, some sufficient conditions on the structure of the systems to guarantee the boundedness and Hölder continuity of weak solutions were found in [15].

Strongly coupled parabolic systems are very difficult to analyze (see [4]). Some counterexamples show that we cannot expect weak solutions for general strongly coupled systems to be regular everywhere (see [14,24]). Therefore, to establish Hölder regularity for the derivatives of weak solutions, we consider strongly coupled systems of special form such as triangular systems. In addition, we note that in some model problems, the systems are of triangular form. For example, consider a two-species ecological model with cross-diffusion and self-diffusion on a bounded domain Ω ⊂ R n ( n = 1 , 2 , … ). Let u 1 = u 1 ( x , t ) and u 2 = u 2 ( x , t ) be the population densities of the two species, respectively, where x = ( x 1 , … , x n ) . Assume that the cross-diffusion pressures of the first species is zero. According to [3, 21], the nonnegative vector function ( u 1 , u 2 ) is governed by parabolic system in the following form:

(1.0) u 1 t = Δ [ ( κ 1 ( x , t ) + γ 11 ( x , t ) u 1 ) u 1 ] + div [ e 1 ( x , t ) u 1 ∇ φ ] + [ d 11 ( x , t ) + d 12 ( x , t ) u 1 + d 13 ( x , t ) u 2 ] u 1 ( ( x , t ) ∈ D T ) , u 2 t = Δ [ ( κ 2 ( x , t ) + γ 21 ( x , t ) u 1 + γ 22 ( x , t ) u 2 ) u 2 ] + div [ e 2 ( x , t ) u 2 ∇ φ ] + [ d 21 ( x , t ) + d 22 ( x , t ) u 1 + d 23 ( x , t ) u 2 ] u 2 ( ( x , t ) ∈ D T ) ,

where D T ≔ Ω × ( 0 , T ] for T > 0 , u l t ≔ ∂ u l / ∂ t , ∇ is the gradient operator, and Δ is the Laplace operator, and where γ 21 ( x , t ) is the cross-diffusion rate of the second species, κ l ( x , t ) , γ l l ( x , t ) are the diffusion rate and self-diffusion rate of the l -species, respectively, and φ is a known outer potential. This system is of triangular form because the cross-diffusion terms occur only in the second equation, and therefore, the diffusion matrix is triangular. In general, the known functions κ l , γ l l , γ 21 , e l , and d l j ( l = 1 , 2 ; j = 1 , 2 , 3 ) are allowed to be dependent on x and t .

Motivated by the aforementioned ecological models, in this paper, we consider a class of n-dimensional strongly coupled parabolic systems in the following form:

(1.1) u l t = div ∑ j = 1 l a l j ( x , t , [ U ] l ) ∇ u j + b l ( x , t , U , ∇ [ U ] l ) ( ( x , t ) ∈ D T ) , l = 1 , 2 ,

where U ≔ ( u 1 , u 2 ) , [ U ] 1 ≔ u 1 , [ U ] 2 ≔ ( u 1 , u 2 ) , ∇ [ U ] 1 ≔ ∇ u 1 and ∇ [ U ] 2 ≔ ( ∇ u 1 , ∇ u 2 ) .

For the triangular systems, sufficient conditions for the global existence of solutions were established in the previous work [2], the existence of solutions for some model problems was obtained in [9,18,20,28], and the Hölder regularity of bounded solutions and the uniform boundedness of global solutions were investigated in [14,23]. In [26], to study a free boundary problem describing S-K-T competition ecological model, by using Steklov average method and L ∞ and Hölder estimates in [12], we investigated L 2 and Hölder regularity of the derivatives of weak solutions for two special one-dimensional triangular systems (see [26, Lemmas 3.5, 4.1]).

In this paper, by using difference ratios and Steklov averages methods and various estimates, we will prove that if U = ( u 1 , u 2 ) is a weak solution from V 2 1 , 0 ( D T ) to (1.1), then for any Ω ′ ⊂ ⊂ Ω and t ′ ∈ ( 0 , T ) , u 1 , u 2 belong to C α ′ , α ′ / 2 ( Ω ¯ ′ × [ t ′ , T ] ) and W 2 2 , 1 ( Ω ′ × ( t ′ , T ] ) under certain conditions, and u 1 , u 2 belong to C 2 + α ′ , 1 + α ′ / 2 ( Ω ¯ ′ × [ t ′ , T ] ) under stronger assumptions. We investigate the regularity in the following two cases: one is that the boundedness of u 1 and u 2 on D ¯ T has not been shown in existence result of solutions (see [6, Theorem 1.1], [8, Theorem 2.1], [10, Theorem 1], and [26, Proposition 3.1]); the other is that the boundedness of u 1 or u 2 has been shown in existence result of solutions (see [10, Theorem 2], [11, Theorem 1.1], and [26, Proposition 3.8]). According to the form of the equations in (1.1), to establish the regularity of ( u 1 , u 2 ) , we first study the regularity of u 1 from the first equation of (1.1), and then apply the obtained result to investigate the regularity of u 2 from the second equation.

This paper is organized as follows: in Section 2, we give the definitions, hypotheses, notations, main theorems, and some preliminaries. Sections 3 and 4 are devoted to the proofs of the two main theorems, respectively. Finally, in Section 5, we give applications of the aforementioned results to two ecological models with cross-diffusion.

2 Hypotheses, main results, and preliminaries

2.1 Definitions, hypotheses, and main results

In this paper, we follow the function space notations adopted in [12]. The symbol Ω ∗ ⊂ ⊂ Ω means that Ω ∗ ⊂ Ω and Ω ¯ ∗ ⊂ Ω , where Ω ¯ ∗ denotes the closure of Ω ∗ .

Since we study only interior regularity of solutions, then the weak solutions of system (1.1) are defined as follows:

Definition 2.1

A vector function U = ( u 1 , u 2 ) is called a weak solution of system (1.1) if for each l = 1 , 2 , u l ∈ V 2 1 , 0 ( D T ) , and u l t ∈ L 2 ( [ 0 , T ] ; ( H 1 ( Ω ) ) ∗ ) , and if U satisfies

(2.1) ∫ τ ′ τ ⟨ u l t , η ⟩ d t + ∫ τ ′ τ ∫ Ω ′ ∑ j = 1 l a l j ( x , t , [ U ] l ) ∇ u j ⋅ ∇ η − b l ( x , t , U , ∇ [ U ] l ) η d x d t = 0 , l = 1 , 2 ,

for any τ ′ , τ ∈ ( 0 , T ] , Ω ′ ⊂ ⊂ Ω , and for any η ∈ V ∘ 2 1 , 0 ( Ω ′ × ( τ ′ , T ] ) , where ⟨ ⋅ , ⋅ ⟩ denotes the duality parring between H 1 ( Ω ′ ) and its dual space ( H 1 ( Ω ′ ) ) ∗ .

In the following discussions, we always use U = ( u 1 , u 2 ) to denote a nonconstant weak solution of (1.1) given by existence result of solutions. In addition, for each l = 1 , 2 , we define a interval S l corresponding to function u l . S l is the closure of ( a ̲ l , a ¯ l ) , where

a ̲ l = ess inf D T u l if it is known that u l is bounded in D T from below , a ̲ l = − ∞ if it is not known that u l is bounded in D T from below ,

and

a ¯ l = ess sup D T u l if it is known that u l is bounded in D T from above , a ¯ l = + ∞ if it is not known that u l is bounded in D T from above .

Set

S ≔ S 1 × S 2 , [ S ] 1 ≔ S 1 , [ S ] 2 ≔ S 1 × S 2 , W ≔ ( w 1 , w 2 ) , [ W ] 1 ≔ w 1 , [ W ] 2 ≔ ( w 1 , w 2 ) , p ≔ ( p 1 , … , p n ) , p k ≔ ( p k 1 , … , p k n ) , P ≔ ( p 1 , p 2 ) , [ P ] 1 ≔ p 1 , [ P ] 2 ≔ ( p 1 , p 2 ) ,

and set

(2.2) q 0 ≔ 2 ( 2 + n ) n , q 1 ≔ 2 + n 2 ( 1 − χ 1 )

for some χ 1 ∈ ( 0 , 1 / 4 ) . Then ( q 0 , q 0 ) satisfies [12, Chapter II, equality (3.3)] with q = r = q 0 , and ( q 1 , q 1 ) satisfies [12, Chapter III, equality (7.2)] with q = r = q 1 . Specially, q 0 = 6 and q 1 = 3 / [ 2 ( 1 − χ 1 ) ] ∈ ( 3 / 2 , 2 ) for n = 1 .

To investigate the regularity of the weak solutions, we will make the hypotheses ( H 1 ) and ( H 2 ):

  1. For l = 1 , 2 , i = 1 , … , n , assume that a l l ( x , t , [ W ] l ) ∈ C 1 ( D T × [ S ] l ) , a l l x i x i ( x , t , [ W ] l ) , ∈ C ( D T × [ S ] l ) , and that a 22 x i w 1 ( x , t , W ) , a 22 w 1 x i , a 22 w 1 w 1 ∈ C ( D T × S ) . In addition, for ( x , t ) ∈ D T , W ∈ S and [ P ] l ∈ ( R n ) l , functions a 21 ( x , t , W ) and b l ( x , t , W , [ P ] l ) have the first partial derivatives with respect to their variables in a pointwise sense.

    1. If S 1 and S 2 are all unbounded, we assume that for ( x , t ) ∈ D T , W ∈ S , and P ∈ ( R n ) 2 ,

      (2.3) ∣ a 21 ( x , t , W ) ∣ ≤ μ ( 0 ) , ν ( ∣ [ W ] l − 1 ∣ ) ≤ a l l ( x , t , [ W ] l ) ≤ μ ( ∣ [ W ] l − 1 ∣ ) , l = 1 , 2

      and

      (2.4) ∣ b l ( x , t , W , [ P ] l ) ∣ ≤ μ ( ∣ [ W ] l − 1 ∣ ) ( 1 + ∣ w l ∣ δ l ) ∑ k = 1 l ∣ p k ∣ + ∑ m = l 2 ∣ w m ∣ σ l m + ϕ l ( x , t ) , l = 1 , 2 ,

      where ν ( θ ) is a positive nonincreasing continuous function for θ ≥ 0 and μ ( θ ) is a positive nondecreasing continuous function for θ ≥ 0 and ∣ [ W ] l − 1 ∣ = 0 for l = 1 , where δ l , σ l l , and σ 12 are all nonnegative constants satisfying

      (2.5) δ l ≤ 2 ( 1 − χ 1 ) / n , σ l l ≤ 4 ( 1 − χ 1 ) / n , l = 1 , 2 ,

      (2.6) σ 12 ≤ 3 for n = 1 , σ 12 ≤ 2 ( 1 − χ 1 ) / n for n = 2 , 3 , … ,

      and ϕ l ( x , t ) is nonnegative function with the finite norm

      (2.7) ‖ ϕ 1 ‖ L 2 q 1 ( D T ) ≤ ς , ‖ ϕ 2 ‖ L q 1 ( D T ) + ‖ ϕ 2 ‖ L 2 ( D T ) ≤ ς , sup 0 ≤ t ≤ T ‖ ϕ 1 , ϕ 2 ‖ L q 2 ( Ω ) ≤ ς .

      Here, q 2 , ς are positive constants with q 2 = 1 for n = 1 , and q 2 > n for n = 2 , 3 , … .

    2. For any fixed l ∈ { 1 , 2 } , if there exists nonempty set E ⊂ { l , 2 } such that for each m ∈ E , S m is bounded, we assume that the conditions in (a) are satisfied except that in (2.4) for b l ( x , t , W , [ P ] l ) , there is no item ∣ w m ∣ σ l m for each m ∈ E , there is no item ∣ w l ∣ δ l if l ∈ E , and μ ( ∣ [ W ] l − 1 ∣ ) is replaced by μ ( ∣ [ W ] l − 1 ∣ + ∑ m ∈ E ∣ w m ∣ ) .

    1. If S 2 is unbounded, assume that for ( x , t ) ∈ D T , W ∈ S and P ∈ ( R n ) 2 ,

      (2.8) ∣ a 21 x i ( x , t , W ) ∣ + ∣ a 21 w k ∣ ≤ μ ( ∣ W ∣ ) , i = 1 , … , n , k = 1 , 2 ,

      and

      (2.9) ∂ ∂ x i b l ( x , t , W , [ P ] l ) ≤ μ ( ∣ [ W ] l ∣ ) ∑ k = 1 l ∣ p k ∣ + ( 2 − l ) ∣ w 2 ∣ σ ˜ 11 + ϕ ˜ l 1 ( x , t ) , ∂ ∂ w l b l ≤ μ ( ∣ [ W ] l ∣ ) ∑ k = 1 l ∣ p k ∣ + ( 2 − l ) ∣ w 2 ∣ σ ˜ 12 + ϕ ˜ l 2 ( x , t ) , ∂ ∂ w r b l ≤ μ ( ∣ [ W ] l ∣ ) ∑ k = 1 l ∣ p k ∣ + ( 2 − l ) ∣ w 2 ∣ σ ˜ 13 + ϕ ˜ l 3 ( x , t ) , r = 1 , 2 , r ≠ l , ∂ ∂ p k i b l ≤ μ ( ∣ [ W ] l ∣ ) [ ( 2 − l ) ∣ w 2 ∣ σ ˜ 14 + ϕ ˜ l 4 ( x , t ) ] , i = 1 , … , n , k = 1 , l ; l = 1 , 2 ,

      where σ ˜ 1 r are nonnegative constants satisfying

      (2.10) σ ˜ 11 , σ ˜ 12 ≤ 4 ( 1 − χ 1 ) / n , σ ˜ 13 , σ ˜ 14 ≤ 2 ( 1 − χ 1 ) / n ,

      and where ϕ ˜ l k are nonnegative functions satisfying

      (2.11) sup 0 ≤ t ≤ T ‖ ϕ ˜ 11 1 / 2 , ϕ ˜ 12 1 / 2 , ϕ ˜ 13 , ϕ ˜ 14 ‖ L q ˜ 2 ( Ω ) ≤ ς ˜ for n = 2 , 3 , … , sup 0 ≤ t ≤ T ‖ ϕ ˜ 21 1 / 2 , ϕ ˜ 22 1 / 2 , ϕ ˜ 23 1 / 2 , ϕ ˜ 24 ‖ L q ˜ 2 ( Ω ) ≤ ς ˜ for n = 1 , 2 , … .

      Here, q ˜ 2 and ς ˜ are also positive constants with q ˜ 2 = 1 for n = 1 , and q ˜ 2 > n for n = 2 , 3 , … .

    2. If S 2 is bounded, we assume that the conditions in (A) are satisfied except that in (2.9) with l = 1 , there are no items ∣ w 2 ∣ σ ˜ 1 r , r = 1 , … , 4 , and μ ( ∣ [ W ] 1 ∣ ) is replaced by μ ( ∣ W ∣ ) .

  1. For any Ω ′ ⊂ ⊂ Ω , t ′ ∈ ( 0 , T ] and for any bounded domains S ′ = S 1 ′ × S 2 ′ ⊂ S and E ′ ⊂ ( R n ) l , there exists α 0 ′ ∈ ( 0 , 1 ) , such that a l j x i ( x , t , [ W ] l ) , a l j w k ∈ C α 0 ′ ( Ω ¯ ′ × [ t ′ , T ] × [ S ¯ ′ ] l ) ( i = 1 , … , n ; j , k = 1 , l ) , and b l ( x , t , W , [ P ] l ) ∈ C 1 ( Ω ¯ ′ × [ t ′ , T ] × S ¯ ′ × E ¯ ′ ) .

The main results of this paper are the following two theorems:

Theorem 2.1

Let U = U ( x , t ) be a weak solution of system (1.1). If hypothesis ( H 1 ) holds, then for any Ω ′ ⊂ ⊂ Ω and t ′ ∈ ( 0 , T ) , u 1 and u 2 are Hölder continuous in Ω ¯ ′ × [ t ′ , T ] , u 1 belongs to W q 3 2 , 1 ( Ω ′ × ( t ′ , T ] ) , and u 2 belongs to W 2 2 , 1 ( Ω ′ × ( t ′ , T ] ) , where q 3 = 2 q 1 if n ≥ 2 , q 3 = min { 2 q 1 , 6 } if n = 1 .

Theorem 2.2

Let U = U ( x , t ) be a weak solution of system (1.1). If hypotheses ( H 1 ) and ( H 2 ) all hold, then for any Ω ′ ⊂ ⊂ Ω and t ′ ∈ ( 0 , T ) , there exists α ′ ∈ ( 0 , 1 ) , such that u 1 and u 2 belong to C 2 + α ′ , 1 + α ′ / 2 ( Ω ¯ ′ × [ t ′ , T ] ) .

2.2 Some more notations and preliminaries

We introduce some more functions and notations used throughout the paper. For function w = w ( x , t ) , we denote

(2.12) w ( σ ) ≔ max { w − σ , 0 } , A w , σ ( t ) ≔ { x : w ( x , t ) > σ , x ∈ Ω } ,

(2.13) w h ( x , t ) ≔ 1 h ∫ t t + h w ( x , θ ) d θ , w ( t ) ≔ w ( x , t + h ) − w ( x , t ) h ,

and

(2.14) w ( x k ) ( x , t ) ≔ w ( x + h k , t ) − w ( x , t ) h k ,

where k ∈ { 1 , … , n } and x + h k ≔ ( x 1 , … , x k − 1 , x k + h k , x k + 1 , … , x n ) . Then w h ( x , t ) is the Steklov average of w in t , and w ( t ) , w ( x k ) are difference ratios. The expression [ f ( x , t , w ( x , t ) , ∇ w ( x , t ) ) ] x i means that

[ f ( x , t , w ( x , t ) , ∇ w ( x , t ) ) ] x i = ∂ f ∂ x i + ∂ f ∂ w w x i + ∑ j = 1 n ∂ f ∂ w x j w x j x i .

Let

(2.15) Λ 1 ( U ) ≔ ∣ u 2 ∣ σ ˜ 11 / 2 + ∣ u 2 ∣ σ ˜ 12 / 2 + ∣ u 2 ∣ σ ˜ 13 + ∣ u 2 ∣ σ ˜ 14 , F l ( U ) ≔ ∑ m = l 2 ∣ u m ∣ σ l m , l = 1 , 2 , Φ ˜ 1 ( x , t ) ≔ ϕ ˜ 11 1 / 2 + ϕ ˜ 12 1 / 2 + ϕ ˜ 13 + ϕ ˜ 14 , Φ ˜ 2 ( x , t ) ≔ ϕ ˜ 21 1 / 2 + ϕ ˜ 22 1 / 2 + ϕ ˜ 23 1 / 2 + ϕ ˜ 24 ,

and let

(2.16) q ˜ 0 ≔ 2 q 0 q 0 − 2 = n + 2 , q ˆ 0 ≔ q 0 q 0 − 1 = 2 n + 4 n + 4 .

For ρ > 0 and P ¯ ≔ ( x ¯ , t ¯ ) ∈ D T , we denote

B ρ ( x ¯ ) ≔ { x : ∣ x − x ¯ ∣ < ρ } , Q ρ , τ ( P ¯ ) ≔ B ρ ( x ¯ ) × ( t ¯ − τ , t ¯ ) , Q ρ ( P ¯ ) ≔ Q ρ , ρ 2 ( P ¯ ) .

We define a function ξ ρ = ξ ρ ( x , t ) corresponding to the two cylinders Q ρ ( P ¯ ) and Q 2 ρ ( P ¯ ) as follows:

(2.17) ξ ρ is a smooth function taking values in [ 0 , 1 ] , such that ξ ρ = 0 on the lateral and base of Q 2 ρ ( P ¯ ) , ξ ρ = 1 for ( x , t ) ∈ Q ρ ( P ¯ ) and ∣ ξ ρ t ∣ + ∣ ∇ ξ ρ ∣ 2 ≤ C / ρ 2 for all ( x , t ) ∈ Q 2 ρ ( P ¯ ) .

In the following discussions, let t ′ ∈ ( 0 , T ) be an arbitrary fixed number, and let Ω ′ ⊂ ⊂ Ω be an arbitrary fixed domain. Set t j = j t ′ / 20 , j = 1 , … , 20 . Choose subdomains Ω j with smooth boundaries, such that Ω ′ ⊂ ⊂ Ω 20 ⊂ ⊂ … ⊂ ⊂ Ω 1 ⊂ ⊂ Ω , dist ( Ω j , ∂ Ω j − 1 ) = d ≔ dist ( Ω ′ , ∂ Ω ) / 20 . Denote

D j , T ≔ Ω j × ( t j , T ] , D ¯ j , T ≔ Ω ¯ j × [ t j , T ] , D j , τ ≔ Ω j × ( t j , τ ] .

For each j = 1 , … , 20 , we define function λ j = λ j ( x , t ) as follows:

(2.18) λ j is a smooth function with values between 0 and 1, such that λ j = 0 for x ∉ Ω j − 1 or t ≤ t j − 1 , λ j = 1 for ( x , t ) ∈ D ¯ j , T and ∣ ∇ λ j ∣ + ∣ λ j t ∣ ≤ C ( d , t ′ ) for all ( x , t ) ∈ D ¯ T .

Lemma 2.3

There exist positive constants C and α 0 ∈ ( 0 , 1 ) such that for each l = 1 , 2 ,

(2.19) ‖ u l ‖ L q 0 ( D T ) ≤ C ‖ u l ‖ V 2 1 , 0 ( D T ) ,

(2.20) ‖ ∣ u l ∣ δ l ‖ L q ˜ 0 ( D T ) + ‖ F l ( U ) ‖ L q ˆ 0 ( D T ) ≤ C , ‖ ∣ u l ∣ 2 δ l , F l ( U ) , ∣ u 2 ∣ σ 12 ‖ L q 1 ( D T ) ≤ C ,

(2.21) ‖ ∣ u 2 ∣ σ 12 ‖ L 2 ( D T ) ≤ C ,

and for n ≥ 2 and any B ρ ( x ¯ ) ⊂ Ω ,

(2.22) ∫ B ρ ( x ¯ ) [ ∣ u 2 ∣ 2 σ 12 + Λ 1 2 ( U ) + ϕ l 2 + Φ ˜ l 2 ] d x ≤ C ρ n − 2 + 2 α 0 .

Proof

It follows from [12, Chapter II, inequality (3.4)] that (2.19) holds. By a direct computation, we see from (2.2) and (2.16) that

min { q 0 / q ˜ 0 , q 0 / ( 2 q 1 ) } = 2 ( 1 − χ 1 ) / n , min { q 0 / q ˆ 0 , q 0 / q 1 } = 4 ( 1 − χ 1 ) / n .

Thus, by (2.5) and (2.6), we further obtain the relations

2 σ 12 ≤ q 0 , 2 δ l q 1 , δ l q ˜ 0 ≤ q 0 , σ l m q 1 , σ l m q ˆ 0 ≤ q 0 for m = l , 2 .

These, together with (2.15) and (2.19), give (2.20) and (2.21).

We next prove that (2.22) holds for n ≥ 2 . Conditions (2.6) and (2.7) show that 1 − n / q 2 > 0 and 2 − n σ 12 ≥ 2 χ 1 > 0 . Hence,

∫ B ρ ( x ¯ ) ϕ l 2 d x ≤ ‖ ϕ l 2 ‖ L q 2 / 2 ( B ρ ( x ¯ ) ) ∫ B ρ ( x ¯ ) d x 1 − 2 / q 2 ≤ C ρ n − 2 + 2 ( 1 − n / q 2 ) , ∫ B ρ ( x ¯ ) ∣ u 2 ∣ 2 σ 12 d x ≤ ‖ u 2 ‖ L 2 ( B ρ ( x ¯ ) ) 2 σ 12 ∫ B ρ ( x ¯ ) d x 1 − σ 12 ≤ C ρ n − 2 + ( 2 − n σ 12 ) .

Moreover, by (2.15), (2.10), and (2.11), ∫ B ρ ( x ¯ ) Λ 1 2 ( U ) d x and ∫ B ρ ( x ¯ ) Φ ˜ l 2 d x satisfy the similar inequalities. Setting α 0 ≔ min { χ 1 , 1 − n / q 2 , 1 − n / q ˜ 2 } , we obtain (2.22).□

The following preliminary lemma will be used to investigate L ∞ estimates.

Lemma 2.4

Assume that w belongs to V 2 ( D T ) and σ ˆ is a nonnegative constant. Then the following statements hold true:

(i) Let Q ρ , τ ( P ¯ ) ⊂ D T be an arbitrary cylinder, and let ζ = ζ ( x , t ) be an arbitrary piecewise-smooth continuous nonnegative function that does not exceed 1 and is equal to zero on the lateral surface and the lower base of the cylinder Q ρ , τ ( P ¯ ) . Assume that for any τ 1 , τ 2 , and σ , t ¯ − τ < τ 1 < τ 2 < t ¯ , σ ≥ σ ˆ ,

(2.23) ess sup τ 1 ≤ t ≤ τ 2 ‖ w ( σ ) ( x , t ) ζ ( x , t ) ‖ L 2 ( Ω ) 2 + ς ̲ ∫ τ 1 τ 2 ∫ A w , σ ( t ) ∣ ∇ w ∣ 2 ζ 2 d x d t ≤ ‖ w ( σ ) ( x , τ 1 ) ζ ( x , τ 1 ) ‖ L 2 ( Ω ) 2 + ∫ τ 1 τ 2 ∫ A w , σ ( t ) { ς ¯ ( ∣ ∇ ζ ∣ 2 + ζ ∣ ζ t ∣ ) ( w ( σ ) ) 2 + G ( x , t ) ζ 2 [ ( w − σ ) 2 + σ 2 ] } d x d t ,

where ς ̲ and ς ¯ are positive constants, and function G ( x , t ) is in L q , r ( D T ) for some q , r satisfying [12, Chapter III, equality (7.2)]. Then for any Ω ″ ⊂ ⊂ Ω and t ″ ∈ ( 0 , T ) , ess sup Ω ″ × ( t ″ , T ] w does not exceed a constant M ˜ determined only by T , σ ˆ , ‖ G ( x , t ) ‖ L q , r ( D T ) , ‖ w ‖ L 2 ( D T ) , t ″ and dist ( Ω ″ , ∂ Ω ) .

(ii) Assume that for any τ ∈ ( 0 , T ] ,

(2.24) ∫ Ω ( w ( σ ) ( x , τ ) ) 2 d x + ∫ 0 τ ∫ A w , σ ( t ) ∣ ∇ w ∣ 2 d x d t ≤ ∫ 0 τ ∫ A w , σ ( t ) G ( x , t ) [ ( w − σ ) 2 + σ 2 ] d x d t ,

where function G ( x , t ) is in L q , r ( D T ) for some q , r satisfying [12,Chapter III, equality (7.2)]. Then ess sup D T w does not exceed a constant M ˜ determined only by T , σ ˆ and ‖ G ( x , t ) ‖ L q , r ( D T ) .

Proof

By inequality (2.23), we can conclude that w satisfies [12, Chapter III, inequality (8.2)]. The deduction is the same as that of [12, Chapter III, inequality (7.14)] from [12, Chapter III, inequality (7.8)]. Using [12, Chapter II, Theorem 6.2 and Remark 6.4], we further obtain the result of part (i) of this lemma.

Since inequality (2.24) has the same property as that of [12, Chapter III, inequality (7.8)], then by [12, Chapter II, Theorem 6.1 and Remark 6.2], the similar proof as that of [12, Chapter III, estimate (7.15)] gives the result of part (ii) of this lemma.□

3 The proof of Theorem 2.1

In this section, assume that hypothesis ( H 1 ) holds. We will only prove Theorem 2.1 for the case that S 1 and S 2 are all unbounded, because the proofs of Theorem 2.1 for the other cases are more simple.

3.1 Hölder estimate of u 1

For simplicity, in this section, we will use M 1 , M 2 , C , C ( ⋯ ) , C j and α j ( j = 1 , 2 , … ) to denote positive constants depending only on the parameters

(3.1) T , mes Ω , dist ( Ω ′ , ∂ Ω ) , t ′ , ν ( 0 ) , μ ( 0 ) , ς , ς ˜ , χ 1 , n , q 2 , q ˜ 2 , ‖ u l ‖ V 2 1 , 0 ( D T ) , δ l , σ l l , σ 12 , σ ˜ 1 r , r = 1 , … , 4 , l = 1 , 2 ,

and the quantities appearing in parentheses. In the same lemma, the same letter C will be used to denote different constants depending on the same set of arguments. We first give the Hölder estimate of u 1 .

Lemma 3.1

The integrals in the first equality of (2.1) exist and are finite. There exist positive constants C and α 1 such that

(3.2) ‖ u 1 ‖ C α 1 , α 1 / 2 ( D ¯ 2 , T ) ≤ C , α 1 ∈ ( 0 , 1 ) .

Proof

Step 1. We show that the integrals in the first equality of (2.1) exist and are finite. For any τ ∈ ( t 1 , T ] and η ∈ V ∘ 2 1 , 0 ( D 1 , T ) , using condition (2.4) and [12, Chapter II, inequalities (1.5) and (3.4)], we have

∬ D 1 , τ ∣ b 1 ( x , t , U , ∇ [ U ] 1 ) η ∣ d x d t ≤ ∬ D 1 , τ { μ ( 0 ) [ ( 1 + ∣ u 1 ∣ δ 1 ) ∣ ∇ u 1 ∣ + F 1 ( U ) ] + ϕ 1 ( x , t ) } ∣ η ∣ d x d t ≤ C { [ ‖ ∣ u 1 ∣ δ 1 ‖ L q ˜ 0 ( D 1 , T ) ‖ ∇ u 1 ‖ L 2 ( D 1 , T ) + ‖ F 1 ( U ) ‖ L q ˆ 0 ( D 1 , T ) ] ‖ η ‖ L q 0 ( D 1 , T ) + [ ‖ ∇ u 1 ‖ L 2 ( D 1 , T ) + ‖ ϕ 1 ‖ L 2 ( D 1 , T ) ] ‖ η ‖ L 2 ( D 1 , T ) } ,

where F 1 ( U ) , q ˜ 0 , and q ˆ 0 are defined by (2.15) and (2.16). Therefore, by (2.20) and (2.7), ∬ D 1 , τ ∣ b 1 η ∣ d x d t is finite. Besides, it follows from condition (2.3) that ∫ t 1 τ ⟨ u 1 t , η ⟩ d t + ∬ D 1 , τ a 11 ( x , t , [ U ] 1 ) ∇ u 1 ⋅ ∇ η d x d t is also finite. Since t 1 = t ′ / 20 is an arbitrary fixed number in ( 0 , T ) and Ω 1 is an arbitrary subdomain of Ω , then the integrals in the first equality of (2.1) exist and are finite.

Step 2. We prove that

(3.3) ess sup D 1 , T ∣ u 1 ∣ ≤ M 1 .

Let cylinder Q ρ , τ ( P ¯ ) and function ζ = ζ ( x , t ) be same as those in Lemma 2.4, and let σ ˆ ≥ 1 . Setting η = u 1 ( σ ) ζ 2 ( x , t ) for σ ≥ σ ˆ in the first equality of (2.1) yields, for any τ 1 and τ ∗ , t ¯ − τ < τ 1 ≤ τ ∗ < t ¯ ,

(3.4) 1 2 ∫ Ω ( u 1 ( σ ) ζ ) 2 d x t = τ 1 t = τ ∗ + ∫ τ 1 τ ∗ ∫ A u 1 , σ ( t ) a 11 ( x , t , u 1 ) ∣ ∇ u 1 ∣ 2 ζ 2 d x d t = ∫ τ 1 τ ∗ ∫ A u 1 , σ ( t ) { − a 11 ( x , t , u 1 ) u 1 ( σ ) ∇ u 1 ⋅ ( 2 ζ ∇ ζ ) + ( u 1 ( σ ) ) 2 ζ ζ t + b 1 ( x , t , U , ∇ u 1 ) u 1 ( σ ) ζ 2 } d x d t ,

where notations u 1 ( σ ) and A u 1 , σ are defined by (2.12). By using conditions (2.3), (2.4), and Cauchy’s inequality with ε , we find from (3.4) that for any ε ∈ ( 0 , 1 ) ,

1 2 ∫ Ω ( u 1 ( σ ) ζ ) 2 d x t = τ 1 t = τ ∗ + ∫ τ 1 τ ∗ ∫ A u 1 , σ ( t ) a 11 ( x , t , u 1 ) ∣ ∇ u 1 ∣ 2 ζ 2 d x d t ≤ C ∫ τ 1 τ ∗ ∫ A u 1 , σ ( t ) { ( u 1 ( σ ) ) 2 ζ ∣ ζ t ∣ + ∣ ∇ u 1 ∣ u 1 ( σ ) ζ ∣ ∇ ζ ∣ + [ ( 1 + ∣ u 1 ∣ δ 1 ) ∣ ∇ u 1 ∣ + F 1 ( U ) + ϕ 1 ] u 1 ( σ ) ζ 2 } d x d t ≤ ε ∫ τ 1 τ ∗ ∫ A u 1 , σ ( t ) ∣ ∇ u 1 ∣ 2 ζ 2 d x d t + C ε ∫ τ 1 τ ∗ ∫ A u 1 , σ ( t ) { ( u 1 ( σ ) ) 2 ( ζ ∣ ζ t ∣ + ∣ ∇ ζ ∣ 2 ) + [ ( 1 + ∣ u 1 ∣ 2 δ 1 ) ( u 1 ( σ ) ) 2 + ( F 1 ( U ) + ϕ 1 ) u 1 ( σ ) ] ζ 2 } d x d t .

Note that ∣ u 1 − σ ∣ ≤ max { 1 , ( u 1 − σ ) 2 } and σ ≥ 1 . Then

u 1 ( σ ) ≤ ∣ u 1 − σ ∣ ≤ ( u 1 − σ ) 2 + σ 2 .

In view of a 11 ( x , t , u 1 ) ≥ ν ( 0 ) , taking ε = min { 1 / 2 , ν ( 0 ) / 2 } , we further obtain, for any t ¯ − τ < τ 1 < τ 2 < t ¯ ,

sup τ 1 ≤ t ≤ τ 2 ‖ u 1 ( σ ) ( x , t ) ζ ( x , t ) ‖ L 2 ( Ω ) 2 + ν ( 0 ) 2 ∫ τ 1 τ 2 ∫ A u 1 , σ ( t ) ∣ ∇ u 1 ∣ 2 ζ 2 d x d t ≤ ‖ u 1 ( σ ) ( x , τ 1 ) ζ ( x , τ 1 ) ‖ L 2 ( Ω ) 2 + ∫ τ 1 τ 2 ∫ A u 1 , σ ( t ) { G 1 ( x , t ) ζ 2 [ ( u 1 − σ ) 2 + σ 2 ] + C ( ∣ ∇ ζ ∣ 2 + ζ ∣ ζ t ∣ ) ( u 1 ( σ ) ) 2 } d x d t ,

where G 1 ( x , t ) = C [ 1 + ∣ u 1 ∣ 2 δ 1 + F 1 ( U ) + ϕ 1 ] . It follows from (2.7) and (2.20) that ‖ G 1 ( x , t ) ‖ L q 1 ( D T ) is bounded from above by C . Then u 1 satisfies inequality (2.23) with w replaced by u 1 . Lemma 2.4 shows that ess sup D 1 , T u 1 does not exceed a constant C . The similar argument implies that ess sup D 1 , T ( − u 1 ) also does not exceed C . Hence, (3.3) holds.

Step 3. We show Hölder estimate (3.2). Set

(3.5) B 1 , i ( x , t , w , p ) ≔ a 11 ( x , t , w ) p i , B 1 ( x , t , w , p ) ≔ − b 1 ( x , t , w , u 2 , p ) .

It follows from the first integral equality of (2.1) that for any τ 0 , τ ∈ ( t 1 , T ] and for any η ∈ W ∘ 2 1 , 1 ( D 1 , T ) ,

(3.6) ∫ Ω 1 u 1 ( x , t ) η ( x , t ) d x τ 0 τ + ∫ τ 0 τ ∫ Ω 1 − u 1 η t + ∑ i = 1 n B 1 , i ( x , t , u 1 , ∇ u 1 ) η x i + B 1 ( x , t , u 1 , ∇ u 1 ) η d x d t = 0 .

From estimate (3.3) and conditions (2.3) and (2.4), we obtain, for ( x , t ) ∈ D 1 , T , w ∈ [ − M 1 , M 1 ] ∩ S 1 and p ∈ R n ,

(3.7) ∑ i = 1 n B 1 , i ( x , t , w , p ) p i ≥ ν ( 0 ) ∣ p ∣ 2 , ∣ B 1 , i ( x , t , w , p ) ∣ ≤ μ ( 0 ) ∣ p ∣ ,

and

(3.8) ∣ B 1 ( x , t , w , p ) ∣ ≤ C ( ∣ p ∣ + ∣ u 2 ∣ σ 12 + ϕ 1 + 1 ) .

In view of (2.7) and (2.20), this equality has the same form as [12, Chapter V, equality (1.6)]. Then by using (3.3), (3.6)–(3.8), and [12, Chapter V, Theorem 1.1], we obtain estimate (3.2).□

3.2 Estimate of ‖ u 1 ‖ W 2 2 , 1 ( Ω ′ × ( t ′ , T ] )

Based on Hölder estimate of u 1 , we will prove that u 1 has weak derivative u 1 t by using the Steklov average method. In view of hypothesis ( H 1 )(I), we obtain, for some positive constant Θ 1 ,

(3.9) ‖ a 11 ( x , t , w ) ‖ C 1 ( Ξ 1 ) , ‖ a 11 x i x i ( x , t , w ) ‖ C ( Ξ 1 ) ≤ Θ 1 , i = 1 , … , n ,

where Ξ 1 ≔ D ¯ 1 , T × ( S 1 ∩ [ − M 1 , M 1 ] ) .

Lemma 3.2

Function u 1 has weak derivative u 1 t in D 3 , T , and satisfies the inequality

(3.10) ess sup t 3 ≤ t ≤ T ∫ Ω 3 ∣ ∇ u 1 ∣ 2 d x + ∬ D 3 , T u 1 t 2 d x d t ≤ C ( Θ 1 ) .

Proof

Let x 2 ∈ Ω 2 be fixed, and let u 1 ∗ = u 1 ( x 2 , t 2 ) . Define

(3.11) w ˆ 1 = w ˆ 1 ( x , t ) = ∫ u 1 ∗ u 1 a 11 ( x , t , ω ) d ω .

Then

(3.12) w ˆ 1 x i = u 1 x i a 11 ( x , t , u 1 ) + ∫ u 1 ∗ u 1 a 11 x i ( x , t , ω ) d ω , i = 1 , … , n .

By using integration by parts, we see from the first equality of (2.1) that for any η ∈ W ∘ 2 1 , 1 ( D 2 , T ) and τ ∈ ( t 2 , T ] ,

(3.13) ∫ Ω 2 u 1 η d x t 2 τ − ∬ D 2 , τ u 1 η t d x d t = ∬ D 2 , τ [ − ∇ w ˆ 1 ⋅ ∇ η + f ˆ 1 ( x , t ) η ] d x d t ,

where

(3.14) f ˆ 1 ( x , t ) = − ∑ i = 1 n ∫ u 1 ∗ u 1 a 11 x i ( x , t , ω ) d ω x i + b 1 ( x , t , U , ∇ u 1 ) .

From the similar arguments as those of [12, Chapter III, Section 2], it follows that for any given h ∈ ( 0 , T − t 3 ) and τ ∈ ( t 2 , T − h ] and for any η ∈ V ∘ 2 1 , 0 ( D 2 , T ) ,

(3.15) ∬ D 2 , τ u 1 ( t ) η d x d t = ∬ D 2 , τ { − ∇ w ˆ 1 h ⋅ ∇ η + [ f ˆ 1 ( x , t ) ] h η } d x d t .

Here and below, notations w 1 h and w 1 ( t ) are defined by (2.13). Note that by (3.11),

w ˆ 1 ( t ) = u 1 ( t ) ∫ 0 1 a 11 ( x , t ϑ , u 1 ϑ ) d ϑ + ∫ 0 1 ∫ u 1 ∗ u 1 ϑ a 11 t ϑ ( x , t ϑ , ω ) d ω d ϑ ,

where t ϑ = t + ϑ h and U ϑ = ϑ U ( x , t + h ) + ( 1 − ϑ ) U ( x , t ) . Thus,

(3.16) u 1 ( t ) = w ˆ 1 ( t ) − ∫ 0 1 ∫ u 1 ∗ u 1 ϑ a 11 t ϑ ( x , t ϑ , ω ) d ω d ϑ ∫ 0 1 a 11 ( x , t ϑ , u 1 ϑ ) d ϑ .

Let the vertex P ¯ = ( x ¯ , t ¯ ) of cylinders Q ρ ( P ¯ ) and Q 2 ρ ( P ¯ ) be in D 3 , T − h for h ∈ ( 0 , T − t 3 ) , and let ρ ≤ ρ 1 ≔ min { d / 4 , I / 4 , 1 } , where I ≔ t ′ / 20 . Thus, Q 2 ρ ( P ¯ ) ⊂ D 2 , T . Function ξ ρ is defined by (2.17). Setting η = w ˆ 1 ( t ) ξ ρ 2 in (3.15), noting that w ˆ 1 ( t ) = ( w ˆ 1 h ) t , and using (3.16), we deduce that

(3.17) ∬ Q 2 ρ ( P ¯ ) ∫ 0 1 a 11 ( x , t ϑ , u 1 ϑ ) d ϑ − 1 w ˆ 1 ( t ) 2 ξ ρ 2 d x d t + 1 2 ∫ B 2 ρ ( x ¯ ) ∣ ∇ w ˆ 1 h ( x , t ¯ ) ∣ 2 ξ ρ 2 ( x , t ¯ ) d x = ∬ Q 2 ρ ( P ¯ ) ∫ 0 1 ∫ u 1 ∗ u 1 ϑ a 11 t ϑ ( x , t ϑ , ω ) d ω d ϑ ⋅ ∫ 0 1 a 11 ( x , t ϑ , u 1 ϑ ) d ϑ − 1 w ˆ 1 ( t ) ξ ρ 2 d x d t + ∬ Q 2 ρ ( P ¯ ) { − 2 w ˆ 1 ( t ) ξ ρ ∇ w ˆ 1 h ⋅ ∇ ξ ρ + ∣ ∇ w ˆ 1 h ∣ 2 ξ ρ ξ ρ t + [ f ˆ 1 ( x , t ) ] h w ˆ 1 ( t ) ξ ρ 2 } d x d t .

By (3.2), (3.9), and Cauchy’s inequality with ε , from (3.17), we obtain, for any ε ∈ ( 0 , 1 ) ,

∬ Q 2 ρ ( P ¯ ) ∫ 0 1 a 11 ( x , t ϑ , u 1 ϑ ) d ϑ − 1 w ˆ 1 ( t ) 2 ξ ρ 2 d x d t + 1 2 ∫ B 2 ρ ( x ¯ ) ∣ ∇ w ˆ 1 h ( x , t ¯ ) ∣ 2 ξ ρ 2 ( x , t ¯ ) d x ≤ ε ∬ Q 2 ρ ( P ¯ ) w ˆ 1 ( t ) 2 ξ ρ 2 d x d t + C ε ∬ Q 2 ρ ( P ¯ ) { ∣ ∇ w ˆ 1 h ∣ 2 ( ∣ ∇ ξ ρ ∣ 2 + ∣ ξ ρ ξ ρ t ∣ ) + [ f ˆ 1 ( x , t ) ] h 2 ξ ρ 2 + ξ ρ 2 } d x d t .

Using (2.3) and choosing ε = min { 1 / 2 , 1 / ( 2 μ ( 0 ) ) } , we further have

(3.18) ∬ Q 2 ρ ( P ¯ ) w ˆ 1 ( t ) 2 ξ ρ 2 d x d t + ∫ B 2 ρ ( x ¯ ) ∣ ∇ w ˆ 1 h ( x , t ¯ ) ∣ 2 ξ ρ 2 ( x , t ¯ ) d x ≤ C ∬ Q 2 ρ ( P ¯ ) { [ ∣ ∇ w ˆ 1 h ∣ 2 ( ∣ ∇ ξ ρ ∣ 2 + ∣ ξ ρ ξ ρ t ∣ ) + [ f ˆ 1 ( x , t ) ] h 2 ξ ρ 2 + ξ ρ 2 } d x d t .

In addition, it follows from (3.12), (3.14), (2.4), (3.2), and (3.9) that

(3.19) ∣ ∇ w ˆ 1 ∣ 2 ≤ C ( 1 + ∣ ∇ u 1 ∣ 2 ) , f ˆ 1 2 ( x , t ) ≤ C ( ∣ ∇ u 1 ∣ 2 + ∣ u 2 ∣ 2 σ 12 + ϕ 1 2 + 1 ) ( ( x , t ) ∈ D 2 , T ) .

Hence, using (2.21) and (2.7) yields that ∣ ∇ w ˆ 1 ∣ , f ˆ 1 ( x , t ) ∈ L 2 ( D 2 , T ) . Moreover, by [12, Chapter II, Lemma 4.7],

‖ ∇ w ˆ 1 h ‖ L 2 ( D 2 , T ) + ‖ [ f ˆ 1 ( x , t ) ] h ‖ L 2 ( D 2 , T ) ≤ C .

Therefore, inequality (3.18) and the definition of function ξ ρ yield

∬ Q ρ ( P ¯ ) w ˆ 1 ( t ) 2 d x d t + ∫ B ρ ( x ¯ ) ∣ ∇ w ˆ 1 h ∣ 2 ( x , t ¯ ) d x ≤ C / ρ 2 .

Hence,

∬ D 3 , T − h w ˆ 1 ( t ) 2 d x d t + sup t 3 ≤ t ≤ T − h ∫ Ω 3 ∣ ∇ w ˆ 1 h ∣ 2 d x ≤ C ,

where C is independent of h . [12, Chapter II, Lemma 4.11] further shows that w ˆ 1 has weak derivative w ˆ 1 t in D 3 , T and satisfies

(3.20) ess sup t 3 ≤ t ≤ T ∫ Ω 3 ∣ ∇ w ˆ 1 ∣ 2 ( x , t ) d x + ∬ D 3 , T w ˆ 1 t 2 d x d t ≤ C .

Then u 1 has weak derivative u 1 t = [ w ˆ 1 t − ∫ u 1 ∗ u 1 a 11 t ( x , t , ω ) d ω ] / a 11 ( x , t , u 1 ) , and estimate (3.20), together with (3.2) and (3.9), leads to estimate (3.10).□

To prove that u 1 has the second derivatives with respect to x , we need the following lemma:

Lemma 3.3

Let n ≥ 2 , and let P ¯ ∈ D 5 , T , ρ ≤ ρ 2 ≔ ρ 1 / 2 . If ζ = ζ ( x , t ) is an arbitrary bounded function from V ∘ 2 ( Q ρ ( P ¯ ) ) , then there exist positive constants C = C ( Θ 1 ) and α 2 = α 2 ( Θ 1 ) , α 2 ∈ ( 0 , 1 ) , such that

(3.21) ∬ Q ρ ( P ¯ ) [ ∣ ∇ u 1 ∣ 2 + ∣ u 2 ∣ 2 σ 12 + Λ 1 2 ( U ) + ϕ 1 2 + Φ ˜ 1 2 ] ζ 2 d x d t ≤ C ρ 2 α 2 ∬ Q ρ ( P ¯ ) ∣ ∇ ζ ∣ 2 d x d t .

Proof

We divide the proof into three steps.

Step 1. We prove that if P ¯ ∈ D 4 , T and ρ ≤ ρ 1 , then for some α ˜ ∈ ( 0 , 1 ) ,

(3.22) ∬ Q ρ ( P ¯ ) ∣ ∇ u 1 ∣ 2 d x d t ≤ C ρ n + 2 α ˜ .

We see that Q 2 ρ ( P ¯ ) ⊂ D 3 , T . Let ( x ∗ , t ∗ ) be a given point in Q ρ ( P ¯ ) . Choosing η = ( u 1 ( x , t ) − u 1 ( x ∗ , t ∗ ) ) ξ ρ 2 and τ = t ¯ in the first equality of (2.1) yields

(3.23) 1 2 ∫ B 2 ρ ( x ¯ ) ( u 1 ( x , t ¯ ) − u 1 ( x ∗ , t ∗ ) ) 2 ξ ρ 2 d x + ∬ Q 2 ρ ( P ¯ ) a 11 ( x , t , u 1 ) ∣ ∇ u 1 ∣ 2 ξ ρ 2 d x d t = ∬ Q 2 ρ ( P ¯ ) { ( u 1 ( x , t ) − u 1 ( x ∗ , t ∗ ) ) 2 ξ ρ ξ ρ t − 2 ( u 1 ( x , t ) − u 1 ( x ∗ , t ∗ ) ) ξ ρ a 11 ( x , t , u 1 ) ∇ u 1 ⋅ ∇ ξ ρ + b 1 ( x , t , U , ∇ u 1 ) ( u 1 ( x , t ) − u 1 ( x ∗ , t ∗ ) ) ξ ρ 2 } d x d t .

By using (2.3), (2.4), (3.2), Cauchy’s inequality with ε , and [12, Chapter II, formula (1.5)], we deduce from (3.23) that for any ε > 0 ,

(3.24) 1 2 ∫ B 2 ρ ( x ¯ ) ( u 1 ( x , t ¯ ) − u 1 ( x ∗ , t ∗ ) ) 2 ξ ρ 2 d x + ∬ Q 2 ρ ( P ¯ ) ν ( 0 ) ∣ ∇ u 1 ∣ 2 ξ ρ 2 d x d t ≤ ε ∬ Q 2 ρ ( P ¯ ) ∣ ∇ u 1 ∣ 2 ξ ρ 2 d x d t + C ε ∬ Q 2 ρ ( P ¯ ) ( u 1 ( x , t ) − u 1 ( x ∗ , t ∗ ) ) 2 ( ∣ ξ ρ ξ ρ t ∣ + ∣ ∇ ξ ρ ∣ 2 + ξ ρ 2 ) d x d t + C ∬ Q 2 ρ ( P ¯ ) ( ∣ u 2 ∣ σ 12 + ϕ 1 ) ∣ u 1 ( x , t ) − u 1 ( x ∗ , t ∗ ) ∣ ξ ρ 2 d x d t ≤ ε ∬ Q 2 ρ ( P ¯ ) ∣ ∇ u 1 ∣ 2 ξ ρ 2 d x d t + C ε ρ 2 α 1 ρ 2 + n max Q 2 ρ ( P ¯ ) [ ∣ ξ ρ ξ ρ t ∣ + ∣ ∇ ξ ρ ∣ 2 + ξ ρ 2 ] + C ρ α 1 [ ‖ ∣ u 2 ∣ σ 12 ‖ L q 1 ( Q 2 ρ ( P ¯ ) ) + ‖ ϕ 1 ‖ L q 1 ( D T ) ] ( ρ n + 2 ) 1 − 1 / q 1 .

Note that

max Q 2 ρ ( P ¯ ) [ ∣ ξ ρ ξ ρ t ∣ + ∣ ∇ ξ ρ ∣ 2 + ξ ρ 2 ] ≤ C / ρ 2 , ( ρ n + 2 ) 1 − 1 / q 1 = ρ n + 2 χ 1 .

Setting ε = ν ( 0 ) / 2 and α ˜ = min { χ 1 , α 1 } , and using (2.20) and (2.7), we further obtain inequality (3.22).

Step 2. We show that if P ¯ ∈ D 5 , T and ρ ≤ ρ 2 , then

(3.25) ess sup t 5 ≤ t ≤ T ∫ B ρ ( x ¯ ) ∣ ∇ u 1 ( x , t ) ∣ 2 d x ≤ C ρ n − 2 + 2 α ˜ .

We find that Q 2 ρ ( P ¯ ) ⊂ D 4 , T . Letting h → 0 in (3.18) and using (3.19) and (2.7), we have

∬ Q 2 ρ ( P ¯ ) w ˆ 1 t 2 ξ ρ 2 d x d t + ∫ B 2 ρ ( x ¯ ) ∣ ∇ w ˆ 1 ( x , t ¯ ) ∣ 2 ξ ρ 2 ( x , t ¯ ) d x ≤ C ∬ Q 2 ρ ( P ¯ ) { ∣ ∇ w ˆ 1 ∣ 2 ( ∣ ∇ ξ ρ 2 ∣ + ∣ ξ ρ ξ ρ t ∣ ) + f ˆ 1 2 ξ ρ 2 + ξ ρ 2 } d x d t ≤ C ρ 2 ∬ Q 2 ρ ( P ¯ ) ( 1 + ∣ ∇ u 1 ∣ 2 ) d x d t + C ∬ Q 2 ρ ( P ¯ ) [ ∣ ∇ u 1 ∣ 2 + ∣ u 2 ∣ 2 σ 12 + ϕ 1 2 + 1 ] ξ ρ 2 d x d t .

Thus, (3.22) and (2.22) further imply that

∬ Q ρ ( P ¯ ) w ˆ 1 t 2 d x d t + ∫ B ρ ( x ¯ ) ∣ ∇ w ˆ 1 ( x , t ¯ ) ∣ 2 d x ≤ C ρ n − 2 + 2 α ˜ ,

which, together with (3.12), leads to inequality (3.25).

Step 3. Employing [12, Chapter II, Lemma 5.2], we find from (3.25) and (2.22) that (3.21) holds.□

We next investigate the second partial derivatives of u 1 with respect to x by dealing with difference ratios of u 1 x i .

Lemma 3.4

Function u 1 has weak derivatives u 1 x i x k in D 6 , T for i , k = 1 , … , n , and satisfies

(3.26) ‖ u 1 ‖ W 2 2 , 1 ( D 6 , T ) ≤ C ( Θ 1 , μ ( M 1 ) ) .

Proof

Step 1. We prove that (3.26) holds for n = 1 . Let w ˆ 1 = w ˆ 1 ( x , t ) be defined by (3.11). From equality (3.13) and estimate (3.10), we find that w ˆ 1 has the second partial derivative w ˆ 1 x x and satisfies

w ˆ 1 x x = u 1 t − f ˆ 1 ( x , t ) ( ( x , t ) ∈ D 5 , T ) ,

which, together with inequalities (3.19) and (3.20), implies that w ˆ 1 x x ∈ L 2 ( D 5 , T ) and w ˆ 1 x ∈ V 2 ( D 5 , T ) . It follows from [12, Chapter II, inequality (3.4)] with n = 1 that ‖ w ˆ 1 x ‖ L 6 ( D 5 , T ) and ‖ w ˆ 1 x ‖ L ∞ , 4 ( D 5 , T ) are estimated from above by C . Thus, by (3.12), (3.2), and (3.9), we have

(3.27) ‖ u 1 x ‖ L 6 ( D 5 , T ) + ‖ u 1 x ‖ L ∞ , 4 ( D 5 , T ) ≤ C

and

u 1 x x = [ a 11 ( x , t , u 1 ) ] − 1 w ˆ 1 x x − u 1 x a 11 x ( x , t , u 1 ) − ∫ u 1 ∗ u 1 a 11 x x ( x , t , ω ) d ω − [ a 11 ( x , t , u 1 ) ] − 2 w ˆ 1 x − ∫ u 1 ∗ u 1 a 11 x ( x , t , ω ) d ω [ u 1 x a 11 u 1 ( x , t , u 1 ) + a 11 x ] .

Hence, (2.3) and (3.9) lead to the inequality ∣ u 1 x x ∣ ≤ C [ ∣ w ˆ 1 x x ∣ + w ˆ 1 x 2 + u 1 x 2 + 1 ] . Estimate (3.27) further shows that ‖ u 1 x x ‖ L 2 ( D 5 , T ) is estimated from above by C . In view of (3.10), we find that (3.26) holds for n = 1 .

Step 2. We prove that (3.26) holds for n ≥ 2 . Let P ¯ ∈ D 6 , T and ρ ≤ ρ 3 ≔ ρ 2 / 2 . Then Q 2 ρ ( P ¯ ) ⊂ D 5 , T . For any fixed k ∈ { 1 , … , n } and for ∣ h k ∣ ≤ min { d / 4 , ρ 3 / 4 } , taking η = [ ( u 1 ( x k ) ξ ρ 2 ) ( x − h k , t ) ] ( x k ) and τ = t ¯ in the first equality of (2.1). Here and below, notation w ( x k ) ( x , t ) is defined by (2.14). Then employing [12, Chapter 2, formula (4.9)] and integrating by parts leads to the equality:

(3.28) 1 2 ∫ B 2 ρ ( x ¯ ) u 1 ( x k ) 2 ( x , t ¯ ) ξ ρ 2 ( x , t ¯ ) d x + ∬ Q 2 ρ ( P ¯ ) − u 1 ( x k ) 2 ξ ρ ξ ρ t + ∑ i = 1 n [ a 11 ( x , t , u 1 ) u 1 x i ] ( x k ) ( u 1 ( x k ) ξ ρ 2 ) x i − [ b 1 ( x , t , U , ∇ u 1 ) ] ( x k ) u 1 ( x k ) ξ ρ 2 d x d t = 0 .

Note that

(3.29) [ a 11 ( x , t , u 1 ) u 1 x i ] ( x k ) = u 1 ( x k ) x i ∫ 0 1 a 11 ( x θ , t , u 1 θ ) d θ + u 1 ( x k ) ∫ 0 1 ∂ a 11 ∂ u 1 θ u 1 x i θ d θ + ∫ 0 1 ∂ a 11 ∂ x k θ u 1 x i θ d θ

and

(3.30) [ b 1 ( x , t , U , ∇ u 1 ) ] ( x k ) = ∑ i = 1 n u 1 ( x k ) x i ∫ 0 1 ∂ b 1 ( x θ , t , U θ , ∇ u 1 θ ) ∂ u 1 x i θ d θ + ∑ m = 1 2 u m ( x k ) ∫ 0 1 ∂ b 1 ∂ u m θ d θ + ∫ 0 1 ∂ b 1 ∂ x k θ d θ ,

where x θ = ( x 1 θ , … , x n θ ) ≔ x + θ h k and U θ ( x , t ) ≔ ( 1 − θ ) U ( x , t ) + θ U ( x + h k , t ) and u 1 ( x k ) x i = ( u 1 ( x k ) ) x i . Using (3.28)–(3.30), (2.9), (3.2), (3.9), and Cauchy’s inequality, we deduce that for any ε ∈ ( 0 , 1 ) ,

(3.31) 1 2 ∫ B 2 ρ ( x ¯ ) u 1 ( x k ) 2 ( x , t ¯ ) ξ ρ 2 ( x , t ¯ ) d x + ∬ Q 2 ρ ( P ¯ ) ∣ ∇ u 1 ( x k ) ∣ 2 ξ ρ 2 ∫ 0 1 a 11 ( x θ , t , u 1 θ ) d θ d x d t ≤ ∬ Q 2 ρ ( P ¯ ) u 1 ( x k ) 2 ∣ ξ ρ ξ ρ t ∣ d x d t + ∬ Q 2 ρ ( P ¯ ) ∣ u 1 ( x k ) ∣ ∫ 0 1 ∣ ∇ u 1 θ ∣ d θ + ∫ 0 1 ∣ ∇ u 1 θ ∣ d θ ∣ ∇ u 1 ( x k ) ∣ ξ ρ 2 + ∣ ∇ u 1 ( x k ) ∣ + ∣ u 1 ( x k ) ∣ ∫ 0 1 ∣ ∇ u 1 θ ∣ d θ + ∫ 0 1 ∣ ∇ u 1 θ ∣ d θ 2 ∣ u 1 ( x k ) ξ ρ ∣ ∣ ∇ ξ ρ ∣ d x d t + ∬ Q 2 ρ ( P ¯ ) ∣ ∇ u 1 ( x k ) ∣ ∫ 0 1 ( ∣ u 2 θ ∣ σ ˜ 14 + ϕ ˜ 14 ( x θ , t ) ) d θ + ∣ u 2 ( x k ) ∣ ∫ 0 1 ( ∣ ∇ u 1 θ ∣ + ∣ u 2 θ ∣ σ ˜ 13 + ϕ ˜ 13 ( x θ , t ) ) d θ + ∣ u 1 ( x k ) ∣ ∫ 0 1 ( ∣ ∇ u 1 θ ∣ + ∣ u 2 θ ∣ σ ˜ 12 + ϕ ˜ 12 ( x θ , t ) ) d θ + ∫ 0 1 ( ∣ ∇ u 1 θ ∣ + ∣ u 2 θ ∣ σ ˜ 11 + ϕ ˜ 11 ( x θ , t ) ) d θ ∣ u 1 ( x k ) ∣ ξ ρ 2 d x d t ≤ ε ∬ Q 2 ρ ( P ¯ ) ∣ ∇ u 1 ( x k ) ∣ 2 ξ ρ 2 d x d t + C ε ( J 1 + J 2 ) ,

where

J 1 ≔ ∬ Q 2 ρ ( P ¯ ) { u 1 ( x k ) 2 [ ∣ ξ ρ ξ ρ t ∣ + ∣ ∇ ξ ρ ∣ 2 + ξ ρ 2 ] + u 2 ( x k ) 2 ξ ρ 2 } d x d t , J 2 ≔ ∫ 0 1 ∬ Q 2 ρ ( P ¯ ) [ ∣ ∇ u 1 θ ∣ 2 + Λ 1 2 ( U θ ) + Φ ˜ 1 2 ( x θ , t ) ] ( u 1 ( x k ) 2 + 1 ) ξ ρ 2 d x d t d θ .

Setting ε = min { 1 / 2 , ν ( 0 ) / 2 } in (3.31) leads to the inequality

(3.32) 1 2 ∫ B 2 ρ ( x ¯ ) u 1 ( x k ) 2 ( x , t ¯ ) ξ ρ 2 ( x , t ¯ ) d x + ν ( 0 ) 2 ∬ Q 2 ρ ( P ¯ ) ∣ ∇ u 1 ( x k ) ∣ 2 ξ ρ 2 d x d t ≤ C ¯ 1 ( J 1 + J 2 ) .

We next estimate J 1 , J 2 . Since u j x k ∈ L 2 ( D T ) for j = 1 , 2 , then by (2.17), we obtain the inequality J 1 ≤ C / ρ 2 . Taking ζ = u 1 ( x k ) ξ ρ and ζ = ξ ρ in (3.21), respectively, we obtain

J 2 ≤ C 1 ∗ ρ 2 α 2 ∬ Q 2 ρ ( P ¯ ) [ ∣ ∇ u 1 ( x k ) ∣ 2 ξ ρ 2 + ( u 1 ( x k ) 2 + 1 ) ∣ ∇ ξ ρ ∣ 2 ] d x d t ≤ C 1 ∗ ρ 2 α 2 ∬ Q 2 ρ ( P ¯ ) ∣ ∇ u 1 ( x k ) ∣ 2 ξ ρ 2 d x d t + C ρ 2 α 2 − 2 .

Let ρ ≤ ρ 4 ≔ min { ρ 3 , [ ν ( 0 ) / ( 4 C ¯ 1 C 1 ∗ ) ] 1 / ( 2 α 2 ) } . Then C ¯ 1 C 1 ∗ ρ 2 α 2 ≤ ν ( 0 ) / 4 . Substituting the estimates of J 1 and J 2 into inequality (3.32) yields

∫ B 2 ρ ( x ¯ ) u 1 ( x k ) 2 ( x , t ¯ ) ξ ρ 2 ( x , t ¯ ) d x + ∬ Q 2 ρ ( P ¯ ) ∣ ∇ u 1 ( x k ) ∣ 2 ξ ρ 2 d x d t ≤ C / ρ 2 ,

which implies

∫ B ρ ( x ¯ ) u 1 ( x k ) 2 ( x , t ¯ ) d x + ∬ Q ρ ( P ¯ ) ∣ ∇ u 1 ( x k ) ∣ 2 d x d t ≤ C / ρ 2 .

Thus, ∬ D 6 , T ∣ ∇ u 1 ( x k ) ∣ 2 d x d t is estimated from above by constant C . [13, Chapter 2, Lemma 4.6] further shows that u 1 has weak derivatives u 1 x i x k in D 6 , T ( i = 1 , … , n ) , and ‖ u 1 x i x k ‖ L 2 ( D 6 , T ) is estimated from above by C . In view of (3.10), u 1 belongs to W 2 2 , 1 ( D 6 , T ) and estimate (3.26) holds for n ≥ 2 . Hence, we complete the proof of the lemma.□

3.3 Estimate of ‖ u 2 ‖ C α , α / 2 ( Ω ¯ ′ × [ t ′ , T ] )

To investigate the boundedness of ∣ u 2 ∣ , we need the estimate of ‖ ∣ ∇ u 1 ∣ ‖ L p for any p > 2 when n ≥ 2 .

Lemma 3.5

Let n ≥ 2 . For any given positive integer K, there exists positive constant C = C ( Θ 1 , μ ( M 1 ) , K ) such that

(3.33) ∬ D 7 , T [ ∣ ∇ u 1 ∣ 4 + 2 K + ( 1 + ∣ ∇ u 1 ∣ ) 2 K ∣ ∇ 2 u 1 ∣ 2 ] d x d t ≤ C ,

where ∣ ∇ 2 u 1 ∣ ≔ ∑ i , j = 1 n u 1 x i x j 2 1 / 2 .

Proof

Step 1. Let B 1 , i ( x , t , w , q ) and B 1 ( x , t , w , q ) also be defined by (3.5), and let ψ ( x , t ) be an arbitrary sufficiently smooth function in D T such that ψ ( x , t ) = 0 for x ∉ Ω 6 . For any given r ∈ { 1 , … , n } , choosing η = ψ x r in the first equality of (2.1) and integrating by parts yields

(3.34) ∬ D 6 , τ − u 1 t ψ x r + ∑ i = 1 n ∑ j = 1 n ∂ B 1 , i ( x , t , u 1 , ∇ u 1 ) ∂ u 1 x j u 1 x j x r + ∂ B 1 , i ∂ u 1 u 1 x r + ∂ B 1 , i ∂ x r ψ x i − B 1 ( x , t , u 1 , ∇ u 1 ) ψ x r d x d t = 0 .

This equality also holds for ψ ∈ W ∘ 2 1 , 0 ( D 6 , T ) . Then from (3.9), we obtain, for ( x , t ) ∈ D 6 , T , w ∈ [ − M 1 , M 1 ] ∩ S 1 and p ∈ R n ,

(3.35) ∑ i = 1 n ∂ B 1 , i ( x , t , w , p ) ∂ w + ∣ B 1 , i ∣ + ∑ i , j = 1 n ∂ B 1 , i ∂ x j ≤ C ( 1 + ∣ p ∣ ) .

Let P ¯ = ( x ¯ , t ¯ ) ∈ D 7 , T and ρ ≤ ρ 4 / 2 . Set g = min { ∣ ∇ u 1 ∣ 2 , L } , where L is a large positive number. For any given s ∈ { 0 , 1 , … , K } , choosing ψ = g s u 1 x r ξ ρ 2 in (3.34) and summing this equality with respect to r from 1 to n , we obtain

(3.36) ∑ r = 1 n ∬ Q 2 ρ ( P ¯ ) − u 1 t ( g s u 1 x r ξ ρ 2 ) x r + ∑ i = 1 n ∑ j = 1 n ∂ B 1 , i ∂ u 1 x j u 1 x j x r + ∂ B 1 , i ∂ u 1 u 1 x r + ∂ B 1 , i ∂ x r × ( g s u 1 x r x i ξ ρ 2 + s g s − 1 g x i u 1 x r ξ ρ 2 + g s u 1 x r 2 ξ ρ ξ ρ x i ) d x d t = ∑ r = 1 n ∬ Q 2 ρ ( P ¯ ) B 1 ( x , t , u 1 , ∇ u 1 ) ( g s u 1 x r x r ξ ρ 2 + s g s − 1 g x r u 1 x r ξ ρ 2 + g s u 1 x r 2 ξ ρ ξ ρ x r ) d x d t .

Note that by (3.5),

∑ r = 1 n ∬ Q 2 ρ ( P ¯ ) ∑ i , j = 1 n ∂ B 1 , i ∂ u 1 x j u 1 x j x r ( g s u 1 x r x i ξ ρ 2 + s g s − 1 g x i u 1 x r ξ ρ 2 ) d x d t = ∬ Q 2 ρ ( P ¯ ) a 11 ( x , t , u 1 ) g s ξ ρ 2 ∣ ∇ 2 u 1 ∣ 2 + s 2 g s − 1 ∣ ∇ g ∣ 2 ξ ρ 2 d x d t ,

and by integration by parts,

(3.37) ∑ r = 1 n ∬ Q 2 ρ ( P ¯ ) − u 1 t ( g s u 1 x r ξ ρ 2 ) x r d x d t = 1 2 ∬ Q 2 ρ ( P ¯ ) g s ( ∣ ∇ u 1 ∣ 2 ) t ξ ρ 2 d x d t = 1 2 ∬ Q 2 ρ ( P ¯ ) { [ g s ∣ ∇ u 1 ∣ 2 ξ ρ 2 ] t − ( g s ) t g ξ ρ 2 − g s ∣ ∇ u 1 ∣ 2 2 ξ ρ ξ ρ t } d x d t = 1 2 ∫ B 2 ρ ( x ¯ ) g s ∣ ∇ u 1 ∣ 2 ξ ρ 2 − s s + 1 g s + 1 ξ ρ 2 ( x , t ¯ ) d x + ∬ Q 2 ρ ( P ¯ ) s s + 1 g s + 1 − g s ∣ ∇ u 1 ∣ 2 ξ ρ ξ ρ t d x d t ≥ 1 2 ( s + 1 ) ∫ B 2 ρ ( x ¯ ) g s + 1 ( x , t ¯ ) ξ ρ 2 ( x , t ¯ ) d x − ∬ Q 2 ρ ( P ¯ ) ( 1 + ∣ ∇ u 1 ∣ 2 ) g s ∣ ξ ρ ξ ρ t ∣ d x d t .

Inequality (3.37) is correct, although in deriving it we made use the partial derivatives u 1 x i t . Indeed, we can first take the sequence infinitely differentiable functions v m converging to u 1 in the norm of W 2 2 , 1 ( D 6 , T ) as m → + ∞ and obtain (3.37) for v m . Then letting m → + ∞ we obtain (3.37) for u 1 .

Thus, substituting the aforementioned two relations into (3.36) and using (3.8), (3.35) leads to

1 2 ( s + 1 ) ∫ B 2 ρ ( x ¯ ) g s + 1 ( x , t ¯ ) ξ ρ 2 ( x , t ¯ ) d x + ∬ Q 2 ρ ( P ¯ ) a 11 ( x , t , u 1 ) [ g s ∣ ∇ 2 u 1 ∣ 2 + s 2 g s − 1 ∣ ∇ g ∣ 2 ] ξ ρ 2 d x d t ≤ C ∬ Q 2 ρ ( P ¯ ) ( 1 + ∣ ∇ u 1 ∣ 2 ) g s ∣ ξ ρ ξ ρ t ∣ d x d t + C ∬ Q 2 ρ ( P ¯ ) [ g s ∣ ∇ 2 u 1 ∣ ∣ ∇ u 1 ∣ ξ ρ ∣ ∇ ξ ρ ∣ + ( ∣ ∇ u 1 ∣ 2 + ∣ u 2 ∣ σ 12 + ϕ 1 + 1 ) ( s g s − 1 ∣ ∇ g ∣ ∣ ∇ u 1 ∣ ξ ρ 2 + g s ∣ ∇ 2 u 1 ∣ ξ ρ 2 + g s ∣ ∇ u 1 ∣ ξ ρ ∣ ∇ ξ ρ ∣ ) ] d x d t .

Note that s g s − 1 ∣ ∇ g ∣ ∣ ∇ u 1 ∣ = s g s − 1 2 ∣ ∇ g ∣ . Then Cauchy’s inequality with ε further shows that for any ε ∈ ( 0 , 1 ) ,

(3.38) ∫ B 2 ρ ( x ¯ ) g s + 1 ( x , t ¯ ) ξ ρ 2 ( x , t ¯ ) d x + ∬ Q 2 ρ ( P ¯ ) [ g s ∣ ∇ 2 u 1 ∣ 2 + s g s − 1 ∣ ∇ g ∣ 2 ] ξ ρ 2 d x d t ≤ ε ∬ Q 2 ρ ( P ¯ ) [ g s ∣ ∇ 2 u 1 ∣ 2 ξ ρ 2 + s g s − 1 ∣ ∇ g ∣ 2 ξ ρ 2 ] d x d t + C ε [ J 3 , s + J 4 , s + J 5 , s ] ,

where

J 3 , s = ∬ Q 2 ρ ( P ¯ ) ∣ ∇ u 1 ∣ 2 ( 1 + ∣ ∇ u 1 ∣ 2 ) ( 1 + g ) s ξ ρ 2 d x d t , J 4 , s = ∬ Q 2 ρ ( P ¯ ) ( ∣ u 2 ∣ 2 σ 12 + ϕ 1 2 ) ( 1 + g ) s ξ ρ 2 d x d t , J 5 , s = ∬ Q 2 ρ ( P ¯ ) ( 1 + ∣ ∇ u 1 ∣ 2 ) ( 1 + g ) s ( ∣ ∇ ξ ρ ∣ 2 + ∣ ξ ρ ξ ρ t ∣ ) d x d t .

Here and below, for s = 0 and g ≥ 0 , the expression g s is taken equal to unity and the expression s g s − 1 is taken equal to 0.

We next estimate J 3 , s , J 4 , s . By taking ζ = ( 1 + ∣ ∇ u 1 ∣ 2 ) 1 / 2 ( 1 + g ) s / 2 ξ ρ and ζ = ( 1 + g ) s / 2 ξ ρ in (3.21), respectively, we have

J 3 , s ≤ C ρ 2 α 2 ∬ Q 2 ρ ( P ¯ ) { [ ∣ ∇ 2 u 1 ∣ 2 ( 1 + g ) s + s ( 1 + g ) s − 1 ∣ ∇ g ∣ 2 ] ξ ρ 2 + ( 1 + ∣ ∇ u 1 ∣ 2 ) ( 1 + g ) s ∣ ∇ ξ ρ ∣ 2 } d x d t

and

J 4 , s ≤ C ρ 2 α 2 ∬ Q 2 ρ ( P ¯ ) [ s ( 1 + g ) s − 2 ∣ ∇ g ∣ 2 ξ ρ 2 + ( 1 + g ) s ∣ ∇ ξ ρ ∣ 2 ] d x d t ≤ C ρ 2 α 2 ∬ Q 2 ρ ( P ¯ ) [ s ( 1 + g ) s − 1 ∣ ∇ g ∣ 2 ξ ρ 2 + ( 1 + g ) s ∣ ∇ ξ ρ ∣ 2 ] d x d t .

Substituting the estimates of J 3 , s , J 4 , s into (3.38) and setting ε = 1 / 2 further yields

(3.39) ∫ B 2 ρ ( x ¯ ) g s + 1 ( x , t ¯ ) ξ ρ 2 ( x , t ¯ ) d x + ∬ Q 2 ρ ( P ¯ ) [ g s ∣ ∇ 2 u 1 ∣ 2 + s g s − 1 ∣ ∇ g ∣ 2 ] ξ ρ 2 d x d t ≤ C 2 ∗ ρ 2 α 2 ∬ Q 2 ρ ( P ¯ ) [ ∣ ∇ 2 u 1 ∣ 2 ( 1 + g ) s + s ( 1 + g ) s − 1 ∣ ∇ g ∣ 2 ] ξ ρ 2 d x d t + C J 5 , s .

Specially, letting s = 1 in (3.39) and ρ ≤ min { ρ 4 / 2 , [ 1 / ( 8 C 2 ∗ ) ] 1 / ( 2 α 2 ) } leads to the inequality

(3.40) ∬ Q 2 ρ ( P ¯ ) [ g ∣ ∇ 2 u 1 ∣ 2 + ∣ ∇ g ∣ 2 ] ξ ρ 2 d x d t ≤ C ∬ Q 2 ρ ( P ¯ ) ∣ ∇ 2 u 1 ∣ 2 ξ ρ 2 d x d t + C J 5 , 1 .

We find that there exists a constant C 3 ∗ such that ( 1 + g ) s ≤ C 3 ∗ ( 1 + g s ) and ( 1 + g ) s − 1 ≤ C 3 ∗ ( 1 + s g s − 1 ) for s ∈ { 0 , 1 , … , K } . If ρ ≤ ρ 5 ≔ min { ρ 4 / 2 , [ 1 / ( 8 C 2 ∗ ) ] 1 / ( 2 α 2 ) , [ 1 / ( 8 K C 2 ∗ C 3 ∗ ) ] 1 / ( 2 α 2 ) } , then K C 2 ∗ C 3 ∗ ρ 2 α 2 ≤ 1 / 8 and C 2 ∗ ρ 2 α 2 ≤ 1 / 8 . Inequality (3.39) implies that

∬ Q 2 ρ ( P ¯ ) [ g s ∣ ∇ 2 u 1 ∣ 2 + s g s − 1 ∣ ∇ g ∣ 2 ] ξ ρ 2 d x d t ≤ C ∬ Q 2 ρ ( P ¯ ) [ ∣ ∇ 2 u 1 ∣ 2 + s ∣ ∇ g ∣ 2 ] ξ ρ 2 d x d t + C J 5 , s ,

which, together with (3.40), yields

(3.41) ∬ Q 2 ρ ( P ¯ ) [ g s ∣ ∇ 2 u 1 ∣ 2 + s g s − 1 ∣ ∇ g ∣ 2 ] ξ ρ 2 d x d t ≤ C ∬ Q 2 ρ ( P ¯ ) ∣ ∇ 2 u 1 ∣ 2 ξ ρ 2 d x d t + C J 5 , s .

Step 2. In another aspect, taking ζ = ( 1 + g ) ( s + 1 ) / 2 ξ ρ in (3.21) we obtain, for s ∈ { 0 , 1 , … , K } ,

(3.42) ∬ Q 2 ρ ( P ¯ ) ∣ ∇ u 1 ∣ 2 ( 1 + g ) s + 1 ξ ρ 2 d x d t ≤ C ρ 2 α 2 ∬ Q 2 ρ ( P ¯ ) [ ( 1 + g ) s − 1 ∣ ∇ g ∣ 2 ξ ρ 2 + ( 1 + g ) s + 1 ∣ ∇ ξ ρ ∣ 2 ] d x d t ≤ C ρ 2 α 2 ∬ Q 2 ρ ( P ¯ ) ( 1 + g ) s − 1 ∣ ∇ g ∣ 2 ξ ρ 2 d x d t + C J 5 , s .

Step 3. Set ρ ¯ s = ρ / 2 s . Let ξ ρ ¯ s = ξ ρ ¯ s ( x , t ) be defined by (2.17) with ρ replaced by ρ ¯ s , and let Q 2 ρ ( P ¯ ) and ξ ρ in (3.41) and (3.42) be replaced by Q 2 ρ ¯ s ( P ¯ ) and ξ ρ ¯ s , respectively.

Specially, by setting s = 0 in (3.42) and using (3.26), (2.17) leads to

∬ Q 2 ρ ¯ 0 ( P ¯ ) ∣ ∇ u 1 ∣ 2 ( 1 + g ) ξ ρ ¯ 0 2 d x d t ≤ C ∬ Q 2 ρ ¯ 0 ( P ¯ ) ∩ { ∣ ∇ u 1 ∣ 2 ≤ L } ∣ ∇ u 1 ∣ 2 ∣ ∇ 2 u 1 ∣ 2 ξ ρ ¯ 0 2 ( 1 + ∣ ∇ u 1 ∣ 2 ) d x d t + C ∬ Q 2 ρ ¯ 0 ( P ¯ ) ( 1 + ∣ ∇ u 1 ∣ 2 ) [ ∣ ∇ ξ ρ ¯ 0 ∣ 2 + ∣ ξ ρ ¯ 0 ξ ρ ¯ 0 t ∣ ] d x d t ≤ C / ρ 2 .

Thus,

(3.43) ∬ Q 2 ρ ¯ 1 ( P ¯ ) ∣ ∇ u 1 ∣ 2 ( 1 + g ) d x d t = ∬ Q ρ ¯ 0 ( P ¯ ) ∣ ∇ u 1 ∣ 2 ( 1 + g ) d x d t ≤ C / ρ 2 .

Inequality (3.41) with s = 1 , together with (3.43), implies that

(3.44) ∬ Q 2 ρ ¯ 1 ( P ¯ ) ∣ ∇ g ∣ 2 ξ ρ ¯ 1 2 d x d t ≤ C ∬ Q 2 ρ ¯ 1 ( P ¯ ) ∣ ∇ 2 u 1 ∣ 2 d x d t + C ρ 2 ∬ Q 2 ρ ¯ 1 ( P ¯ ) ( 1 + ∣ ∇ u 1 ∣ 2 ) ( 1 + g ) d x d t ≤ C ( 1 / ρ 2 ) .

Moreover, for s = 1 , … , K , it follows from (3.41) that

(3.45) ∬ Q 2 ρ ¯ s ( P ¯ ) [ g s ∣ ∇ 2 u 1 ∣ 2 + g s − 1 ∣ ∇ g ∣ 2 ] ξ ρ ¯ s 2 d x d t ≤ C ( 1 / ρ 2 ) + C ∬ Q 2 ρ ¯ s ( P ¯ ) ( 1 + ∣ ∇ u 1 ∣ ) 2 ( 1 + g ) s ( ∣ ∇ ξ ρ ¯ s ∣ 2 + ∣ ξ ρ ¯ s ( ξ ρ ¯ s ) t ∣ ) d x d t ,

and from (3.42), (3.44), and (3.45) that

(3.46) ∬ Q 2 ρ ¯ s ( P ¯ ) ∣ ∇ u 1 ∣ 2 ( 1 + g ) s + 1 ξ ρ ¯ s 2 d x d t ≤ C ∬ Q 2 ρ ¯ s ( P ¯ ) ∣ ∇ g ∣ 2 ξ ρ ¯ s 2 d x d t + C ∬ Q 2 ρ ¯ s ( P ¯ ) g s − 1 ∣ ∇ g ∣ 2 ξ ρ ¯ s 2 d x d t + C ∬ Q 2 ρ ¯ s ( P ¯ ) ( 1 + ∣ ∇ u 1 ∣ ) 2 ( 1 + g ) s ( ∣ ∇ ξ ρ ¯ s ∣ 2 + ∣ ξ ρ ¯ s ( ξ ρ ¯ s ) t ∣ ) d x d t ≤ C ( 1 / ρ 2 ) + C ∬ Q 2 ρ ¯ s ( P ¯ ) ( 1 + ∣ ∇ u 1 ∣ ) 2 ( 1 + g ) s ( ∣ ∇ ξ ρ ¯ s ∣ 2 + ∣ ξ ρ ¯ s ( ξ ρ ¯ s ) t ∣ ) d x d t .

In view of (3.43), by considering in succession (3.46) for s = 1 , … , K , we can find from inequalities (3.45) and (3.46) that

∬ Q ρ ¯ K [ g K ∣ ∇ 2 u 1 ∣ 2 + g K − 1 ∣ ∇ g ∣ 2 + ∣ ∇ u 1 ∣ 2 ( 1 + g ) K + 1 ] d x d t ≤ C ( 1 / ρ 2 ) .

Letting L → + ∞ leads to the estimate

∬ Q ρ ¯ K [ ∣ ∇ u 1 ∣ 2 K ∣ ∇ 2 u 1 ∣ 2 + ∣ ∇ u 1 ∣ 4 + 2 K ] d x d t ≤ C ( 1 / ρ 2 ) .

This gives estimate (3.33).□

We next use (3.33) to estimate ess sup D 8 , T ∣ u 2 ∣ from the second equality of (2.1).

Lemma 3.6

There exists positive constant M 2 = M 2 ( Θ 1 , ν ( M 1 ) , μ ( M 1 ) ) such that

(3.47) ess sup D 8 , T ∣ u 2 ∣ ≤ M 2 .

Proof

Step 1. From (2.3) and (3.3), we see that for ( x , t ) ∈ D ¯ 1 , T and W ∈ ( S 1 ∩ [ − M 1 , M 1 ] ) × S 2 ,

(3.48) ν ( M 1 ) ≤ a 22 ( x , t , W ) ≤ μ ( M 1 ) , ∣ a 21 ( x , t , W ) ∣ ≤ μ ( M 1 ) .

Using (3.48), (2.4), (2.7), (2.20), and the similar argument as that of Step 1 in Lemma 3.1, we can prove that the integrals in the second equality of (2.1) exist and are finite.

Step 2. We show that u 2 has estimate (3.47). For any given P ¯ ∈ D 8 , T and Q ρ , τ ( P ¯ ) ⊂ D 7 , T , let ζ = ζ ( x , t ) be same as that in Lemma 2.4. Choosing σ ˆ ≥ 1 and taking η = u ( σ ) ζ 2 in the second equality of (2.1) for σ ≥ σ ˆ , we obtain, for any τ 1 and τ ∗ , t ¯ − τ < τ 1 ≤ τ ∗ < t ¯ ,

1 2 ∫ Ω ( u 2 ( σ ) ζ ) 2 d x t = τ 1 t = τ ∗ + ∫ τ 1 τ ∗ ∫ A u 2 , σ ( t ) a 22 ( x , t , U ) ∣ ∇ u 2 ∣ 2 ζ 2 d x d t = ∫ τ 1 τ ∗ ∫ A u 2 , σ ( t ) { − [ ζ 2 ∇ u 2 + 2 u 2 ( σ ) ζ ∇ ζ ] ⋅ a 21 ( x , t , U ) ∇ u 1 − a 22 ( x , t , U ) u 2 ( σ ) ∇ u 2 ⋅ ( 2 ζ ∇ ζ ) + ( u 2 ( σ ) ) 2 ζ ζ t + b 2 ( x , t , U , ∇ U ) u 2 ( σ ) ζ 2 } d x d t .

By (2.4), (3.2), (3.48) and Cauchy’s inequality with ε , we find that

1 2 ∫ Ω [ u 2 ( σ ) ζ ( x , t ) ] 2 d x τ 1 τ ∗ + ∫ τ 1 τ ∗ ∫ A u 2 , σ ( t ) a 22 ( x , t , U ) ∣ ∇ u 2 ∣ 2 ζ 2 d x d t ≤ C ∫ τ 1 τ ∗ ∫ A u 2 , σ ( t ) { [ ∣ ∇ u 2 ∣ ζ 2 + u 2 ( σ ) ζ ∣ ∇ ζ ∣ ] ∣ ∇ u 1 ∣ + ∣ ∇ u 2 ∣ u 2 ( σ ) ζ ∣ ∇ ζ ∣ + [ u 2 ( σ ) ] 2 ζ ∣ ζ t ∣ + [ ( 1 + ∣ u 2 ∣ δ 2 ) ( ∣ ∇ u 2 ∣ + ∣ ∇ u 1 ∣ ) + F 2 ( U ) + ϕ 2 ] u 2 ( σ ) ζ 2 } d x d t ≤ ε ∫ τ 1 τ ∗ ∫ A u 2 , σ ( t ) ∣ ∇ u 2 ∣ 2 ζ 2 d x d t + C ε ∫ τ 1 τ ∗ ∫ A u 2 , σ ( t ) { ( ζ ∣ ζ t ∣ + ∣ ∇ ζ ∣ 2 ) [ u 2 ( σ ) ] 2 + ( 1 + ∣ u 2 ∣ 2 δ 2 ) [ u 2 ( σ ) ] 2 ζ 2 + ∣ ∇ u 1 ∣ 2 ζ 2 + [ F 2 ( U ) + ϕ 2 ] u 2 ( σ ) ζ 2 } d x d t .

Setting ε = min { 1 / 2 , ν ( M 1 ) / 2 } further yields, for any τ 1 and τ 2 , t ¯ − τ < τ 1 < τ 2 < t ¯ ,

sup τ 1 ≤ t ≤ τ 2 ‖ u 2 ( σ ) ( x , t ) ζ ( x , t ) ‖ L 2 ( Ω ) 2 + ν ( M 1 ) ∫ τ 1 τ 2 ∫ A u 2 , σ ( t ) ∣ ∇ u 2 ∣ 2 ζ 2 d x d t ≤ ‖ u 2 ( σ ) ( x , τ 1 ) ζ ( x , τ 1 ) ‖ L 2 ( Ω ) 2 + ∫ τ 1 τ 2 ∫ A u 2 , σ ( t ) { G 2 ( x , t ) ζ 2 [ ( u 2 − σ ) 2 + σ 2 ] + C ( ∣ ∇ ζ ∣ 2 + ζ ∣ ζ t ∣ ) [ u 2 ( σ ) ] 2 } d x d t ,

where G 2 ( x , t ) = C ( ∣ ∇ u 1 ∣ 2 + ∣ u 2 ∣ 2 δ 2 + F 2 ( U ) + ϕ 2 + 1 ) . It follows from (2.7), (2.20), (3.27), and (3.33) that G 2 ( x , t ) ∈ L q 1 ( D 7 , T ) . Then Lemma 2.4 shows that ess sup D 8 , T u 2 does not exceed constant C . The similar argument implies that ess sup D 8 , T ( − u 2 ) also does not exceed C . Thus, (3.47) holds.□

To obtain Hölder estimate for u 1 , we need to estimate ‖ u 1 ‖ W q 3 2 , 1 ( D 10 , T ) .

Lemma 3.7

We have

(3.49) ‖ u 1 x i ‖ C α 3 , α 3 / 2 ( D ¯ 10 , T ) ≤ ϒ 1 , ‖ u 1 ‖ W q 3 2 , 1 ( D 10 , T ) ≤ C , α 3 ∈ ( 0 , 1 ) , i = 1 , … , n ,

where constants ϒ 1 and C depend on Θ 1 , ν ( M 1 ) and μ ( ∣ ( M 1 , M 2 ) ∣ ) , and q 3 is same as that in Theorem 2.1.

Proof

Step 1. We show that when n ≥ 2 ,

(3.50) ess sup D 9 , T ∣ ∇ u 1 ∣ ≤ C .

Let z ˜ = ∣ ∇ u 1 ∣ 2 λ 9 2 , where function λ 9 is defined by (2.18), and let σ ≥ 1 . Set ψ = u 1 x r λ 9 2 max { z ˜ − σ , 0 } = u 1 x r λ 9 2 z ˜ ( σ ) in (3.34), and sum the resulting equations with respect to r from 1 to n . Then

(3.51) ∑ r = 1 n ∫ 0 τ ∫ Ω − u 1 t [ u 1 x r λ 9 2 z ˜ ( σ ) ] x r d x d t + ∑ r = 1 n ∫ 0 τ ∫ A z ˜ , σ ( t ) ∑ i = 1 n ∑ j = 1 n ∂ B 1 , i ( x , t , u 1 , ∇ u 1 ) ∂ u 1 x j u 1 x j x r + ∂ B 1 , i ∂ u 1 u 1 x r + ∂ B 1 , i ∂ x r ( u 1 x r x i λ 9 2 ( z ˜ − σ ) + u 1 x r λ 9 2 z ˜ x i + 2 u 1 x r ( z ˜ − σ ) λ 9 λ 9 x i ) − B 1 ( x , t , u 1 , ∇ u 1 ) [ u 1 x r x r λ 9 2 ( z ˜ − σ ) + u 1 x r λ 9 2 z ˜ x r + 2 u 1 x r ( z ˜ − σ ) λ 9 λ 9 x r ] } d x d t = 0 .

Note that by integration by parts,

(3.52) ∑ r = 1 n ∫ 0 τ ∫ Ω − u 1 t [ u 1 x r λ 9 2 z ˜ ( σ ) ] x r d x d t = 1 4 ∫ Ω ( z ˜ ( σ ) ( x , τ ) ) 2 d x − ∫ 0 τ ∫ Ω ∣ ∇ u 1 ∣ 2 z ˜ ( σ ) λ 9 λ 9 t d x d t ≥ 1 4 ∫ Ω ( z ˜ ( σ ) ( x , τ ) ) 2 d x − ∫ 0 τ ∫ A z ˜ , σ ( t ) ∣ ∇ u 1 ∣ 2 ( z ˜ − σ ) ∣ λ 9 λ 9 t ∣ d x d t ,

and by (3.5),

(3.53) ∑ r = 1 n ∑ i , j = 1 n ∂ B 1 , i ( x , t , u 1 , ∇ u 1 ) ∂ u 1 x j u 1 x j x r [ u 1 x r x i λ 9 2 ( z ˜ − σ ) + u 1 x r λ 9 2 z ˜ x i ] = a 11 ( x , t , u 1 ) ∣ ∇ 2 u 1 ∣ 2 ( z ˜ − σ ) λ 9 2 + 1 2 ∣ ∇ z ˜ ∣ 2 − ∣ ∇ u 1 ∣ 2 λ 9 ∇ λ 9 ⋅ ∇ z ˜ .

Substituting (3.52) and (3.53) into (3.51) and using (3.8), (3.35), (3.47), and Cauchy’s inequality with ε , we can obtain

1 4 ∫ Ω ( z ˜ ( σ ) ) 2 ( x , τ ) d x + ν ( 0 ) 2 ∫ 0 τ ∫ A z ˜ , σ ( t ) ∣ ∇ 2 u 1 ∣ 2 ( z ˜ − σ ) λ 9 2 + 1 2 ∣ ∇ z ˜ ∣ 2 d x d t ≤ C ∫ 0 τ ∫ A z ˜ , σ ( t ) { ( 1 + ∣ ∇ u 1 ∣ ) 6 ( λ 9 2 + ∣ ∇ λ 9 ∣ 2 ) + ∣ ∇ u 1 ∣ 2 ( z ˜ − σ ) ( λ 9 2 + ∣ ∇ λ 9 ∣ 2 + ∣ λ 9 λ 9 t ∣ ) + ( ϕ 1 2 + 1 ) [ ( z ˜ − σ ) λ 9 2 + z ˜ ] } d x d t .

Note that λ 8 = 1 for ( x , t ) ∈ Ω 8 × ( t 8 , T ] and λ 9 = 0 for x ∉ Ω 8 or t ≤ t 8 . Then we further obtain { ( x , t ) : x ∈ A z ˜ , σ ( t ) , t ∈ ( 0 , τ ] } ⊆ D 8 , τ , and

∫ Ω ( z ˜ ( σ ) ( x , τ ) ) 2 d x + ∫ 0 τ ∫ A z ˜ , σ ( t ) ∣ ∇ z ˜ ∣ 2 d x d t ≤ C ∫ 0 τ ∫ A z ˜ , σ ( t ) { ( 1 + ∣ ∇ u 1 ∣ ∣ λ 8 ∣ ) 6 ( λ 9 2 + ∣ ∇ λ 9 ∣ 2 ) + ∣ ∇ u 1 ∣ 2 λ 8 2 ( z ˜ − σ ) ( λ 9 2 + ∣ ∇ λ 9 ∣ 2 + ∣ λ 9 λ 9 t ∣ ) + ( ϕ 1 2 + 1 ) [ ( z ˜ − σ ) λ 9 2 + z ˜ ] } d x d t ≤ C ∫ 0 τ ∫ A z ˜ , σ ( t ) G 3 ( x , t ) [ ( z ˜ − σ ) 2 + σ 2 ] d x d t ,

where G 3 ( x , t ) = C ( ∣ ∇ u 1 ∣ 6 λ 8 2 + ϕ 1 2 + 1 ) . From (2.7) and (3.33) we find that G 3 ( x , t ) ∈ L q 1 ( D T ) . Result (ii) of Lemma 2.4 shows that sup D T z ˜ does not exceed constant C . Therefore, estimate (3.50) holds for n ≥ 2 .

Step 2. We prove that estimate (3.49) holds for n ≥ 1 . By estimates (3.2) and (3.26), the first equality of (2.1) yields

(3.54) u 1 t = div [ a 11 ( x , t , u 1 ) ∇ u 1 ] + b 1 ( x , t , U , ∇ u 1 ) ( ( x , t ) ∈ D 9 , T ) .

For fixed x 9 ∈ Ω 9 , set u 1 ∗ ∗ = u 1 ( x 9 , t 9 ) . Let

z ˇ 1 = λ 10 2 ∫ u 1 ∗ ∗ u 1 a 11 ( x , t , ω ) d ω .

By a direct computation we find from (3.54) that z ˇ 1 is a solution in W 2 2 , 1 ( D 9 , T ) for a linear problem in the form

(3.55) z t = a ¯ 1 ( x , t ) Δ z + f ¯ 1 ( x , t ) ( ( x , t ) ∈ D 9 , T ) , z ( x , t ) = 0 ( x ∈ ∂ Ω 9 , t ∈ [ t 9 , T ] ) , z ( x , t 9 ) = 0 ( x ∈ Ω 9 ) ,

where

(3.56) a ¯ 1 ( x , t ) = a 11 ( x , t , u 1 ) , f ¯ 1 ( x , t ) = λ 10 2 a 11 ( x , t , u 1 ) b 1 ( x , t , U , ∇ u 1 ) + ( λ 10 2 ) t ∫ u 1 ∗ ∗ u 1 a 11 ( x , t , ω ) d ω − a 11 ( x , t , u 1 ) Δ ( λ 10 2 ) ∫ u 1 ∗ ∗ u 1 a 11 ( x , t , ω ) d ω + 2 a 11 ( x , t , u 1 ) ∇ ( λ 10 2 ) ⋅ ∇ u 1 + 2 ∇ ( λ 10 2 ) ⋅ ∫ u 1 ∗ ∗ u 1 ∇ a 11 ( x , t , ω ) d ω + λ 10 2 div ∫ u 1 ∗ ∗ u 1 ∇ a 11 ( x , t , ω ) d ω .

Hence,

(3.57) ∇ u 1 = ∇ z ˇ 1 − ∫ u 1 ∗ ∗ u 1 ∇ a 11 ( x , t , ω ) d ω a 11 ( x , t , u 1 ) ( ( x , t ) ∈ D 10 , T ) , ( ∇ u 1 ) x k = ( ∇ z ˇ 1 ) x k − [ a 11 ( x , t , u 1 ) ] x k ∇ u 1 − ∫ u 1 ∗ ∗ u 1 ∇ a 11 ( x , t , ω ) d ω x k a 11 ( x , t , u 1 ) ( ( x , t ) ∈ D 10 , T ) .

Estimate (3.2) shows that a ¯ 1 ( x , t ) ∈ C ( D ¯ 9 , T ) , and (2.4), (3.56), (3.2), (3.9), and (3.47) yield

∣ ∇ a ¯ 1 ( x , t ) ∣ ≤ C ( ∣ ∇ u 1 ∣ + 1 ) , ∣ f ¯ 1 ( x , t ) ∣ ≤ C ( ∣ ∇ u 1 ∣ + ϕ 1 + 1 ) .

Thus, it follows from (2.7) and (3.27) that ∣ ∇ a ¯ 1 ( x , t ) ∣ ∈ L ∞ , 4 ( D 9 , T ) ∩ L 6 ( D 9 , T ) and f ¯ 1 ( x , t ) ∈ L q 3 ( D 9 , T ) for n = 1 , and from (2.7) and (3.50) that ∣ ∇ a ¯ 1 ( x , t ) ∣ ∈ L ∞ ( D 9 , T ) and f ¯ 1 ( x , t ) ∈ L q 3 ( D 9 , T ) for n ≥ 2 . Then applying [12, Chapter IV, Section 9, Theorem 9.1], we see that problem (3.55) has a unique solution z in W q 3 2 , 1 ( D 9 , T ) , and

(3.58) ‖ z ‖ W q 3 2 , 1 ( D 9 , T ) ≤ C .

Since z ˇ 1 is also a solution in W 2 2 , 1 ( D 9 , T ) for problem (3.55), then

(3.59) ( z − z ˇ 1 ) t = a ¯ 1 ( x , t ) Δ ( z − z ˇ 1 ) ( ( x , t ) ∈ D 9 , T ) , ( z − z ˇ 1 ) ( x , t ) = 0 ( x ∈ ∂ Ω 9 , t ∈ [ t 9 , T ] ) , ( z − z ˇ 1 ) ( x , t 9 ) = 0 ( x ∈ Ω 9 ) .

We multiply the equation in (3.59) by z − z ˇ 1 , integrate it over D 9 , τ and use Cauchy’s inequality to find that for any ε ∈ ( 0 , 1 ) ,

∫ Ω 9 ( z − z ˇ 1 ) 2 ( x , τ ) d x + ∫ t 9 τ ∫ Ω 9 a ¯ 1 ( x , t ) ∣ ∇ ( z − z ˇ 1 ) ∣ 2 d x d t = ∫ t 9 τ ∫ Ω 9 − [ ∇ a ¯ 1 ( x , t ) ⋅ ∇ ( z − z ˇ 1 ) ] ( z − z ˇ 1 ) d x d t ≤ ε ∫ t 9 τ ∫ Ω 9 ∣ ∇ ( z − z ˇ 1 ) ∣ 2 d x d t + C ε ∫ t 9 τ ‖ ∇ a ¯ 1 ( x , t ) ‖ L ∞ ( Ω 9 ) 2 ∫ Ω 9 ( z − z ˇ 1 ) 2 d x d t .

Setting ε = ν ( 0 ) / 2 leads to the inequality

∫ Ω 9 ( z − z ˇ 1 ) 2 ( x , τ ) d x + ∫ t 9 τ ∫ Ω 9 ∣ ∇ ( z − z ˇ 1 ) ∣ 2 d x d t ≤ C ∫ t 9 τ ‖ ∇ a ¯ 1 ( x , t ) ‖ L ∞ ( Ω 9 ) 2 ∫ Ω 9 ( z − z ˇ 1 ) 2 d x d t .

Gronwall’s inequality further implies that z = z ˇ 1 in D ¯ 9 , T . Since (2.2) leads to the relation q 3 > n + 2 , then it follows from (3.58) and Sobolev embedding theorem that for some α 3 ∈ ( 0 , 1 ) ,

‖ z ˇ 1 x i ‖ C α 3 , α 3 / 2 ( D ¯ 9 , T ) ≤ C ′ ‖ z ˇ 1 ‖ W q 3 2 , 1 ( D 9 , T ) ≤ C , i = 1 , … , n .

This, together with (3.57), shows that (3.49) holds.□

Lemma 3.8

We have

(3.60) ‖ u 2 ‖ C α 4 , α 4 / 2 ( D ¯ 11 , T ) ≤ C , α 4 ∈ ( 0 , 1 ) ,

where α 4 = α 4 ( Θ 1 , ν ( M 1 ) , μ ( ∣ ( M 1 , M 2 ) ∣ ) ) and C = C ( Θ 1 , ν ( M 1 ) , μ ( ∣ ( M 1 , M 2 ) ∣ ) ) .

Proof

Let

(3.61) B 2 , i ( x , t , w , p ) ≔ a 22 ( x , t , u 1 , w ) p i + a 21 ( x , t , u 1 , w ) u 1 x i , B 2 ( x , t , w , p ) ≔ − b 2 ( x , t , u 1 , w , ∇ u 1 , p ) .

From (3.47) and (3.49), it follows that for ( x , t ) ∈ D 10 , T , w ∈ [ − M 2 , M 2 ] ∩ S 2 and p ∈ R n ,

(3.62) ∑ i = 1 n B 2 , i ( x , t , w , p ) p i ≥ ν ( M 1 ) 2 ∣ p ∣ 2 − C , ∣ B 2 , i ( x , t , w , p ) ∣ ≤ C ( ∣ p ∣ + 1 ) ,

(3.63) ∣ B 2 ( x , t , w , p ) ∣ ≤ C ( ∣ p ∣ + ϕ 2 + 1 ) .

Moreover, it follows from the second integral equality of (2.1) that for any τ 0 , τ ∈ ( t 10 , T ] and for any η ∈ W ∘ 2 1 , 1 ( D 10 , T ) ,

(3.64) ∫ Ω 10 u 2 ( x , t ) η ( x , t ) d x τ 0 τ + ∫ τ 0 τ ∫ Ω 10 − u 2 η t + ∑ i = 1 n B 2 , i ( x , t , u 2 , ∇ u 2 ) η x i + B 2 ( x , t , u 2 , ∇ u 2 ) η d x d t = 0 .

Then employing [12, Chapter V, Section 1, Theorem 1.1] again, we see from (2.7), (2.20), (3.47), and (3.62)–(3.64) that (3.60) holds.□

3.4 Estimate of ‖ u 2 ‖ W 2 2 , 1 ( Ω ′ × ( t ′ , T ] )

In this subsection, by using the similar methods as those in Section 3.2, we will establish the estimate of ‖ u 2 ‖ W 2 2 , 1 ( Ω ′ × ( t ′ , T ] ) from the second equation of (1.1). Based on estimates (3.47) and (3.49), we first use Steklov average method to prove that u 2 has weak derivative u 2 t .

By hypothesis ( H 1 )(I), for some positive constant Θ 2 ,

(3.65) ‖ a 22 ( x , t , W ) ‖ C 1 ( Ξ 2 ) , ‖ a 22 x i x i , a 22 x i w j , a 22 w 1 w 1 ‖ C ( Ξ 2 ) ≤ Θ 2 ,

where i = 1 , … , n , j = 1 , 2 and Ξ 2 ≔ D ¯ 8 , T × ( S 1 ∩ [ − M 1 , M 1 ] ) × ( S 2 ∩ [ − M 2 , M 2 ] ) .

Lemma 3.9

Function u 2 has weak derivative u 2 t in D 12 , T , and satisfies

(3.66) ess sup t 12 ≤ t ≤ T ‖ ∇ u 2 ( ⋅ , t ) ‖ L 2 ( Ω 12 ) + ‖ u 2 t ‖ L 2 ( D 12 , T ) ≤ C .

Moreover, if the vertex P ¯ = ( x ¯ , t ¯ ) of cylinders Q ρ ( P ¯ ) and Q 2 ρ ( P ¯ ) is in Ω ¯ 13 × [ t 13 , T ] and if ρ ≤ ρ 5 , then

(3.67) ∬ Q 2 ρ ( P ¯ ) u 2 t 2 ξ ρ 2 d x d t + ∫ B 2 ρ ( x ¯ ) ∣ ∇ u 2 ( x , t ¯ ) ∣ 2 ξ ρ 2 ( x , t ¯ ) d x ≤ C ∫ B 2 ρ ( x ¯ ) ξ ρ 2 ( x , t ¯ ) d x + C ∬ Q 2 ρ ( P ¯ ) [ ( 1 + ∣ ∇ u 2 ∣ 2 ) ( ∣ ∇ ξ ρ ∣ 2 + ∣ ξ ρ ξ ρ t ∣ + ξ ρ 2 ) + ( u 1 t 2 + ∣ ∇ 2 u 1 ∣ 2 + ϕ 2 2 ) ξ ρ 2 ] d x d t .

Here, C = C ( Θ 1 , Θ 2 , ν ( M 1 ) , μ ( ∣ ( M 1 , M 2 ) ∣ ) ) .

Proof

Fix x 11 ∈ Ω 11 and set u 2 ∗ = u 2 ( x 11 , t 11 ) . Let

(3.68) w ˆ 2 = w ˆ 2 ( x , t ) = ∫ u 2 ∗ u 2 a 22 ( x , t , u 1 , ω ) d ω .

Hence, for each i = 1 , … , n ,

(3.69) w ˆ 2 x i = u 2 x i a 22 ( x , t , U ) + u 1 x i ∫ u 2 ∗ u 2 a 22 u 1 ( x , t , u 1 , ω ) d ω + ∫ u 2 ∗ u 2 a 22 x i ( x , t , u 1 , ω ) d ω .

It follows from the second equality of (2.1) and (3.69) that for any η ∈ W ∘ 2 1 , 1 ( D 11 , T ) and τ ∈ ( t 11 , T ] ,

∫ Ω 11 u 2 η d x t 11 τ − ∬ D 11 , τ u 2 η t d x d t = ∬ D 11 , τ [ − ∇ w ˆ 2 ⋅ ∇ η + f ˆ 2 ( x , t ) η ] d x d t ,

where

f ˆ 2 ( x , t ) = − ∑ i = 1 n u 1 x i ∫ u 2 ∗ u 2 a 22 u 1 ( x , t , u 1 , ω ) d ω + ∫ u 2 ∗ u 2 a 22 x i ( x , t , u 1 , ω ) d ω x i + ∑ i = 1 n [ a 21 ( x , t , U ) u 1 x i ] x i + b 2 ( x , t , U , ∇ U ) .

Inequalities (2.4), (2.8), and (3.65), and estimates (3.2), (3.47), and (3.49) yield

(3.70) ∣ ∇ w ˆ 2 ∣ ≤ C ( 1 + ∣ ∇ u 2 ∣ ) , ∣ ∇ u 2 ∣ ≤ C ( ∣ ∇ w ˆ 2 ∣ + 1 ) , ∣ f ˆ 2 ∣ ≤ C ( ∣ ∇ u 2 ∣ + ∣ ∇ 2 u 1 ∣ + ϕ 2 ) .

Furthermore, by (3.49) and (2.7),

(3.71) ‖ w ˆ 2 ‖ L ∞ ( D 11 , T ) ≤ C , ‖ ∣ ∇ w ˆ 2 ∣ , u 1 t , f ˆ 2 ‖ L 2 ( D 11 , T ) ≤ C .

Again from the similar arguments as those of [12, Chapter III, Section 2], it follows that for any given h ∈ ( 0 , T − t 12 ) and τ ∈ ( t 11 , T − h ] and for any η ∈ V ∘ 2 1 , 0 ( D 11 , T ) ,

(3.72) ∬ D 11 , τ u 2 ( t ) η d x d t = ∬ D 11 , τ { − ∇ w ˆ 2 h ⋅ ∇ η + [ f ˆ 2 ( x , t ) ] h η } d x d t .

(3.68) yields

w ˆ 2 ( t ) = u 2 ( t ) ∫ 0 1 a 22 ( x , t ϑ , U ϑ ) d ϑ + u 1 ( t ) ∫ 0 1 ∫ u 2 ∗ u 2 ϑ a 22 u 1 ϑ ( x , t ϑ , u 1 ϑ , ω ) d ω d ϑ + ∫ 0 1 ∫ u 2 ∗ u 2 ϑ a 22 t ϑ ( x , t ϑ , u 1 ϑ , ω ) d ω d ϑ ,

where t ϑ = t + ϑ h and U ϑ = ϑ U ( x , t + h ) + ( 1 − ϑ ) U ( x , t ) . Hence,

(3.73) u 2 ( t ) = w ˆ 2 ( t ) − u 1 ( t ) ∫ 0 1 ∫ u 2 ∗ u 2 ϑ a 22 u 1 ϑ ( x , t ϑ , u 1 ϑ , ω ) d ω d ϑ − ∫ 0 1 ∫ u 2 ∗ u 2 ϑ a 22 t ϑ ( x , t ϑ , u 1 ϑ , ω ) d ω d ϑ ∫ 0 1 a 22 ( x , t ϑ , U ϑ ) d ϑ .

For any given P ¯ ∈ D ¯ 12 , T − h and ρ ≤ ρ 5 , set η = w ˆ 2 ( t ) ξ ρ 2 in (3.72) and use (3.73) to find that

∬ Q 2 ρ ( P ¯ ) ∫ 0 1 a 22 ( x , t ϑ , U ϑ ) d ϑ − 1 w ˆ 2 ( t ) 2 ξ ρ 2 d x d t + 1 2 ∫ B 2 ρ ( x ¯ ) ∣ ∇ w ˆ 2 h ( x , t ¯ ) ∣ 2 ξ ρ 2 ( x , t ¯ ) d x = ∬ Q 2 ρ ( P ¯ ) u 1 ( t ) ∫ 0 1 ∫ u 2 ∗ u 2 ϑ a 22 u 1 ϑ ( x , t ϑ , u 1 ϑ , ω ) d ω d ϑ + ∫ 0 1 ∫ u 2 ∗ u 2 ϑ a 22 t ϑ ( x , t ϑ , u 1 ϑ , ω ) d ω d ϑ × ∫ 0 1 a 22 ( x , t ϑ , U ϑ ) d ϑ − 1 w ˆ 2 ( t ) ξ ρ 2 d x d t + ∬ Q 2 ρ ( P ¯ ) { − 2 w ˆ 2 ( t ) ξ ρ ∇ w ˆ 2 h ⋅ ∇ ξ ρ + ∣ ∇ w ˆ 2 h ∣ 2 ξ ρ ξ ρ t + [ f ˆ 2 ( x , t ) ] h w ˆ 2 ( t ) ξ ρ 2 } d x d t .

By (3.48), (3.65), and cauchy’s inequality with ε , we can further obtain

(3.74) ∬ Q 2 ρ ( P ¯ ) w ˆ 2 ( t ) 2 ξ ρ 2 d x d t + ∫ B 2 ρ ( x ¯ ) ∣ ∇ w ˆ 2 h ( x , t ¯ ) ∣ 2 ξ ρ 2 ( x , t ¯ ) d x ≤ C ∬ Q 2 ρ ( P ¯ ) { ∣ ∇ w ˆ 2 h ∣ 2 ( ∣ ∇ ξ ρ ∣ 2 + ∣ ξ ρ ξ ρ t ∣ ) + ( u 1 ( t ) 2 + 1 ) ξ ρ 2 + [ f ˆ 2 ( x , t ) ] h 2 ξ ρ 2 } d x d t .

This, together with (3.10) and (3.71), implies that u 2 has weak derivative u 2 t in D 12 , T and estimate (3.66) holds. The deduction is the same as that of (3.10) from (3.18).

If P ¯ ∈ D ¯ 13 , T and ρ ≤ ρ 5 , then Q 2 ρ ( P ¯ ) ⊂ D 12 , T . Letting h → 0 in (3.74) leads to the inequality

∬ Q 2 ρ ( P ¯ ) w ˆ 2 t 2 ξ ρ 2 d x d t + ∫ B 2 ρ ( x ¯ ) ∣ ∇ w ˆ 2 ( x , t ¯ ) ∣ 2 ξ ρ 2 ( x , t ¯ ) d x ≤ C ∬ Q 2 ρ ( P ¯ ) [ ∣ ∇ w ˆ 2 ∣ 2 ( ∣ ∇ ξ ρ ∣ 2 + ∣ ξ ρ ξ ρ t ∣ ) + ( u 1 t 2 + 1 ) ξ ρ 2 + f ˆ 2 2 ξ ρ 2 ] d x d t ,

which, together with (3.70), gives (3.67).□

As we have done in the derivation of (3.26), to show that u 2 has the second partial derivatives u x i x k , we need the following lemma:

Lemma 3.10

For any given P ¯ ∈ D 13 , T and ρ ≤ ρ 6 ≔ ρ 5 / 2 , if ζ ( x , t ) be an arbitrary bounded function from V ∘ 2 ( Q ρ ( P ¯ ) ) , then there exist constants C , α 5 depending on Θ 1 , Θ 2 , ν ( M 1 ) , and μ ( ∣ ( M 1 , M 2 ) ∣ ) , such that

(3.75) ∬ Q ρ ( P ¯ ) ( ∣ ∇ u 2 ∣ 2 + ϕ 2 2 + Φ ˜ 2 2 ) ζ 2 d x d t ≤ C ρ 2 α 5 ∬ Q ρ ( P ¯ ) ∣ ∇ ζ ∣ 2 d x d t , α 5 ∈ ( 0 , 1 ) .

Proof

Step 1. We prove that there exists constant α ˆ ∈ ( 0 , 1 ) such that

(3.76) ∬ Q ρ ∣ ∇ u 2 ∣ 2 d x d t ≤ C ρ n + 2 α ˆ

for ρ ≤ ρ 5 . Let ( x ∗ , t ∗ ) be a given point in Q ρ ( P ¯ ) . Set η = ( u 2 ( x , t ) − u 2 ( x ∗ , t ∗ ) ) ξ ρ 2 and τ = t ¯ in the second equality of (2.1) and use (2.4), (3.48), (3.49), (3.60), and Cauchy’s inequality to obtain, for any ε ∈ ( 0 , 1 ) ,

1 2 ∫ B 2 ρ ( x ¯ ) ( u 2 ( x , t ¯ ) − u 2 ( x ∗ , t ∗ ) ) 2 ξ ρ 2 d x + ∬ Q 2 ρ ( P ¯ ) a 22 ( x , t , U ) ∣ ∇ u 2 ∣ 2 ξ ρ 2 d x d t = ∬ Q 2 ρ ( P ¯ ) ( u 2 ( x , t ) − u 2 ( x ∗ , t ∗ ) ) 2 ξ ρ ξ ρ t − a 21 ( x , t , U ) ξ ρ 2 ∇ u 1 ⋅ ∇ u 2 − 2 ( u 2 ( x , t ) − u 2 ( x ∗ , t ∗ ) ) ξ ρ ∑ j = 1 2 a 2 j ( x , t , U ) ∇ u j ⋅ ∇ ξ ρ + b 2 ( x , t , U , ∇ U ) ( u 2 ( x , t ) − u 2 ( x ∗ , t ∗ ) ) ξ ρ 2 } d x d t ≤ ε ∬ Q 2 ρ ( P ¯ ) ∣ ∇ u 2 ∣ 2 ξ ρ 2 d x d t + C ε ∬ Q 2 ρ ( P ¯ ) { ( u 2 ( x , t ) − u 2 ( x ∗ , t ∗ ) ) 2 [ ξ ρ 2 + ∣ ∇ ξ ρ ∣ 2 + ∣ ξ ρ ξ ρ t ∣ ] + ξ ρ 2 + ∣ u 2 ( x , t ) − u 2 ( x ∗ , t ∗ ) ∣ [ ∣ ∇ ξ ρ ∣ 2 + ( 1 + ϕ 2 ) ξ ρ 2 ] } d x d t .

This inequality has the similar property as (3.24) for α ˆ = min { α 4 , 2 χ 1 } . Then we can obtain (3.76). The deduction is the same as that of (3.22) from (3.24).

Step 2. We show that (3.75) holds for ρ ≤ ρ 6 .

By (3.66), (2.7), and (2.11), we find from [12, Chapter II, Lemma 5.3’] that (3.75) holds for n = 1 .

We next consider the case n ≥ 2 . In view of condition (2.7) and estimates (3.49) and (3.76), it follows from (3.67) and (2.22) that

∬ Q 2 ρ ( P ¯ ) u 2 t 2 ξ ρ 2 d x d t + ∫ B 2 ρ ( x ¯ ) ∣ ∇ u 2 ( x , t ¯ ) ∣ 2 ξ ρ 2 ( x , t ¯ ) d x ≤ C ρ n + C ρ 2 ∬ Q 2 ρ ( P ¯ ) ( 1 + ∣ ∇ u 2 ∣ 2 ) d x d t + C ‖ u 1 ‖ W q 3 2 , 1 ( Q 2 ρ ( P ¯ ) ) 2 ( ρ 2 + n ) 1 − 2 / q 3 + ∬ Q 2 ρ ( P ¯ ) ϕ 2 2 d x d t ≤ C [ ρ n + ρ n − 2 + 2 α ˆ + ( ρ 2 + n ) 1 − 2 / q 3 + ρ n + 2 α 0 ] ≤ C ρ n − 2 + 2 α 5 ,

where α 5 = min { α ˆ , 2 − ( n + 2 ) / q 3 } = min { α ˆ , 1 + χ 1 } = α ˆ . Thus,

ess sup t 13 ≤ t ≤ T ∫ B ρ ∣ ∇ u 2 ∣ 2 d x ≤ C ρ n − 2 + 2 α 5 .

Using [12, Chapter II, Lemma 5.2] and (2.22) again leads to inequality (3.75) for the case n ≥ 2 . Hence, we complete the proof of the lemma.□

Lemma 3.11

Function u 2 has weak derivatives u 2 x i x k in D 14 , T for i , k = 1 , … , n , and satisfies

(3.77) ‖ u 2 ‖ W 2 2 , 1 ( D 14 , T ) ≤ C ,

where C = C ( Θ 1 , Θ 2 , ν ( M 1 ) , μ ( ∣ ( M 1 , M 2 ) ∣ ) ) .

Proof

Let P ¯ ∈ D 14 , T and ρ ≤ ρ 6 . Thus Q 2 ρ ( P ¯ ) ⊂ D 13 , T . By using the similar proof as that of (3.28), we see from the second equality of (2.1) that for any given k ∈ { 1 , … , n } ,

(3.78) 1 2 ∫ B 2 ρ ( x ¯ ) u 2 ( x k ) 2 ( x , t ¯ ) ξ ρ 2 ( x , t ¯ ) d x + ∬ Q 2 ρ ( P ¯ ) − u 2 ( x k ) 2 ξ ρ ξ ρ t + ∑ i = 1 n [ a 22 ( x , t , U ) u 2 x i ] ( x k ) ( u 2 ( x k ) ξ ρ 2 ) x i d x d t = ∬ Q 2 ρ ( P ¯ ) − ∑ i = 1 n [ a 21 ( x , t , U ) u 1 x i ] ( x k ) ( u 2 ( x k ) ξ ρ 2 ) x i + [ b 2 ( x , t , U , ∇ U ) ] ( x k ) u 2 ( x k ) ξ ρ 2 d x d t .

Since

[ a 2 j ( x , t , U ) u j x i ] ( x k ) = u j ( x k ) x i ∫ 0 1 a 2 j ( x θ , t , U θ ) d θ + ∑ r = 1 2 u r ( x k ) ∫ 0 1 ∂ a 2 j ∂ u r θ u j x i θ d θ + ∫ 0 1 ∂ a 2 j ∂ x k θ u j x i θ d θ ,

and

[ b 2 ( x , t , U , ∇ U ) ] ( x k ) = ∑ m = 1 2 ∑ i = 1 n u m ( x k ) x i ∫ 0 1 ∂ b 2 ( x θ , t , U θ , ∇ U θ ) ∂ u m x i θ d θ + ∑ m = 1 2 u m ( x k ) ∫ 0 1 ∂ b 2 ∂ u m θ d θ + ∫ 0 1 ∂ b 2 ∂ x k θ d θ ,

then by (2.8), (2.9), (3.47)–(3.49), (3.65), and Cauchy’s inequality, we further obtain, for any ε ∈ ( 0 , 1 ) ,

∑ i = 1 n [ a 22 ( x , t , U ) u 2 x i ] ( x k ) ( u 2 ( x k ) x i ξ ρ 2 + 2 u 2 ( x k ) ξ ρ ξ ρ x i ) ≥ ∣ ∇ u 2 ( x k ) ∣ 2 ξ ρ 2 ∫ 0 1 a 22 ( x θ , t , U θ ) d θ − C ∣ u 2 ( x k ) ∣ ∣ ∇ u 2 ( x k ) ∣ ξ ρ 2 ∫ 0 1 ∣ ∇ u 2 θ ∣ d θ − C ∣ u 2 ( x k ) ξ ρ ∣ ∣ ∇ ξ ρ ∣ ∣ ∇ u 2 ( x k ) ∣ + ( ∣ u 2 ( x k ) ∣ + 1 ) ∫ 0 1 ∣ ∇ u 2 θ ∣ d θ ≥ ν ( M 1 ) ∣ ∇ u 2 ( x k ) ∣ 2 ξ ρ 2 − ε ∣ ∇ u 2 ( x k ) ∣ 2 ξ ρ 2 − C ε ( ∣ u 2 ( x k ) ∣ + 1 ) 2 ξ ρ 2 ∫ 0 1 ∣ ∇ u 2 θ ∣ 2 d θ + ∣ u 2 ( x k ) ∣ 2 ∣ ∇ ξ ρ ∣ 2 , ∑ i = 1 n ∣ [ a 21 ( x , t , U ) u 1 x i ] ( x k ) ( u 2 ( x k ) x i ξ ρ 2 + 2 u 2 ( x k ) ξ ρ ξ ρ x i ) ∣ ≤ C ( ∣ ∇ u 1 ( x k ) ∣ + ∣ u 2 ( x k ) ∣ + 1 ) ( ∣ ∇ u 2 ( x k ) ∣ ξ ρ 2 + ∣ u 2 ( x k ) ∣ ξ ρ ∣ ∇ ξ ρ ∣ ) ≤ ε ∣ ∇ u 2 ( x k ) ∣ 2 ξ ρ 2 + C ε ∣ ∇ u 1 ( x k ) ∣ 2 ξ ρ 2 + C ε ( u 2 ( x k ) 2 + 1 ) ( ξ ρ 2 + ∣ ∇ ξ ρ ∣ 2 ) ,

and

∣ [ b 2 ( x , t , U , ∇ U ) ] ( x k ) u 2 ( x k ) ξ ρ 2 ∣ ≤ C ( ∣ ∇ u 2 ( x k ) ∣ + ∣ ∇ u 1 ( x k ) ∣ ) ∫ 0 1 ( ϕ ˜ 24 ( x θ , t ) + 1 ) d θ + ∣ u 2 ( x k ) ∣ ∫ 0 1 ( ∣ ∇ u 2 θ ∣ + ϕ ˜ 22 ( x θ , t ) + 1 ) d θ + ∫ 0 1 ( ∣ ∇ u 2 θ ∣ + ϕ ˜ 23 ( x θ , t ) + ϕ ˜ 21 ( x θ , t ) + 1 ) d θ ∣ u 2 ( x k ) ∣ ξ ρ 2 ≤ ε ∣ ∇ u 2 ( x k ) ∣ 2 ξ ρ 2 + C ε ∣ ∇ u 1 ( x k ) ∣ 2 ξ ρ 2 + ( ∣ u 2 ( x k ) ∣ + 1 ) 2 ξ ρ 2 ∫ 0 1 ( ∣ ∇ u 2 θ ∣ 2 + Φ ˜ 2 2 ( x θ , t ) + 1 ) d θ .

Choosing ε = min { 1 / 2 , ν ( M 1 ) / 6 } and substituting the aforementioned three inequalities into (3.78) yields

(3.79) ∫ B 2 ρ ( x ¯ ) u 2 ( x k ) 2 ( x , t ¯ ) ξ ρ 2 ( x , t ¯ ) d x + ∬ Q 2 ρ ( P ¯ ) ∣ ∇ u 2 ( x k ) ∣ 2 ξ ρ 2 d x d t ≤ C ∬ Q 2 ρ ( P ¯ ) u 2 ( x k ) 2 [ ξ ρ 2 + ∣ ∇ ξ ρ ∣ 2 + ∣ ξ ρ ξ ρ t ∣ ] d x d t + C ∬ Q 2 ρ ( P ¯ ) ∣ ∇ u 1 ( x k ) ∣ 2 ξ ρ 2 d x d t + C ∫ 0 1 ∬ Q 2 ρ ( P ¯ ) ( ∣ ∇ u 2 θ ∣ 2 + Φ ˜ 2 2 ( x θ , t ) ) ( u 2 ( x k ) 2 + 1 ) ξ ρ 2 d x d t d θ .

In view of estimate (3.49), inequality (3.79) has the similar property as (3.31). In addition, note that (3.75) holds for all n ≥ 1 . Then by using (3.75) and the deduction similar to that of (3.26) from (3.31), we can conclude that u 2 has weak derivatives u 2 x i x k in D 14 , T for i , k = 1 , … , n , and estimate (3.77) holds.□

Proof of Theorem 2.1

It follows from estimates (3.2) and (3.60) that for any Ω ′ ⊂ ⊂ Ω and t ′ ∈ ( 0 , T ) , u 1 , u 2 ∈ C α ′ , α ′ / 2 ( Ω ¯ ′ × [ t ′ , T ] ) for some α ′ ∈ ( 0 , 1 ) , and from (3.49), (3.77) that u 1 ∈ W q 3 2 , 1 ( Ω ′ × ( t ′ , T ] ) and u 2 ∈ W 2 2 , 1 ( Ω ′ × ( t ′ , T ] ) .□

4 The proof of Theorem 2.2

In this section, assume that hypotheses ( H 1 ) and ( H 2 ) all hold. We will use L p estimates for parabolic equations, Sobolev embedding theorem and [12, Chapter III, Theorem 12.2] to complete the proof of Theorem 2.2. We use C and α j ( j = 6 , 7 , … ) to denote positive constants depending only on Θ 1 , Θ 2 , ν ( M 1 ) , μ ( ∣ ( M 1 , M 2 ) ∣ ) , ‖ b 1 ( x , t , W , p ) ‖ C 1 ( Ξ 3 × [ − ϒ 1 , ϒ 1 ] n ) , ‖ a 11 x i ( x , t , w 1 ) , a 11 w 1 ‖ C α 0 ′ ( Ξ 4 ) , ‖ a 2 j x i ( x , t , W ) , a 2 j w k ‖ C α 0 ′ ( Ξ 3 ) ( i = 1 , … , n ; j , k = 1 , 2 ), and the quantities appearing (3.1) and in parentheses, where Ξ 3 ≔ D ¯ 14 , T × ( S 1 ∩ [ − M 1 , M 1 ] ) × ( S 2 ∩ [ − M 2 , M 2 ] ) and Ξ 4 ≔ D ¯ 14 , T × ( S 1 ∩ [ − M 1 , M 1 ] ) .

Lemma 4.1

There exist positive constants C and α 6 such that

(4.1) ‖ u 1 ‖ C 2 + α 6 , 1 + α 6 / 2 ( D ¯ 15 , T ) ≤ C , α 6 ∈ ( 0 , 1 ) .

Proof

Consider the following linear problem

(4.2) w t = a ˘ 1 ( x , t ) Δ w + f ˘ 1 ( x , t ) ( ( x , t ) ∈ D 14 , T ) , w ( x , t ) = 0 ( x ∈ ∂ Ω 14 , t ∈ [ t 14 , T ] ) , w ( x , t 14 ) = 0 ( x ∈ Ω 14 ) ,

where

a ˘ 1 ( x , t ) = a 11 ( x , t , u 1 ) , f ˘ 1 ( x , t ) = { ∇ [ a 11 ( x , t , u 1 ) ] ⋅ ∇ u 1 + b 1 ( x , t , U , ∇ u 1 ) } λ 15 2 − a 11 ( x , t , u 1 ) [ 2 ∇ u 1 ⋅ ∇ ( λ 15 2 ) + u 1 Δ ( λ 15 2 ) ] + u 1 ( λ 15 2 ) t .

By a direct computation, we find that w ˘ 1 = u 1 λ 15 2 is the weak solution of (4.2) in W q 3 2 , 1 ( D 14 , T ) . Moreover, hypothesis ( H 2 ) and estimates (3.49) and (3.60) imply that a ˘ 1 ( x , t ) and f ˘ 1 ( x , t ) are Hölder continuous in Ω ¯ 14 × [ t 14 , T ] . Then [12, Chapter III, Theorem 12.2] shows that w ˘ 1 belongs to C 2 + α 6 , 1 + α 6 / 2 ( D ¯ 14 , T ) and ‖ w ˘ 1 ‖ C 2 + α 6 , 1 + α 6 / 2 ( D ¯ 14 , T ) is bounded from above by a constant C . This further leads to estimate (4.1).□

To show that u 2 ∈ C 2 + α , 1 + α / 2 ( Ω ¯ ′ × [ t ′ , T ] ) , we need the estimate of ‖ u 2 ‖ W q 2 , 1 ( D 18 , T ) .

Lemma 4.2

For any q > n + 2 , we have

(4.3) ‖ u 2 ‖ W q 2 , 1 ( D 18 , T ) ≤ C .

Proof

Step 1. We prove that for any positive integer K ,

(4.4) ∬ D 16 , T [ ∣ ∇ u 2 ∣ 4 + 2 K + ( 1 + ∣ ∇ u 2 ∣ ) 2 K ∣ ∇ 2 u 2 ∣ 2 ] d x d t ≤ C ( K ) .

Let B 2 , i ( x , t , w , p ) and B 2 ( x , t , w , p ) be defined by (3.61). It follows from (3.48), (3.60), and (4.1) that for ( x , t ) ∈ D ¯ 15 , T , w ∈ [ − M 2 , M 2 ] ∩ S 2 and p ∈ R n , B 2 ( x , t , w , p ) satisfies (3.63), and B 2 , i ( x , t , w , p ) ( i = 1 , … , n ) satisfy

(4.5) ν ( M 1 ) ∑ j = 1 n Ψ j 2 ≤ ∑ i , j = 1 n ∂ B 2 , i ( x , t , w , p ) ∂ p j Ψ i Ψ j ≤ C ∑ j = 1 n Ψ j 2 ,

(4.6) ∑ i = 1 n ∂ B 2 , i ( x , t , w , p ) ∂ w + ∣ B 2 , i ∣ + ∑ i , j = 1 n ∂ B 2 , i ∂ x j ≤ C ( 1 + ∣ p ∣ ) .

Let ψ = ψ ( x , t ) be smooth function satisfying ψ = 0 for x ∉ Ω 15 . As we have done in Lemma 3.5, for any given r ∈ { 1 , … , n } setting η = ψ x r in the second equality of (2.1), we have

∬ D 14 , T − u 2 t ψ x r + ∑ i = 1 n ∑ j = 1 n ∂ B 2 , i ( x , t , u 2 , ∇ u 2 ) ∂ u 2 x j u 2 x j x r + ∂ B 2 , i ∂ u 2 u 2 x r + ∂ B 2 , i ∂ x r ψ x i − B 2 ( x , t , u 2 , ∇ u 2 ) ψ x r d x d t = 0 .

Note that (3.75) holds for n ≥ 1 . Then by inequalities (3.63), (3.75), (4.5), and (4.6), and by the proof similar to that of (3.33), we can obtain estimate (4.4).

Step 2. Based on (4.4), the proof similar to that of (3.50) further yields

(4.7) ess sup D 17 , T ∣ ∇ u 2 ∣ ≤ ϒ 2

for some positive constant ϒ 2 .

Step 3. Consider the following linear problem:

(4.8) z t = a ¯ 2 ( x , t ) Δ z + f ¯ 2 ( x , t ) ( ( x , t ) ∈ D 17 , T ) , z ( x , t ) = 0 ( x ∈ ∂ Ω 17 , t ∈ [ t 17 , T ] ) , z ( x , t 17 ) = 0 ( x ∈ Ω 17 ) ,

where

a ¯ 2 ( x , t ) = a 22 ( x , t , U ) , f ¯ 2 ( x , t ) = λ 18 2 a 22 ( x , t , U ) [ b 2 ( x , t , U , ∇ U ) + div ( a 21 ( x , t , U ) ∇ u 1 ) ] + ( λ 18 2 ) t ∫ u 2 ∗ ∗ u 2 a 22 ( x , t , u 1 , ω ) d ω − a 22 ( x , t , U ) × Δ ( λ 18 2 ) ∫ u 2 ∗ ∗ u 2 a 22 ( x , t , u 1 , ω ) d ω + 2 a 22 ( x , t , U ) ∇ ( λ 18 2 ) ⋅ ∇ u 2 + 2 ∇ ( λ 18 2 ) ⋅ ∫ u 2 ∗ ∗ u 2 ∇ a 22 ( x , t , u 1 , ω ) d ω + λ 18 2 div ∫ u 2 ∗ ∗ u 2 ∇ a 22 ( x , t , u 1 , ω ) d ω ,

where u 2 ∗ ∗ = u 2 ( x 17 , t 17 ) for fixed x 17 ∈ Ω 17 . Hence, a ¯ 2 ( x , t ) is Hölder continuous in Ω ¯ 17 × [ t 17 , T ] . It follows from estimates (4.1) and (4.7) that ∣ ∇ a ¯ 2 ∣ , f ¯ 2 ∈ L ∞ ( D 17 , T ) . Then problem (4.8) has a unique solution z in W q 2 , 1 ( D 17 , T ) for any q > n + 2 , and

(4.9) ‖ z ‖ W q 2 , 1 ( D 17 , T ) ≤ C ( q ) .

Let z ˇ 2 = λ 18 2 ∫ u 2 ∗ ∗ u 2 a 22 ( x , t , u 1 , ω ) d ω . A direct computation shows that z ˇ 2 is also a solution in W 2 2 , 1 ( D 17 , T ) of (4.8). The similar argument as that of Lemma 3.7 shows that z ˇ 2 = z in D ¯ 17 , T . Note that z ˇ 2 = ∫ u 2 ∗ ∗ u 2 a 22 ( x , t , u 1 , ω ) d ω for ( x , t ) ∈ D 18 , T . Thus,

∇ u 2 = ∇ z ˇ 2 − ∫ u 2 ∗ ∗ u 2 ∇ a 22 ( x , t , u 1 , ω ) d ω a 22 ( x , t , U ) ( ( x , t ) ∈ D 18 , T ) , ( ∇ u 2 ) x k = ( ∇ z ˇ 2 ) x k − [ a 22 ( x , t , U ) ] x k ∇ u 2 − ∫ u 2 ∗ ∗ u 2 ∇ a 22 ( x , t , u 1 , ω ) d ω x k a 22 ( x , t , U ) ( ( x , t ) ∈ D 18 , T ) .

These, together with (4.9), leads to estimate (4.3).□

Furthermore, estimate (4.3) and Sobolev embedding theorem yield

(4.10) ‖ u 2 x i ‖ C α 7 , α 7 / 2 ( D ¯ 18 , T ) ≤ C , α 7 ∈ ( 0 , 1 ) , i = 1 , … , n .

Lemma 4.3

There exist positive constants C , α 8 such that

(4.11) ‖ u 2 ‖ C 2 + α 8 , 1 + α 8 / 2 ( D ¯ 19 , T ) ≤ C ,

where C , α 8 depend on ‖ b 1 ( x , t , W , p ) ‖ C 1 ( Ξ 5 × [ − ϒ 1 , ϒ 1 ] n ) and ‖ b 2 ( x , t , W , P ) ‖ C 1 ( Ξ 5 × [ − ϒ 1 , ϒ 1 ] n × [ − ϒ 2 , ϒ 2 ] n ) , Ξ 5 ≔ D ¯ 18 , T × ( S 1 ∩ [ − M 1 , M 1 ] ) × ( S 2 ∩ [ − M 2 , M 2 ] ) .

Proof

Let w ˘ 2 = u 2 λ 19 2 . Then a direct computation shows that w ˘ 2 ∈ W q 2 , 1 ( D 18 , T ) satisfies

w ˘ 2 t = a ˘ 2 ( x , t ) Δ w ˘ 2 + f ˘ 2 ( x , t ) ( ( x , t ) ∈ D 18 , T ) , w ˘ 2 ( x , t ) = 0 ( x ∈ ∂ Ω 18 , t ∈ [ t 18 , T ] ) , w ˘ 2 ( x , t 18 ) = 0 ( x ∈ Ω 18 ) ,

where

a ˘ 2 ( x , t ) = a 22 ( x , t , U ) , f ˘ 2 ( x , t ) = { ∇ a 22 ( x , t , U ) ⋅ ∇ u 2 + b 2 ( x , t , U , ∇ U ) + div [ a 21 ( x , t , U ) ∇ u 1 ] } λ 19 2 − a 22 ( x , t , U ) [ 2 ∇ u 2 ⋅ ∇ ( λ 19 2 ) + u 2 Δ ( λ 19 2 ) ] + u 2 ( λ 19 2 ) t ,

By using (4.1), (4.10), and hypothesis ( H 2 ), we find that a ˘ 2 ( x , t ) and f ˘ 2 ( x , t ) are Hölder continuous in D ¯ 18 , T . Again by [12, Chapter III, Theorem 12.2], we conclude that w ˘ 2 is in C 2 + α ˜ , 1 + α ˜ / 2 ( D ¯ 18 , T ) and u 2 satisfies estimate (4.11).□

Proof of Theorem 2.2

Lemmas 4.1 and 4.3 show that for any Ω ′ ⊂ ⊂ Ω and t ′ ∈ ( 0 , T ) , u 1 , u 2 ∈ C 2 + α ′ , 1 + α ′ / 2 ( Ω ¯ ′ × [ t ′ , T ] ) for some α ′ ∈ ( 0 , 1 ) .□

5 Applications

In this section, we give some applications to the regularity of nonnegative weak solutions for two ecological models with cross-diffusion.

5.1 Regularity of bounded nonnegative weak solutions for (1.0)

Consider system (1.0). It is a special case of (1.1) with

a 11 ( x , t , [ U ] 1 ) = κ 1 ( x , t ) + 2 γ 11 ( x , t ) u 1 , a 21 ( x , t , [ U ] 2 ) = γ 21 ( x , t ) u 2 , a 22 ( x , t , [ U ] 2 ) = κ 2 ( x , t ) + γ 21 ( x , t ) u 1 + 2 γ 22 ( x , t ) u 2 ,

and

b 1 ( x , t , U , ∇ u 1 ) = ( ∇ κ 1 + 2 u 1 ∇ γ 11 ) ⋅ ∇ u 1 + e 1 ∇ φ ⋅ ∇ u 1 + u 1 Δ κ 1 + u 1 2 Δ γ 11 + u 1 div ( e 1 ∇ φ ) + [ d 11 ( x , t ) + d 12 ( x , t ) u 1 + d 13 ( x , t ) u 2 ] u 1 , b 2 ( x , t , U , ∇ [ U ] 2 ) = ( ∇ κ 2 + u 1 ∇ γ 21 + 2 u 2 ∇ γ 22 ) ⋅ ∇ u 2 + u 2 ∇ γ 21 ⋅ ∇ u 1 + e 2 ∇ φ ⋅ ∇ u 2 + u 2 Δ κ 2 + u 1 u 2 Δ γ 21 + u 2 2 Δ γ 22 + u 2 div ( e 2 ∇ φ ) + [ d 21 ( x , t ) + d 22 ( x , t ) u 1 + d 23 ( x , t ) u 2 ] u 2 .

Corollary 5.1

Assume that functions κ i , γ 2 i , and γ 11 ( i = 1 , 2 ) belong to C 1 + α ( D T ) for some α ∈ ( 0 , 1 ) , and that functions κ i x k x k , γ 2 i x k x k , γ 11 x k x k , div ( e i ∇ φ ) , and d i j ( i = 1 , 2 ; j = 1 , 2 , 3 ; k = 1 , … , n ) belong to C 1 ( D T ) . Let U = ( u 1 , u 2 ) be a bounded nonnegative weak solution of (1.0). Then for any Ω ′ ⊂ ⊂ Ω and t ′ ∈ ( 0 , T ) , there exists α ′ ∈ ( 0 , 1 ) , such that u 1 , u 2 belong to C 2 + α ′ , 1 + α ′ / 2 ( Ω ¯ ′ × [ t ′ , T ] ) .

Proof

Since U is a bounded nonnegative weak solution of (1.0), then S = [ ess inf D T u 1 , ess sup D T u 1 ] × [ ess inf D T u 2 , ess sup D T u 2 ] ⊆ [ 0 , ess sup D T u 1 ] × [ 0 , ess sup D T u 2 ] . By a direct computation, we find that hypotheses ( H 1 ) and ( H 2 ) all hold. Then employing Theorem 2.2, we obtain this corollary.□

5.2 Regularity of nonnegative weak solutions for a predator–prey model

Consider a predator–prey model with two-species and with cross-diffusion. Let u 1 and u 2 be the population densities of predator and prey, respectively. Assume that the prey exhibits group defense and the cross-diffusion pressure of the predator is zero. If the mankind’s influence is taken into account, function U = ( u 1 , u 2 ) satisfies triangular parabolic system

(5.1) u 1 t = Δ [ κ 1 ( x , t ) u 1 ] + div [ e 1 ( x , t ) u 1 ∇ φ ] + K 1 u 1 u 2 ϱ 1 + u 2 2 − d 11 ( x , t ) u 1 + d 12 ( x , t ) u 1 1 + m ̲ n ϱ 2 + u 1 ( ( x , t ) ∈ D T ) , u 2 t = Δ [ κ 2 ( x , t ) u 2 + γ 21 u 1 u 2 ϱ 0 + u 2 ] + div [ e 2 ( x , t ) u 2 ∇ φ ] + d 21 ( x , t ) u 2 − K 2 u 1 u 2 ϱ 1 + u 2 2 + d 22 ( x , t ) u 2 1 + m ̲ n ϱ 2 + u 2 ( ( x , t ) ∈ D T ) ,

where κ 1 , κ 2 > 0 are the diffusion rates, and φ is a known outer potential, d 11 ( x , t ) > 0 is the death rate of the predator, d 21 ( x , t ) > 0 is the growth rate of the prey, and γ 21 , K 1 , K 2 , ϱ 0 , ϱ 1 , ϱ 2 are positive constants, and where γ 21 u 1 u 2 / ( ϱ 0 + u 2 ) represents the cross-diffusion pressures of the prey, u 2 / ( ϱ 1 + u 2 2 ) represents predator–prey interaction when the prey exhibits group defense (see [7]), and d 12 ( x , t ) u 1 1 + m ̲ n / ( ϱ 2 + u 1 ) and d 22 ( x , t ) u 2 1 + m ̲ n / ( ϱ 2 + u 2 ) represent the mankind’s influence.

Corollary 5.2

Let n be in { 1 , 2 , 3 } , and let m ̲ 1 = 2 , m ̲ 2 = 9 / 5 , and m ̲ 3 = 6 / 5 . Assume that functions κ 1 and κ 2 belong to C 1 + α ( D T ) for some α ∈ ( 0 , 1 ) , and that functions κ i x k x k , div ( e i ∇ φ ) and d i j ( i , j = 1 , 2 ; k = 1 , … , n ) belong to C 1 ( D T ) . Let U = ( u 1 , u 2 ) be a nonnegative weak solution of (5.1). Then for any Ω ′ ⊂ ⊂ Ω and t ′ ∈ ( 0 , T ) , there exists α ′ ∈ ( 0 , 1 ) , such that u 1 , u 2 belong to C 2 + α ′ , 1 + α ′ / 2 ( Ω ¯ ′ × [ t ′ , T ] ) .

Proof

System (5.1) is a special case of (1.1) with

a 11 ( x , t , [ U ] 1 ) = κ 1 ( x , t ) , a 21 ( x , t , [ U ] 2 ) = γ 21 u 2 ϱ 0 + u 2 , a 22 ( x , t , [ U ] 2 ) = κ 2 ( x , t ) + γ 21 ϱ 0 u 1 ( ϱ 0 + u 2 ) 2

and

b 1 ( x , t , U , ∇ u 1 ) = ∇ κ 1 ⋅ ∇ u 1 + e 1 ∇ φ ⋅ ∇ u 1 + u 1 Δ κ 1 + u 1 div ( e 1 ∇ φ ) + K 1 u 1 u 2 ϱ 1 + u 2 2 − d 11 ( x , t ) u 1 + d 12 ( x , t ) u 1 1 + m ̲ n ϱ 2 + u 1 , b 2 ( x , t , U , ∇ U ) = ∇ κ 2 ⋅ ∇ u 2 + e 2 ∇ φ ⋅ ∇ u 2 + u 2 Δ κ 2 + u 2 div ( e 2 ∇ φ ) + d 21 ( x , t ) u 2 − K 2 u 1 u 2 ϱ 1 + u 2 2 + d 22 ( x , t ) u 2 1 + m ̲ n ϱ 2 + u 2 .

Since U is a nonnegative weak solution of (5.1), then S = [ ess inf D T u 1 , + ∞ ) × [ ess inf D T u 2 , + ∞ ) ⊆ [ 0 , + ∞ ) × [ 0 , + ∞ ) . We can take

q 0 = 6 , q 1 = 2 , χ 1 = 1 / 4 , for n = 1 , q 0 = 4 , q 1 = 20 / 9 , χ 1 = 1 / 10 , for n = 2 , q 0 = 10 / 3 , q 1 = 25 / 9 , χ 1 = 1 / 10 , for n = 3 .

Then hypotheses ( H 1 ) and ( H 2 ) are satisfied with

δ l = 0 , σ l l = m ̲ n , σ 12 = 0 , σ ˜ 1 r = 0 , ϕ l = ϕ ˜ l r ( x , t ) = 0 , r = 1 , … , 4 ; l = 1 , 2 .

By Theorem 2.2, we obtain the result of the corollary.□

Acknowledgements

The author would like to thank the anonymous reviewers very much for their helpful suggestions and comments. They have contributed to the improvement of the author’s work.

  1. Funding information: The work was supported by the research fund of Chengdu Normal University (No. CS19ZA09).

  2. Conflict of interest: The author states no conflict of interest.

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Received: 2022-01-12
Revised: 2022-06-18
Accepted: 2022-06-20
Published Online: 2022-07-27

© 2022 Qi-Jian Tan, published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/ans-2022-0015
Keywords for this article
strongly coupled parabolic systems; regularity; weak solutions; triangular systems
Creative Commons
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Articles in the same Issue

  1. Research Articles
  2. Editorial
  3. A sharp global estimate and an overdetermined problem for Monge-Ampère type equations
  4. Non-degeneracy of bubble solutions for higher order prescribed curvature problem
  5. On fractional logarithmic Schrödinger equations
  6. Large solutions of a class of degenerate equations associated with infinity Laplacian
  7. Chemotaxis-Stokes interaction with very weak diffusion enhancement: Blow-up exclusion via detection of absorption-induced entropy structures involving multiplicative couplings
  8. Asymptotic mean-value formulas for solutions of general second-order elliptic equations
  9. Weighted critical exponents of Sobolev-type embeddings for radial functions
  10. Existence and asymptotic behavior of solitary waves for a weakly coupled Schrödinger system
  11. On the Lq-reflector problem in ℝn with non-Euclidean norm
  12. Existence of normalized solutions for the coupled elliptic system with quadratic nonlinearity
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  15. Multiple solutions to multi-critical Schrödinger equations
  16. Existence of solutions to contact mean-field games of first order
  17. The regularity of weak solutions for certain n-dimensional strongly coupled parabolic systems
  18. Uniform stabilization for a strongly coupled semilinear/linear system
  19. Existence of nontrivial solutions for critical Kirchhoff-Poisson systems in the Heisenberg group
  20. Existence of ground state solutions for critical fractional Choquard equations involving periodic magnetic field
  21. Least energy sign-changing solutions for Schrödinger-Poisson systems with potential well
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  23. Global well-posedness analysis for the nonlinear extensible beam equations in a class of modified Woinowsky-Krieger models
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  33. Existence of two solutions for singular Φ-Laplacian problems
  34. Existence and multiplicity results for first-order Stieltjes differential equations
  35. Concentration-compactness principle associated with Adams' inequality in Lorentz-Sobolev space
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