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System of degenerate parabolic p-Laplacian

  • Sunghoon Kim EMAIL logo and Ki-Ahm Lee
Published/Copyright: November 25, 2024
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Open Mathematics
From the journal Open Mathematics Volume 22 Issue 1

Abstract

In this article, we study the mathematical properties of the solution u = ( u 1 , … , u k ) to the degenerate parabolic system

u t = ∇ ⋅ ( ∣ ∇ u ∣ p − 2 ∇ u ) , ( p > 2 ) .

More precisely, we show the existence and uniqueness of solution u and investigate a priori L ∞ boundedness of the gradient of the solution. Assuming that the solution decays quickly at infinity, we also prove that the component u l , ( 1 ≤ l ≤ k ) , converges to the function c l ℬ in space as t → ∞ . Here, the function ℬ is the fundamental or Barenblatt solution of p -Laplacian equation, and the constant c l is determined by the L 1 -mass of u l . The proof is based on the existence of entropy functional. As an application of the asymptotic large-time behaviour, we establish a Harnack-type inequality, which makes the size of the spatial average controlled by the value of the solution at one point.

Keywords: degenerate parabolic system; entropy approach; asymptotic behaviour; Harnack-type inequality
MSC 2010: 35K40; 35K45; 35K65; 35K67

1 Introduction and main results

We consider the solution u = ( u 1 , … , u k ) to the Cauchy problem for the p -Laplacian-type parabolic system

(SPL) ( u l ) t = ∇ ( ∣ ∇ u ∣ p − 2 ∇ u l ) , in  R n × ( 0 , ∞ ) , u l ( x , 0 ) = u 0 l ( x ) , ∀ x ∈ R n , 1 ≤ l ≤ k ,

in the range of p > 2 , with initial data u 0 l nonnegative, integrable, and compactly supported.

This type of system has been studied by many authors because of mathematical interest and wide applications. Local boundedness and local C 1 , α estimates of solution were studied by DiBenedetto [1]. It also arose in geometric application [2], quasiregular mappings [3], and fluid dynamics [4]. In particular, in the mid-1960s, Ladyzhenskaya [5] suggested the aforementioned system as a model of non-Newtonian fluids, which is formulated by a series of equations having a stress tensor determined by the symmetric part of the gradient of the velocity:

v t = ∇ ⋅ A ( D ( v ) ) + ∇ p = ∇ ( v ⊗ v ) − f , ∇ v = 0 ,

where v = ( v 1 , v 2 , v 3 ) denotes the velocity, D ( v ) is the symmetric part of ∇ v , p is the pressure, and A is a monotone vector field.

When k = 1 , the system (SPL) is usually called the parabolic p -Laplacian equation, which is well known in a series of models of gradient-dependent diffusion equations. A large number of studies on the mathematical theory of the parabolic p -Laplacian equation can be found. We refer the readers to the articles [6,7] for asymptotic behaviour and articles [1,8] for regularity theory.

Another well-known model equation for nonlinear diffusion, which is deeply related to the p -Laplacian-type equation, is the porous medium equations (PMEs)

u t = △ u m = m ∇ ⋅ ( u m − 1 ∇ u ) , m > 1 .

The systematic study of the PME can treat many mathematical properties of p -Laplacian equation. For example, the combination of symmetrization, scaling properties, and time discretization can be useful to obtain suitable estimates in the theory of p -Laplacian. We also considered a few problems of porous medium-type system, which were designed to obtain suitable techniques for studying degenerate parabolic p -Laplacian system (SPL). In [9], we investigated the local continuity and asymptotic behaviour of the degenerate parabolic system

( u l ) t = m ∇ ⋅ ∑ l = 1 k u l m − 1 ∇ u l , m > 1 , 1 ≤ l ≤ k ,

which describes the population densities of k -species in a closed system whose diffusion is determined by their total population density. In [10], we studied the mathematical properties of the solution u = ( u 1 , … , u k ) of the degenerate parabolic system

(1.1) ( u l ) t = m ∇ ⋅ ( ∣ u ∣ m − 1 ∇ u l ) , m > 1 , 1 ≤ l ≤ k .

Two different methods were considered for the analysis of asymptotic behaviour of (1.1) and, as a consequence of asymptotic analysis, we established a Harnack-type inequality of solution. We also found one-directional traveling wave-type solutions of (1.1) in [10].

As the first result of this article, we will give the existence and uniqueness of the solution to the parabolic system (SPL). Recently, many approaches to the existence of solutions rely on obtaining suitable regularities, which are available in the data. Following such ideas, we first focus on a priori L ∞ boundedness estimates of diffusion coefficients of the parabolic system (SPL) in the proof of the existence of a solution. The statement of our first result is as follows.

Theorem 1.1

(Existence of solution u and uniform L ∞ boundedness of ∣ ∇ u ∣ ) Let n ≥ 2 and p > 2 . Suppose that u 0 ∈ L 2 ( R n ) ∩ W 1 , p ( R n ) . Then, there exists a weak solution u = ( u 1 , … , u k ) of (SPL). Moreover, there exist a constant C > 0 such that

(1.2) sup x ∈ R n , t ≥ T ∣ ∇ u ∣ ( x , t ) ≤ C ∥ ∇ u 0 ∥ L p T n + 1 n ( p − 2 ) + p .

In general, solutions of parabolic systems continuously lose their information from the initial data as time goes by and evolve only under the laws determined by the system. Therefore, only the diffusion coefficients and external forcing terms will play important roles in the evolution of the solution after a long time. Under this observation, we can expect that each component of the solution u of parabolic system (SPL) satisfies that

(1.3) u l ( x , t ) ≈ c l f ( x , t ) , for sufficiently large  t ,

for some constants c 1 , … , c k and function f . Then, by (SPL) and (1.3) f will satisfy

(1.4) f t ≈ ∣ c ∣ p − 2 ∇ ⋅ ( ∣ ∇ f ∣ p − 2 ∇ f ) , for sufficiently large  t ,

where c = ( c 1 , … , c k ) .

Denote by ℬ M the fundamental solution of p -Laplacian equation, i.e., the function ℬ M is a solution of

g t = div ( ∣ ∇ g ∣ p − 2 ∇ g ) ( p > 1 ) ,

in a weak sense in R n × ( 0 , ∞ ) , and satisfies

ℬ M ( x , 0 ) = M δ ( x ) ( M > 0 ) .

Then, it is expressed by the following self-similar form:

(1.5) ℬ M ( x , t ) = t − a 1 C M − ( p − 2 ) a 2 1 p − 1 p ∣ x ∣ t a 2 p p − 1 + p − 1 p − 2 ,

where the constant C M is determined by L 1 -mass M and the constants a 1 and a 2 are given by

(1.6) a 1 = n ( p − 2 ) n + p and a 2 = 1 ( p − 2 ) n + p .

Since it is well known by [6,7,11] that the solution of PMEs converges to the fundamental profile ℬ M as t → ∞ , it is natural to expect that

u l ≈ c l f → c l ℬ M , as  t → ∞ .

As the second result of our article, we studied the asymptotic large-time behaviour of component u l , ( 1 ≤ l ≤ k ) , through the method of entropy approach. The statement is as follows.

Theorem 1.2

(Asymptotic large time behaviour with entropy approach) Let n ≥ 2 , p > 2 and let u = ( u 1 , … , u k ) be a solution of (SPL) with initial data u 0 = ( u 0 1 , … , u 0 k ) such that

∫ R n 1 + ∣ u 0 ∣ p − 2 p − 1 + ∣ x ∣ p p − 1 ∣ u 0 ∣ d x < ∞ .

Then,

(1.7) lim t → ∞ u l ( ⋅ , t ) − M l ∣ M ∣ B ∣ M ∣ ( ⋅ , t ) L 1 ( R n ) = 0 , 1 ≤ l ≤ k ,

where, ∥ u 0 l ∥ L 1 ( R n ) = M l for l = 1 , … , k and M = ( M 1 , … , M k ) . More precisely, there exists a constant C l > 0 such that

u l ( ⋅ , t ) − M l ∣ M ∣ B ∣ M ∣ ( ⋅ , t ) L 1 ( R n ) ≤ C l t a 2 − a 2 2 , 1 ≤ l ≤ k ,

where the constant a 2 is given by (1.6).

The entropic approach for the degenerate parabolic systems can be found in [12].

The Harnack-type inequality plays an important role in the mathematical theories of parabolic differential equations. It is used to obtain the upper bounds of a solution of the generalized PME [13]. It is also the core of investigating the local behaviour of the solution to nonlinear parabolic equation (see [14] for the local structure of free boundaries and [15] for the boundary behaviour of solutions). Hence, the existence of the system version of Harnack-type inequality will be very helpful in studying a variety of mathematical theories on the system.

As a generalization, we will consider a suitable Harnack-type inequality for the component u l of a continuous weak solution u = ( u 1 , … , u k ) of (SPL) in the last part of our study. One interesting thing in our result is that the size of the spatial average of u l is controlled by the value of it at one point. The result is stated as follows.

Theorem 1.3

(Harnack-type inequality) Let p > 2 , T > 0 and let u = ( u 1 , … , u k ) be a continuous weak solution of

(1.8) ( u l ) t = ∇ ( ∣ ∇ u ∣ p − 2 ∇ u l ) , i n R n × [ 0 , T ] ,

with the initial data u 0 whose components are non-negative, continuous, and compactly supported. Suppose that there exists an uniform constant μ 0 > 0 such that

min 1 ≤ j ≤ k ∫ R n u 0 j ( x ) d x max 1 ≤ j ≤ k ∫ R n u 0 j ( x ) d x ≥ μ 0 > 0 .

Then, for R > T 1 p > 0 , there exists a constant C ∗ = C ∗ ( m , n ) > 0 such that

(1.9) ∫ { ∣ x ∣ < R } u l ( x , 0 ) d x ≤ C ∗ ( μ l ) 1 + n ( p − 2 ) p R n + p p − 2 T 1 p − 2 + T n p ( u l ) 1 + n ( p − 2 ) p ( 0 , T ) , ∀ 1 ≤ l ≤ k ,

where

μ l = ∫ R n u 0 l ( x ) d x max 1 ≤ j ≤ k ∫ R n u 0 j ( x ) d x , ∀ 1 ≤ l ≤ k .

The result of Theorem 1.3 can be used to construct an initial trace of the non-negative continuous weak solution u , which belongs to a specific growth class.

Corollary 1.4

(cf. Theorem 4.1 of [16]) Let p > 2 , T > 0 , and let u = ( u 1 , … , u k ) be a continuous weak solution of

( u l ) t = ∇ ( ∣ ∇ u ∣ p − 2 ∇ u l ) , in  R n × [ 0 , T ] .

Under the hypothesis of Theorem 1.3, there exists a vector ρ = ( ρ 1 , … , ρ k ) of nonnegative Borel measures on R n such that

(1.10) lim t → 0 ∫ R n u l ( x , t ) φ ( x ) d x = ∫ R n φ ( x ) ρ l ( d x ) , ∀ 1 ≤ l ≤ k ,

for all φ ∈ C 0 ( R n ) . Moreover, there exists a constant C = C ( n , m ) > 0 such that

(1.11) ∫ { ∣ x ∣ < R } ρ l ( d x ) < C R n + p p − 2 T 1 p − 2 + T n p ( u l ) 1 + n ( p − 2 ) p ( 0 , T ) , ∀ 1 ≤ l ≤ k ,

for all R > 0 .

We end up this section by introducing the concept of weak solution. Let E be an open set in R n , and for T > 0 , let E T denote the parabolic domain E × ( 0 , T ] . We say that u = ( u 1 , … , u k ) is a weak solution of (SPL) in E T if the component u l , ( 1 ≤ l ≤ k ) , is a locally integrable function satisfying

  1. u belongs to the function space:

    u ∈ C ( 0 , T : L loc 2 ( E ) ) ∩ L p ( 0 , T : W loc 1 , p ( E ) ) .

  2. u l satisfies the identity:

    (1.12) ∫ 0 T ∫ E { ∣ ∇ u ∣ p − 2 ∇ u l ⋅ ∇ φ − u l φ t } d x d t = ∫ E u 0 ( x ) φ ( x , 0 ) d x

    holds for any test function φ ∈ W 1 , 2 ( 0 , T : L 2 ( E ) ) ∩ L 2 ( 0 , T : W 0 1 , p ( E ) ) .

To control the regularity ( u l ) t , we consider the Lebesgue-Steklov average ( u l ) h of the function ( u l ) , which is introduced in [1]:

( u l ) h ( ⋅ , t ) = 1 h ∫ t t + h u l ( ⋅ , τ ) d τ , ( h > 0 ) .

Then, ( u l ) h is well defined, and it converges to u l as h → 0 in L p for all p ≥ 1 . In addition, it is differentiable in time for all h > 0 and its derivative is

u l ( t + h ) − u l ( t ) h .

Fix t ∈ ( 0 , T ) , and let h be a small positive number such that 0 < t < t + h < T . Then, for every compact subset K ⊂ R n , the following formulation is equivalent to (1.12):

(1.13) ∫ K × { t } [ ( ( u l ) h ) t φ + ( ∣ ∇ u ∣ p − 2 ∇ u l ) h ⋅ ∇ φ ] d x = 0 , ∀ 0 < t < T − h ,

for any φ ∈ H 0 1 ( K ) . From now on, we use the limit of (1.13) as h → 0 for the weak formulation (1.12) of (SPL).

A brief outline of this article is as follows. In Section 2, we will give preliminary results such as the law of L 1 -mass conservation, uniqueness, and Hölder estimates of the solution u . Section 3 is devoted to the existence of a solution whose gradient is bounded globally in L ∞ . Section 4 deals with the asymptotic large-time behaviour of solution with the method called entropy approach (Theorem 1.2). Finally, we consider a suitable Harnack-type inequality of continuous weak solution of (SPL). In the last section, we show that the size of the spatial average of the solution is controlled by the value of the solution at one point.

2 Preliminary results

In this section, we establish and prove various preliminary results that will play important roles throughout the coming section. The first preliminary result is the uniqueness of the system (SPL).

Lemma 2.1

(Uniqueness) Let p > 2 , n ≥ 2 and let u 1 = ( u 1 1 , … , u 1 k ) and u 2 = ( u 2 1 , … , u 2 k ) be two solutions of (SPL) such that

u 1 l , u 2 l ∈ L p ( 0 , T : W 0 1 , p ( R n ) ) , ∀ 1 ≤ l ≤ k .

Suppose that

(2.1) u 1 l ( x , 0 ) = u 2 l ( x , 0 ) , ∀ x ∈ R n , 1 ≤ l ≤ k .

Then, for any T > 0 ,

(2.2) u 1 ( x , t ) = u 2 ( x , t ) , ∀ ( x , t ) ∈ R n × [ 0 , T ) .

Proof

Consider the weak formulation (1.12) for u 1 l − u 2 l . Taking u 1 l − u 2 l as a test function and summing it over l = 1 , … , k , we can have

(2.3) 1 2 ∑ i = 1 k sup 0 ≤ t ≤ T ∫ R n ( u 1 l − u 2 l ) 2 ( x , t ) d x + ∫ 0 T ∫ R n ( ∣ u ¯ 1 ∣ p − 2 u ¯ 1 − ∣ u ¯ 2 ∣ p − 2 u ¯ 2 ) ⋅ ( u ¯ 1 − u ¯ 2 ) d x d t ≤ ∑ i = 1 k ∫ R n ( u ¯ 0 l − u 0 l ) 2 ( x ) d x ,

where u ¯ 1 and u ¯ 2 are two vectors such that

u ¯ 1 = ( ∇ u 1 1 , … , ∇ u 1 k ) and u ¯ 2 = ( ∇ u 2 1 , … , ∇ u 2 k ) .

Suppose that both ∣ u ¯ 1 ∣ and ∣ u ¯ 2 ∣ are nonzero. Then, by the theory of vector calculus, we can have

( ∣ u ¯ 1 ∣ p − 2 u ¯ 1 − ∣ u ¯ 2 ∣ p − 2 u ¯ 2 ) ⋅ ( u ¯ 1 − u ¯ 2 ) = ( ∣ u ¯ 1 ∣ p − 2 u ¯ 1 + ∣ − u ¯ 2 ∣ p − 2 ( − u ¯ 2 ) ) ⋅ ( u ¯ 1 + ( − u ¯ 2 ) ) ≥ ∣ ∣ u ¯ 1 ∣ p − 2 u ¯ 1 + ∣ − u ¯ 2 ∣ p − 2 ( − u ¯ 2 ) ∣ ∣ u ¯ 1 + ( − u ¯ 2 ) ∣ cos α 2 ≥ 0 ,

where α is the angle between u ¯ 1 and − u ¯ 2 , which is less than π . Thus,

(2.4) ∫ 0 T ∫ R n ( ∣ u ¯ 1 ∣ p − 2 u ¯ 1 − ∣ u ¯ 2 ∣ p − 2 u ¯ 2 ) ⋅ ( u ¯ 1 − u ¯ 2 ) d x d t ≥ ∫ 0 T ∫ R n ∩ { ∣ u ¯ 1 ∣ = 0 } ∣ u ¯ 2 ∣ p d x d t + ∫ 0 T ∫ R n ∩ { ∣ u ¯ 2 ∣ = 0 } ∣ u ¯ 1 ∣ p d x d t + ∫ 0 T ∫ R n ∩ { ∣ u ¯ 1 ∣ > 0 } ∩ { ∣ u ¯ 2 ∣ > 0 } ∣ ∣ u ¯ 1 ∣ p − 2 u ¯ 1 + ∣ − u ¯ 2 ∣ p − 2 ( − u ¯ 2 ) ∣ ∣ u ¯ 1 + ( − u ¯ 2 ) ∣ cos α 2 d x d t ≥ 0 .

By (2.3), (2.4), and initial condition, we have

(2.5) ∑ l = 1 k sup 0 ≤ t ≤ T ∫ R n ( u ¯ l − u l ) 2 ( x , t ) d x ≤ ∑ l = 1 k ∫ R n ( u ¯ 0 l − u 0 l ) 2 ( x ) d x = 0 ,

and the lemma follows.□

The second preliminary result is the law of L 1 mass conservation on (SPL). In the mathematical theory of standard parabolic PDE, the concept of mass conservation plays an important role in the study of asymptotic large-time behaviour of solutions.

Lemma 2.2

(Law of L 1 mass conservation) Let n ≥ 2 , p > 2 and let u 0 ∈ L 2 ( R n ) . Let u = ( u 1 , … , u k ) be a weak solution of (SPL) in R n × ( 0 , ∞ ) . Then, for any t > 0 ,

(2.6) ∫ R n u l ( x , t ) d x = ∫ R n u 0 l ( x ) d x .

Proof

Let T > 0 . By weak formulation of (SPL), we can obtain

(2.7) 1 2 sup 0 < t < T ∫ R n ( u l ( x , t ) ) 2 d x + ∫ 0 T ∫ R n ∣ ∇ u ∣ p − 2 ∣ ∇ u l ∣ 2 d x d t ≤ ∫ R n ( u 0 l ) 2 d x , ∀ 1 ≤ l ≤ k ⇒ ∫ 0 T ∫ R n ∣ ∇ u ∣ p d x d t ≤ C , for some constant C depending on u 0 .

Let ζ j ⊂ W 1 , p ( R n ) be a sequence of cut-off functions such that

ζ 0 ( x ) = 1 , for  ∣ x ∣ ≤ j , ζ 0 ( x ) = 0 , for  ∣ x ∣ ≥ 2 j , 0 < ζ j ( x ) < 1 , for  j < ∣ x ∣ < 2 j

and ∥ ∇ ζ j ∥ L p ( R n ) being uniformly bounded. Then, by (2.7), we have

∫ R n u l ( x , T ) ζ j ( x ) d x − ∫ R n u 0 l ( x ) ζ j ( x ) d x = ∫ 0 T ∫ R n ( u l ) t ζ j d x d t = − ∫ 0 T ∫ R n w p − 2 ∇ u l ⋅ ∇ ζ j d x d t ≤ T 1 p ∥ ∇ ζ j ∥ L p ( B 2 j \ B j ) ∫ 0 T ∫ B 2 j \ B j w p d x d t p − 1 p ≤ T 1 p ∥ ∇ ζ j ∥ L p ( R n ) ∫ 0 T ∫ B 2 j \ B j w p d x d t p − 1 p → 0 , as  j → ∞ .

where w = ∣ ∇ u ∣ . Thus, (2.6) holds, and the lemma follows.□

We are wrapping up this section by introducing the local continuity of the weak solution of (SPL), especially the Hölder estimates of the solution u and of the gradient of the solution ∇ u . The proofs of the following estimates can be found in [17] (Theorem 1) and [1] (Theorem 1.1 of Chap. IX).

Lemma 2.3

(Local Hölder estimates) Let p > 2 . Then, any weak solution u of (SPL) is locally Hölder continuous in R n × ( 0 , T ] . Moreover, if ∥ ∇ u ∥ L ∞ ( R n × ( 0 , T ] ) < ∞ , then the function u x j l is also locally Hölder continuous in R n × ( 0 , T ] for all 1 ≤ l ≤ k and 1 ≤ j ≤ n .

3 Uniform boundedness of the diffusion coefficients ∣ ∇ u ∣

The evolution of solutions will be influenced mostly by the diffusion coefficients over a large time. Therefore, it is very important to have suitable regularity estimates for the diffusion coefficients. This section is devoted to providing the existence of solution u and a priori L ∞ boundedness of diffusion coefficients ∣ ∇ u ∣ . The proof of L ∞ boundedness is based on a recurrence relation between a series of truncations of ∣ ∇ u ∣ . An energy-type inequality and an embedding play important roles in the proof.

We start this section by stating the well-known inequality.

Lemma 3.1

(cf. Proposition 3.1 of Chap. I of [1]) Let m 1 , m 2 ≥ 1 , and let 0 ≤ t < T . There exists a constant C > 0 depending on n and p such that for every v ∈ L ∞ ( t , T ; L m 1 ( R n ) ) ∩ L m 2 ( t , T ; W 0 1 , m 2 ( R n ) ) ,

∫ t T ∫ R n ∣ v ∣ m 2 ( n + m 1 ) n d x d t ≤ C ∫ t T ∫ R n ∣ ∇ v ∣ m 2 d x d t sup t < τ < T ∫ R n ∣ v ( ⋅ , τ ) ∣ m 1 d x m 2 n .

We now are ready to prove Theorem 1.1.

Proof of Theorem 1.1

For each 1 ≤ l ≤ k , let { u 0 , j ^ l } ⊂ C 0 ∞ ( R n ) be a sequence of functions such that

(3.1) ∣ ∇ u 0 , j ^ l ∣ ≤ j ^ , ∥ u 0 , j ^ l ∥ L 2 ( R n ) ≤ 2 ∥ u 0 ∥ L 2 ( R n ) , ∥ u 0 , j ^ l ∥ W 1 , p ( R n ) ≤ 2 ∥ u 0 ∥ W 1 , p ( R n ) , ∀ j ^ ∈ N ,

and

u 0 , j ^ → u 0 in  L 2 ∩ W 1 , p , as  j ^ → ∞ .

By the standard theory for the uniformly parabolic equation [18], a smooth solution u j ^ = ( u j ^ 1 , … , u j ^ k ) of the system

(3.2) ( u j ^ l ) t = ∇ ⋅ ∣ ∇ u j ^ ∣ p − 2 + 1 j ^ ∇ u j ^ l , ∀ 1 ≤ l ≤ k ,

with the initial data

u j ^ l ( x , 0 ) = u 0 , j ^ l ( x ) , ∀ x ∈ R n ,

exists on a short time interval. Let ( 0 , t 0 ) be the maximal interval of existence of smooth solution and suppose that t 0 < ∞ .

Claim 1: ∣ ∇ u j ^ ∣ is L ∞ bounded at t = t 0 .

We will use a modification of the proof of Theorem 1 of [19] to prove the claim. Let

(3.3) w ¯ j ^ = ∣ ∇ u j ^ ∣ 2 = ∑ l = 1 k ∣ ∇ u j ^ l ∣ 2 = ∑ i = 1 n ∑ l = 1 k [ ( u j ^ l ) x i ] 2 = ∑ i = 1 n ∑ l = 1 k [ ( u j ^ l ) x i ] 2 .

Then, by simple computation, we can obtain

(3.4) ( w ¯ j ^ ) t = δ i j w ¯ j ^ p − 2 2 + ( p − 2 ) a ¯ j ^ i j w ¯ j ^ p − 4 2 ( w ¯ j ^ ) x i x j + 1 j ^ △ w ¯ j ^ − 2 ∑ l = 1 k ∑ i = 1 n ∣ ∇ ( u j ^ l ) x i ∣ 2 − F ¯ , ∀ 1 ≤ i , j ≤ n ,

where

a ¯ j ^ i j ( x , t ) = ∑ l = 1 k ( u j ^ l ) x i ( u j ^ l ) x j and F ¯ = 2 ∑ l = 1 k ∑ i = 1 n w ¯ j ^ p − 2 2 ∣ ∇ ( u j ^ l ) x i ∣ 2 + p − 2 2 w ¯ j ^ p − 4 2 ∣ ∇ w ¯ j ^ ∣ 2 .

The matrix

( u j ^ l ) x 1 ⋮ ( u j ^ l ) x n ( ( u j ^ l ) x 1 , … , ( u j ^ l ) x n ) = ( ∇ u j ^ l ) T ∇ u j ^ l , ∀ 1 ≤ l ≤ k ,

is symmetric and positive semi-definite, having exactly n − 1 zero eigenvalues. Moreover, the remaining eigenvalue is

λ j ^ = ∇ u j ^ l ( ∇ u j ^ l ) T = ∣ ∇ u j ^ l ∣ 2 , 1 ≤ l ≤ k .

Thus, the matrix [ a ¯ j ^ i j ] 1 ≤ i , j ≤ n is also a symmetric and positive semi-definite satisfying

(3.5) 0 ≤ a ¯ j ^ i j ξ i ξ j ≤ w ¯ j ^ ∣ ξ ∣ 2 , ∀ ξ ∈ R n .

By (3.4) and (3.5), w ¯ j ^ satisfies

(3.6) ( w ¯ j ^ ) t = a j ^ i j w ¯ j ^ p 2 x i x j + 1 j ^ △ w ¯ j ^ − 2 ∑ l = 1 k ∑ i = 1 n ∣ ∇ ( u j ^ l ) x i ∣ 2 − F ¯ , ∀ ( x , t ) ∈ R n × ( 0 , ∞ )

for the positive definite matrix [ a j ^ i j ] 1 ≤ i , j ≤ n = 2 p δ i j + ( p − 2 ) ∑ l = 1 k ( u j ^ l ) x i ( u j ^ l ) x j w ¯ j ^ 1 ≤ i , j ≤ n satisfying

(3.7) 2 p ∣ ξ ∣ 2 ≤ a j ^ i j ξ i ξ j ≤ 2 ( p − 1 ) p ∣ ξ ∣ 2 , ∀ ξ = ( ξ 1 , … , ξ n ) ∈ R n .

Let w ¯ j ^ = w j ^ 2 . Then,

(3.8) ( w j ^ ) t = p 2 ( p − 1 ) [ a j ^ i j ( w j ^ p − 1 ) x i ] x j + 1 j ^ △ w j ^ + 1 j ^ w j ^ ∣ ∇ w j ^ ∣ 2 − ∑ l = 1 k ∑ i = 1 n ∣ ∇ ( u j ^ l ) x i ∣ 2 + F ,

where

F = p 2 w j ^ p − 3 a j ^ i j ( w j ^ ) x i ( w j ^ ) x j − ∑ l = 1 k ∑ i = 1 n w j ^ p − 3 ∣ ∇ ( u j ^ l ) x i ∣ 2 − ( p − 2 ) w j ^ p − 3 ∣ ∇ w j ^ ∣ 2 .

Observe that, by the Cauchy-Schwarz inequality, we have

(3.9) ∣ ∇ w j ^ ∣ 2 − ∑ l = 1 k ∑ i = 1 n ∣ ∇ ( u j ^ ) x i l ∣ 2 ≤ 0 .

Thus,

1 j ^ w j ^ ∣ ∇ w j ^ ∣ 2 − ∑ l = 1 k ∑ i = 1 n ∣ ∇ ( u j ^ l ) x i ∣ 2 + F ≤ 0 .

For s ∈ N , let

L s = M 1 − 1 2 s and ( w j ^ ) s = ( w j ^ − L s ) + ,

for a constant M > 2 , which will be determined later. Then,

(3.10) w j ^ ≥ M 2 > 1 on { ( w j ^ ) s ≥ 0 } , ∀ j ∈ N .

Multiply (3.8) by ( w j ^ ) s p − 1 and integrate it over R n . Then, by (3.7) and (3.9), we have the following energy inequality for ( w j ^ ) s :

(3.11) 1 p ∂ ∂ t ∫ R n ( w j ^ ) s p d x + 1 p − 1 ∫ R n ∣ ∇ ( w j ^ ) s p − 1 ∣ 2 d x ≤ 0 .

Let a n = 1 n + p ( n + 1 ) n ( n ( p − 2 ) + p ) . For fixed 0 < t 1 < t 0 , let T s = t 1 1 + a n 1 − 1 2 p s and

A s = sup T s ≤ t ≤ t 0 ∫ R n ( w j ^ ) s p d x + ∫ T s t 0 ∫ R n ∣ ∇ ( w j ^ ) s p − 1 ∣ 2 d x d t .

We first choose the constant M so large that

A 1 ≤ 1 .

Then,

(3.12) A s ≤ 1 , ∀ s ∈ N .

Integrating (3.11) over ( τ , t ) and ( τ , t 0 ) , ( T s − 1 < τ < T s , T s < t < t 0 ) , we have

(3.13) A s ≤ ∫ R n ( w j ^ ) s p ( x , τ ) d x .

Taking the mean value in τ on [ T s − 1 , T s ] , we have

(3.14) A s ≤ 2 p s t 1 1 + a n ∫ T s − 1 t 0 ∫ R n ( w j ^ ) s p d x d t .

By Lemma 3.1 with v , m 1 , and m 2 being replaced by ( w j ^ ) s p − 1 , p p − 1 and 2, respectively, we can obtain

∫ T s − 1 t 0 ∫ R n ( w j ^ ) s − 1 2 p − 1 + p n d x d t n ( p − 1 ) 2 ( n p + p − n ) ≤ 1 C 1 sup T s ≤ t ≤ t 0 ∫ R n ( w j ^ ) s p d x 1 − 1 p + ∫ T s t 0 ∫ R n ∣ ∇ ( w j ^ ) s p − 1 ∣ 2 d x d t 1 2 ,

for some constant C 1 > 0 . Combining this with (3.12), we can obtain

(3.15) ∫ T s − 1 t 0 ∫ R n ( w j ^ ) s − 1 2 p − 1 + p n d x d t n ( p − 1 ) 2 ( n p + p − n ) ≤ C 2 A s − 1 1 2 ,

for some constant C 2 > 0 . Since ( w j ^ ) s − 1 ≥ M 2 s on ( w j ^ ) s ≥ 0 , we can obtain

(3.16) χ { ( w j ^ ) s ≥ 0 } ≤ 2 s M ( w j ^ ) s − 1 p − 2 + 2 p n .

By (3.14), (3.15), and (3.16),

(3.17) A s ≤ 2 2 p − 1 + p n s t 1 1 + a n M p − 2 + 2 p n ∫ T j − 1 t 0 ∫ R n ( w j ^ ) s − 1 2 p − 1 + p n d x d t ≤ C 3 4 p − 1 + p n s t 1 M p − 2 + 2 p n A s − 1 1 + p n ( p − 1 ) ,

for some constant C 3 > 0 . Let M = M t 1 n + 1 n ( p − 2 ) + p and choose the constant M > 0 so large that

(3.18) A 0 ≤ M p − 2 + 2 p n C 3 n ( p − 1 ) p 4 − p − 1 + p n p n ( p − 1 ) 2 .

Then, by Lemma 4.1 of Chapter I of [1], we have

A s → 0 , as  s → ∞ .

We note that A 0 ≤ ∥ ∇ u 0 , j ^ ∥ L p p . This gives that the constant M is determined by ∥ ∇ u 0 , j ^ ∥ L p . Therefore, for some constant C 4 > 0 , we have

(3.19) sup x ∈ R n , t 1 ≤ t ≤ t 0 ∣ ∇ u j ^ ( x , t ) ∣ ≤ C 4 ∥ ∇ u 0 , j ^ ∥ L p t 1 n + 1 n ( p − 2 ) + p ,

and claim 1 follows.

By claim 1, the equation for u j ^ l is still uniformly parabolic at t = t 0 . Thus, by the standard theory for the uniformly parabolic equation, we can extend the smoothness of u j ^ l to the time interval ( 0 , t 0 + t 2 ) for some constant t 2 > 0 . By the maximality of t 0 , a contradiction arises. Hence, t 0 = ∞ .

Let T > 0 . By simple computation, it can be easily checked that

(3.20) 1 2 sup 0 < t < T ∫ R n ( u j ^ l ( x , t ) ) 2 d x + ∫ 0 T ∫ R n ∣ ∇ u j ^ ∣ p − 2 ∣ ∇ u j ^ l ∣ 2 d x d t + 1 j ^ ∫ 0 T ∫ R n ∣ ∇ u j ^ l ∣ 2 d x d t ≤ ∫ R n ( u 0 l ) 2 d x , ∀ 1 ≤ l ≤ k .

By (3.20), { u j ^ l } , { ∇ u j ^ } and 1 j ^ ∇ u j ^ l are uniformly bounded in L 2 ( R n × ( 0 , T ) ) , W 1 , p ( R n × ( 0 , T ) ) , and L 2 ( R n × ( 0 , T ) ) . Thus, they have subsequences which we may assume without loss of generality to be the sequence themselves that converge to some functions u l , ∣ ∇ u ∣ = ∑ i = 1 k ( ∇ u l ) 2 and w l weakly in L 2 ( R n × ( 0 , T ) ) , W 1 , p ( R n × ( 0 , T ) ) , and H 1 ( R n × ( 0 , T ) ) , respectively, as j ^ → ∞ . i.e.,

(3.21) u j ^ l → u l , weakly in L 2 ( R n × ( 0 , T ) ) as j ^ → ∞ .

and

(3.22) ∇ u j ^ l → ∇ u l , ∣ ∇ u j ^ ∣ → ∣ ∇ u ∣ = ∑ i = 1 k ( ∇ u l ) 2 , weakly in L p ( R n × ( 0 , T ) ) as j ^ → ∞ ,

and

(3.23) 1 j ^ ∇ u j ^ l → w l , weakly in L 2 ( R n × ( 0 , T ) ) as j ^ → ∞ .

Let φ ∈ H 1 ( 0 , T : L 2 ( R n ) ) ∩ L 2 ( 0 , T : W 0 1 , p ( R n ) ) be a test function. Then, we can obtain

(3.24) ∫ 0 ∞ ∫ R n ∣ ∇ u j ^ ∣ p − 2 ∇ u j ^ l ⋅ ∇ φ d x d t + ∫ 0 ∞ ∫ R n 1 j ^ ∇ u j ^ l ⋅ ∇ φ d x d t − ∫ 0 ∞ ∫ R n u j ^ l φ t d x d t = 0 , ∀ 1 ≤ l ≤ k .

Letting j ^ → ∞ in (3.24), by (3.21), (3.22), and (3.23) we have

(3.25) ∫ 0 ∞ ∫ R n ∣ ∇ u ∣ p − 2 ∇ u l ⋅ ∇ φ d x d t − ∫ 0 ∞ ∫ R n u l φ t d x d t = 0 , ∀ 1 ≤ l ≤ k ,

We now are going to show that

(3.26) u l ( ⋅ , t ) → u 0 l , in L 1 as t → 0 + , ∀ 1 ≤ l ≤ k .

Let η ( x ) ∈ C 0 2 ( R n ) and let 0 < t < 1 . Then, by (3.20), we have

(3.27) ∫ R n u j ^ l ( x , t ) η ( x ) d x − ∫ R n u 0 , j ^ l ( x ) η ( x ) d x ≤ ∫ 0 t ∫ R n ∣ ∇ u j ^ ∣ p − 2 ∣ ∇ u j ^ l ( x , t ) ∣ ∣ ∇ η ( x ) ∣ d x d t + 1 j ^ ∫ 0 t ∫ R n ∣ ∇ u j ^ l ( x , t ) ∣ ∣ ∇ η ( x ) ∣ d x d t ≤ k p − 2 2 ∑ i = 1 k ∫ 0 t ∫ R n ∣ ∇ u j ^ l ( x , t ) ∣ p − 1 ∣ ∇ η ( x ) ∣ d x d t + 1 j ^ ∫ 0 t ∫ R n ∣ ∇ u j ^ l ( x , t ) ∣ ∣ ∇ η ( x ) ∣ d x d t ≤ C ( ∥ u 0 ∥ L 2 ( R n ) , ∥ ∇ η ∥ L ∞ ) t 1 p + t j ^ 1 2 , ∀ 0 < t < 1 , 1 ≤ l ≤ k .

Thus, letting j ^ → 0 and then t → 0 in (3.27), (3.26) holds.

By (3.25), (3.26), and Proposition 5.1 of [20], weak formulation of (SPL) follows. Hence, u = ( u 1 , … , u k ) is a weak solution of (SPL).

To complete the proof, we let j ^ → ∞ in (3.19). Then, inequality (1.2) holds and the theorem follows.□

Remark 3.2

With the weak differentiability of ∣ ∇ u ∣ p − 2 2 u x j l , ( 1 ≤ l ≤ k , 1 ≤ j ≤ n ) , L loc p -boundedness of ∣ ∇ u ∣ was studied by E. DiBenedetto. We refer the reader to Chapter VIII of [1] for the local boundedness of the gradient of solution of (SPL).

4 Asymptotic large time behaviour: The entropy approach

In this section, we will investigate the convergence between the solution of (SPL) and the fundamental solution of p -Laplacian equation. We use a modification of the techniques used in [20], [21], and [22] to show the convergence. For the constant a 2 given by (1.6), let

R = R ( t ) = t a 2 a 2 , ( t > 0 ) .

For any M > 0 , let

ℬ ˜ M ( η ) = C − ( p − 2 ) p ∣ η ∣ p p − 1 + p − 1 p − 2 .

Then,

(4.1) ℬ M ( x , t ) = t a 2 − a 1 ℬ ˜ M t a 2 − a 2 x = 1 R n ℬ ˜ M ( η ) ,

where the constant C is uniquely determined by the constants p , n , and initial mass M .

For a solution u = ( u 1 , … , u k ) of (SPL), let

M l = ∫ R n u 0 l d x and M = ( M 1 , … , M k ) .

For j ^ ∈ N , we chose the sequence { ( u j ^ l ) 0 } such that

∫ R n ( u j ^ l ) 0 d x = ∫ R n u 0 l d x , ∀ 1 ≤ l ≤ k ,

and let u j ^ be the smooth solution of (3.2) satisfying

(4.2) u j ^ ( x , 0 ) = ( ( u j ^ 1 ) 0 , … , ( u j ^ k ) 0 ) .

Consider the continuous rescaling

(4.3) θ l ( η , τ ) = R n u l ( x , t ) , θ ( η , τ ) = R n u ( x , t )

and

θ j ^ l ( η , τ ) = R n u j ^ l ( x , t ) , θ j ^ ( η , τ ) = R n u j ^ ( x , t ) , η = x R , τ = log t , 1 ≤ l ≤ k .

Then, θ j ^ = ( θ j ^ 1 , … , θ j ^ k ) satisfies

(4.4) ( θ j ^ l ) τ = ∇ ⋅ ( Θ j ^ p − 2 ∇ θ j ^ l ) + ∇ θ j ^ l ⋅ η + n θ j ^ l + 1 j ^ e 2 a 2 − 1 a 2 τ △ θ j ^ l = ∇ ⋅ ( Θ j ^ p − 2 ∇ θ j ^ l ) + ∇ ( η θ j ^ l ) + 1 j ^ e 2 a 2 − 1 a 2 τ △ θ j ^ l , ( Θ j ^ = ∣ ∇ θ j ^ ∣ , 1 ≤ l ≤ k )

and

( ∣ θ j ^ ∣ ) τ = ∇ ⋅ ( Θ j ^ p − 2 ∇ ∣ θ j ^ ∣ ) + ∇ ⋅ ( η ∣ θ j ^ ∣ ) + Θ j ^ p − 2 ∣ ∇ ∣ θ j ^ ∣ ∣ 2 ∣ θ j ^ ∣ − Θ j ^ p ∣ θ j ^ ∣ + 1 j ^ e 2 a 2 − 1 a 2 τ △ ∣ θ j ^ ∣ + 1 j ^ e 2 a 2 − 1 a 2 τ ∣ ∇ ∣ θ j ^ ∣ ∣ 2 ∣ θ j ^ ∣ − Θ j ^ 2 ∣ θ j ^ ∣ .

Note that, by the Cauchy-Schwarz inequality,

∣ ∇ ∣ θ j ^ ∣ ∣ ≤ Θ j ^ .

For a positive function f , we now consider the convex functional

H f ( τ ) = ∫ R n [ σ ( f ) − σ ( ℬ ˜ ∣ M ∣ ) − σ ′ ( ℬ ˜ ∣ M ∣ ) ( f − ℬ ˜ ∣ M ∣ ) ] d η

and

H ^ f ( τ ) = ∫ R n σ ( f ) − σ ( ℬ ˜ ∣ M ∣ ) + ( p − 1 ) p ∣ η ∣ p p − 1 ( f − ℬ ˜ ∣ M ∣ ) d η ,

where σ : R + → R is a function defined by

σ ( s ) = ( p − 1 ) 2 ( 2 p − 3 ) ( p − 2 ) s 2 p − 3 p − 1 .

By the second-order Taylor expansion of σ around 1, we can have

(4.5) σ ( ∣ θ j ^ ∣ ) − σ ( ℬ ˜ ∣ M ∣ ) − σ ′ ( ℬ ˜ ∣ M ∣ ) ( ∣ θ j ^ ∣ − ℬ ˜ ∣ M ∣ ) = σ ∣ θ j ^ ∣ ℬ ˜ ∣ M ∣ − σ ( 1 ) − σ ′ ( 1 ) ∣ θ j ^ ∣ ℬ ˜ ∣ M ∣ − 1 ℬ ˜ ∣ M ∣ 2 p − 3 p − 1 = 1 2 ℬ ˜ ∣ M ∣ − 1 p − 1 ∣ ∣ θ j ^ ∣ − ℬ ˜ ∣ M ∣ ∣ 2 σ ′ ′ 1 + β ∣ θ j ^ ∣ ℬ ˜ ∣ M ∣ − 1 ,

for some β ∈ ( 0 , 1 ) . Thus,

(4.6) H ∣ θ j ^ ∣ ( τ ) ≥ ∫ { ℬ ˜ ∣ M ∣ ≠ 0 } [ σ ( ∣ θ j ^ ∣ ) − σ ( ℬ ˜ ∣ M ∣ ) − σ ′ ( ℬ ˜ ∣ M ∣ ) ( ∣ θ j ^ ∣ − ℬ ˜ ∣ M ∣ ) ] d η = 1 2 ∫ { ℬ ˜ ∣ M ∣ ≠ 0 } ℬ ˜ ∣ M ∣ − 1 p − 1 ∣ ∣ θ j ^ ∣ − ℬ ˜ ∣ M ∣ ∣ 2 1 + β ∣ θ j ^ ∣ ℬ ˜ ∣ M ∣ − 1 − 1 p − 1 d η ≥ 1 2 ∫ { ∣ θ j ^ ∣ < ℬ ˜ ∣ M ∣ } ℬ ˜ ∣ M ∣ − 1 p − 1 ∣ ∣ θ j ^ ∣ − ℬ ˜ ∣ M ∣ ∣ 2 d η

(4.7) ≥ 0 .

Since

(4.8) σ ′ ( ℬ ˜ ∣ M ∣ ) = p − 1 p − 2 C ∣ M ∣ − ( p − 2 ) p ∣ η ∣ p p − 1 + and ∫ R n ∣ θ j ^ ∣ ( η , τ ) d η = ∫ R n ℬ ˜ ∣ M ∣ ( η ) d η ,

we also have

(4.9) H ∣ θ j ^ ∣ ( τ ) = ∫ R n σ ( ∣ θ j ^ ∣ ) − σ ( ℬ ˜ ∣ M ∣ ) + ( p − 1 ) p ∣ η ∣ p p − 1 ( ∣ θ j ^ ∣ − ℬ ˜ ∣ M ∣ ) d η + ∫ { ℬ ˜ ∣ M ∣ = 0 } p − 1 p − 2 C ∣ M ∣ − ( p − 2 ) p ∣ η ∣ p p − 1 ∣ θ j ^ ∣ d η ≤ ∫ R n σ ( ∣ θ j ^ ∣ ) − σ ( ℬ ˜ ∣ M ∣ ) + ( p − 1 ) p ∣ η ∣ p p − 1 ( ∣ θ j ^ ∣ − ℬ ˜ ∣ M ∣ ) d η = H ^ ∣ θ j ^ ∣ ( τ ) .

We first obtain the following variation of entropy H ^ ∣ θ ∣ .

Lemma 4.1

Let n ≥ 2 and p > 2 . Let θ = ( θ 1 , … , θ k ) be a solution of (4.4). If

(4.10) ∫ R n 1 + ∣ u 0 ∣ p − 2 p − 1 + ∣ x ∣ p p − 1 ∣ u 0 ∣ d x < ∞ ,

then

(4.11) H ^ ∣ θ ∣ ( τ ) ≤ e − τ ⋅ H ^ ∣ θ ∣ ( 0 ) , ∀ τ > 0 .

Proof

We will use a modification of the proofs of Theorem 2.2 of [21] and Proposition 1 of [22] to prove the lemma. Let

H ^ j ^ ( τ ) = H ^ ∣ θ j ^ ∣ ( τ ) .

By integration by parts and Young’s inequality, we have

(4.12) d H ^ j ^ d τ = ∫ R n σ ′ ( ∣ θ j ^ ∣ ) + ( p − 1 ) p ∣ η ∣ p p − 1 ( ∣ θ j ^ ∣ ) τ d η = − ∫ R n ∇ σ ′ ( ∣ θ j ^ ∣ ) + ( p − 1 ) p ∣ η ∣ p p − 1 ⋅ ( Θ j ^ p − 2 ∇ ∣ θ j ^ ∣ + η ∣ θ j ^ ∣ ) d η − ∫ R n Θ j ^ p − 2 ∣ θ j ^ ∣ σ ′ ( ∣ θ j ^ ∣ ) + ( p − 1 ) p ∣ η ∣ p p − 1 ( Θ j ^ 2 − ∣ ∇ ∣ θ j ^ ∣ ∣ 2 ) d η − 1 j ^ e 2 a 2 − 1 a 2 τ ∫ R n ∇ σ ′ ( ∣ θ j ^ ∣ ) + ( p − 1 ) p ∣ η ∣ p p − 1 ⋅ ∇ ∣ θ j ^ ∣ d η − 1 j ^ e 2 a 2 − 1 a 2 τ ∫ R n 1 ∣ θ j ^ ∣ σ ′ ( ∣ θ j ^ ∣ ) + ( p − 1 ) p ∣ η ∣ p p − 1 ( Θ j ^ 2 − ∣ ∇ ∣ θ j ^ ∣ ∣ 2 ) d η ≤ − ∫ R n ∣ θ j ^ ∣ − 1 p − 1 ∇ ∣ θ j ^ ∣ + ∣ η ∣ p p − 1 − 2 η ⋅ ( Θ j ^ p − 2 ∇ ∣ θ j ^ ∣ + η ∣ θ j ^ ∣ ) d η − p − 1 p − 2 ∫ R n ∣ θ j ^ ∣ − 1 p − 1 Θ j ^ p − 2 ( Θ j ^ 2 − ∣ ∇ ∣ θ j ^ ∣ ∣ 2 ) d η − p − 1 p ∫ R n ∣ η ∣ p p − 1 Θ j ^ p − 2 ∣ θ j ^ ∣ ( Θ j ^ 2 − ∣ ∇ ∣ θ j ^ ∣ ∣ 2 ) d η + 1 j ^ e 2 − 1 a 2 τ ∫ R n ∣ η ∣ 1 p − 1 ∣ ∇ ∣ θ j ^ ∣ ∣ d η = − ( I 1 + I 2 + I 3 + I 4 ) − 1 p − 2 ∫ R n ∣ θ j ^ ∣ − 1 p − 1 Θ j ^ p − 2 ( Θ j ^ 2 − ∣ ∇ ∣ θ j ^ ∣ ∣ 2 ) d η − p − 1 p ∫ R n ∣ η ∣ p p − 1 Θ j ^ p − 2 ∣ θ j ^ ∣ ( Θ j ^ 2 − ∣ ∇ ∣ θ j ^ ∣ ∣ 2 ) d η + 1 j ^ e 2 − 1 a 2 τ ∫ R n ∣ η ∣ 1 p − 1 ∣ ∇ ∣ θ j ^ ∣ ∣ d η ,

where

I 1 = ∫ R n ∣ θ j ^ ∣ − 1 p − 1 Θ j ^ p d η , I 2 = ∫ R n ∣ η ∣ p p − 1 ∣ θ j ^ ∣ d η ,

and

I 3 = p − 1 p − 2 ∫ R n ∣ θ j ^ ∣ ∇ ∣ θ j ^ ∣ p − 2 p − 1 ⋅ η d η , I 4 = ∫ R n ∣ η ∣ − ( p − 2 ) p − 1 Θ j ^ p − 2 ∇ ∣ θ j ^ ∣ ⋅ η d η .

By the Hölder inequality and condition (4.10), there exists a constant C 1 > 0 such that

(4.13) 1 j ^ e 2 − 1 a 2 τ ∫ R n ∣ η ∣ 1 p − 1 ∣ ∇ ∣ θ j ^ ∣ ∣ d η ≤ 1 j ^ e 2 − 1 a 2 τ ∫ R n ∣ η ∣ p p − 1 ∣ θ j ^ ∣ d η 1 p ∫ R n ∣ θ j ^ ∣ d η p − 2 2 p 4 j ^ ∫ R n ∇ ∣ θ j ^ ∣ 1 2 2 d η 1 2 ≤ C 1 j ^ e 2 − 1 a 2 τ .

By Young’s inequality, we have

(4.14) ∣ I 4 ∣ ≤ p − 1 p ∫ R n ∣ θ j ^ ∣ − 1 p − 1 Θ j ^ p d η + 1 p ∫ R n ∣ η ∣ p p − 1 ∣ θ j ^ ∣ d η ≤ p − 1 p I 1 + 1 p I 2 .

By (4.12), (4.13), (4.14), and the Cauchy-Schwarz inequality, we have

(4.15) d H ^ j ^ d τ ≤ − 1 p I 1 + p − 1 p I 2 + I 3 + C 1 e 1 a 2 − 2 τ j ^ ≤ − 1 p I 1 ′ + p − 1 p I 2 + I 3 + C 1 e 1 a 2 − 2 τ j ^ ,

where

I 1 ′ = ∫ R n ∣ θ j ^ ∣ − 1 p − 1 ∣ ∇ ∣ θ j ^ ∣ ∣ p d η .

On the other hand, by equation (36) in the proof of Theorem 2.2 of [21], the functional H ^ j ^ is also bounded from below by the last term of (4.15), i.e.,

(4.16) H ^ j ^ ≤ 1 p I 1 ′ + p − 1 p I 2 + I 3 .

By (4.15) and (4.16),

(4.17) ( H ^ j ^ ) τ ≤ − H ^ j ^ + C 1 e 1 a 2 − 2 τ j ^ ⇒ e τ H ^ j ^ − C 1 1 a 2 − 1 j ^ e 1 a 2 − 1 τ τ ≤ 0 , ∀ τ > 0 .

Integrating (4.17) over ( 0 , τ ) , we have

H ^ j ^ ( τ ) ≤ e − τ H ^ j ^ ( 0 ) + C 1 1 a 2 − 1 j ^ e 1 a 2 − 1 τ − 1 .

This immediately implies

H ^ ∣ θ ∣ ( τ ) ≤ liminf j ^ → ∞ H ^ j ^ ( τ ) ≤ lim j ^ → ∞ e − τ H ^ j ^ ( 0 ) + C 1 1 a 2 − 1 j ^ e 1 a 2 − 1 τ − 1 = e − τ ⋅ H ^ ∣ θ ∣ ( 0 ) .

Therefore, inequality (4.11) and the lemma follow.□

Next, we investigate the similarity between the components of θ at infinity.

Lemma 4.2

Suppose that the solution θ = ( θ 1 , … , θ k ) converges as τ → ∞ . Then, under the hypotheses of Lemma 4.1 we also have

(4.18) lim τ → ∞ θ l = lim τ → ∞ M l ∣ M ∣ ∣ θ ∣ , a.e. i n R n .

Proof

Let I 1 , I 2 , I 3 , I 4 , and I 1 ′ be given in the proof of Lemma 4.1, and let

lim τ → ∞ θ l = θ ˜ l and lim τ → ∞ θ = θ ˜ .

Let

H j ^ ( τ ) = H ∣ θ j ^ ∣ ( τ ) and H ^ j ^ ( τ ) = H ^ ∣ θ j ^ ∣ ( τ ) .

By (4.7) and (4.9), we have

(4.19) H ^ j ^ ( τ ) ≥ H j ^ ( τ ) ≥ 0 .

By (4.12), (4.13), (4.14), and (4.16), we can obtain

e τ ∫ R n 1 p − 2 ∣ θ j ^ ∣ p − 2 p − 1 + p − 1 p ∣ η ∣ p p − 1 Θ j ^ p − 2 ∣ θ j ^ ∣ ( Θ j ^ 2 − ∣ ∇ ∣ θ j ^ ∣ ∣ 2 ) d η ≤ − e τ H ^ j ^ + C 1 1 a 2 − 1 j ^ e 1 a 2 − 1 τ τ .

By (4.19), integrating over τ > 0 gives

0 ≤ ∫ τ 2 τ e s ∫ R n 1 p − 2 ∣ θ j ^ ∣ p − 2 p − 1 + p − 1 p ∣ η ∣ p p − 1 Θ j ^ p − 2 ∣ θ j ^ ∣ ( Θ j ^ 2 − ∣ ∇ ∣ θ j ^ ∣ ∣ 2 ) d η d s ≤ ∫ 0 τ e s ∫ R n 1 p − 2 ∣ θ j ^ ∣ p − 2 p − 1 + p − 1 p ∣ η ∣ p p − 1 Θ j ^ p − 2 ∣ θ j ^ ∣ ( Θ j ^ 2 − ∣ ∇ ∣ θ j ^ ∣ ∣ 2 ) d η d s ≤ − e τ H ^ j ^ ( τ ) + C 1 1 a 2 − 1 j ^ e 1 a 2 − 1 τ + H ^ j ^ ( 0 ) − C 1 1 a 2 − 1 j ^ ≤ C 1 1 a 2 − 1 j ^ e 1 a 2 − 1 τ + H ^ j ^ ( 0 ) .

This immediately implies that

0 ≤ ∫ τ 2 τ e s ∫ R n 1 p − 2 ∣ θ ∣ p − 2 p − 1 + p − 1 p ∣ η ∣ p p − 1 Θ p − 2 ∣ θ ∣ ( Θ 2 − ∣ ∇ ∣ θ ∣ ∣ 2 ) d η d s ≤ liminf j ^ → ∞ ∫ τ 2 τ e s ∫ R n 1 p − 2 ∣ θ j ^ ∣ p − 2 p − 1 + p − 1 p ∣ η ∣ p p − 1 Θ j ^ p − 2 ∣ θ j ^ ∣ ( Θ j ^ 2 − ∣ ∇ ∣ θ j ^ ∣ ∣ 2 ) d η d s ≤ lim j ^ → ∞ C 1 1 a 2 − 1 j ^ e 1 a 2 − 1 τ + H ^ j ^ ( 0 ) = H ^ ∣ θ ∣ ( 0 ) .

Taking the mean value in s in τ 2 , τ , we have

(4.20) 0 ≤ ∫ R n 1 p − 2 ∣ θ ∣ p − 2 p − 1 + p − 1 p ∣ η ∣ p p − 1 Θ p − 2 ∣ θ ∣ ( Θ 2 − ∣ ∇ ∣ θ ∣ ∣ 2 ) d η ( τ ′ ) ≤ τ H ^ ∣ θ ∣ ( 0 ) 2 e τ 2 τ 2 < τ ′ < τ .

Letting τ → ∞ in (4.20), we can obtain

(4.21) ∣ ∇ θ ˜ ∣ 2 = ∣ ∇ ∣ θ ˜ ∣ ∣ 2 .

By (4.21) and the condition of equality in the Cauchy-Schwarz inequality, there exist constants c > 0 and c i j > 0 , ( 1 ≤ i , j ≤ k ) , such that

∇ θ ˜ i = c i j ∇ θ ˜ j , ∀ 1 ≤ i , j ≤ k , and θ ˜ l = c ∣ ∇ θ ˜ l ∣ , ∀ 1 ≤ l ≤ k .

This immediately implies that

(4.22) ∣ θ ˜ ∣ 2 = ∑ j = 1 k ( θ ˜ j ) 2 = c 2 ∑ j = 1 k ∣ ∇ θ ˜ j ∣ 2 = c 2 ∣ ∇ θ ˜ i ∣ 2 ∑ j = 1 k ( c i j ) 2 = ( θ ˜ i ) 2 ∑ j = 1 k ( c i j ) 2 ≔ C l ( θ ˜ i ) 2 .

By (4.22) and the L 1 mass conservation of θ l , one can easily check that the constant C l = ∣ M ∣ M l . Therefore equation (4.18) holds, and the lemma follows.□

We now are ready to show the asymptotic large time behaviour of solution u = ( u 1 , … , u k ) .

Proof of Theorem 1.2

By Lemma 4.2, it suffices to show that

(4.23) ∥ ∣ u ∣ ( ⋅ , t ) − ℬ ∣ M ∣ ( ⋅ , t ) ∥ L 1 ( R n ) ≤ C t a 2 − a 2 2 ,

for some constant C > 0 . Let θ = ( θ 1 , … , θ k ) be a solution of (4.4) given by (4.3). Since ∣ θ ∣ and ℬ ˜ M have equal mass,

(4.24) 1 2 ∫ R n ∣ ∣ θ ∣ − ℬ ˜ ∣ M ∣ ∣ d η = ∫ { ∣ θ ∣ < ℬ ˜ M } ∣ ∣ θ ∣ − ℬ ˜ ∣ M ∣ ∣ d η .

By (4.6), (4.9), (4.11), (4.24), and Hölder’s inequality,

(4.25) ∫ R n ∣ ∣ θ ∣ − ℬ ˜ ∣ M ∣ ∣ d η ≤ 2 ∫ { ∣ θ ∣ < ℬ ˜ ∣ M ∣ } ℬ ˜ ∣ M ∣ − 1 p − 1 ∣ ∣ θ ∣ − ℬ ˜ ∣ M ∣ ∣ 2 d η 1 2 ∫ R n ℬ ˜ ∣ M ∣ 1 p − 1 d η 1 2 ≤ 2 ( 2 H ∣ θ ∣ ( τ ) ) 1 2 ∫ R n ℬ ˜ ∣ M ∣ 1 p − 1 d η 1 2 ≤ 2 ( 2 e − τ H ^ ∣ θ ∣ ( 0 ) ) 1 2 ∫ R n ℬ ˜ ∣ M ∣ 1 p − 1 d η 1 2 = C e − τ 2 ,

where

C = 2 2 H ^ ∣ θ ∣ ( 0 ) ∫ R n ℬ ˜ ∣ M ∣ 1 p − 1 d η .

Therefore, by (4.1), (4.3), and (4.25), we have

∫ R n ∣ ∣ u ∣ ( x , t ) − ℬ ∣ M ∣ ( x , t ) ∣ d x = ∫ R n ∣ u ∣ ( x , t ) − t a 2 − a 1 ℬ ˜ ∣ M ∣ t a 2 − a 2 x d x = ∫ R n ∣ ∣ θ ∣ ( x , t ) − ℬ ˜ ∣ M ∣ ∣ d η ≤ C t a 2 − a 2 2 .

Hence, the claim holds, and the theorem follows.□

Remark 4.3

By Hölder’s, continuity of component u l , ( l = 1 , … , k ) ,

θ l ( η , τ ) → ℬ ˜ ∣ M ∣ ( η ) , on every compact subset of  R n  as  τ → ∞ .

This immediately implies the uniform convergence between u l and ℬ M in L loc ∞ , i.e.,

(4.26) lim t → ∞ t a 1 u l ( ⋅ , t ) − M l ∣ M ∣ ℬ ∣ M ∣ ( ⋅ , t ) = 0 , uniformly on every compact subset of  R n .

5 Harnack-type inequality of degenerated p -Laplacian system

The last section is devoted to finding a suitable Harnack-type inequality for the component u l , ( 1 ≤ l ≤ k ) , of solution of (SPL) which makes the size of a spatial average of u l under control by the value of u l at one point. We will use a modification of techniques in the proof of Theorem 1.5 of [10] to show the regularity theory.

We start the proof of Harnack-type inequality by reviewing a Harnack-type estimate of the p -Laplacian equation

(5.1) u t = △ p u = ∇ ⋅ ( ∣ ∇ u ∣ p − 2 ∇ u ) .

Lemma 5.1

(cf. Result 5 of [6] and Theorem 3.1 of [16]) Let u be a nonegative solution of (5.1) in R n × [ 0 , T ] for some T > 0 . Then, for every R > 0 , there exists a constant C = C ( m , n ) > 0 such that

∫ { ∣ x ∣ < R } u ( x , 0 ) d x ≤ C R n + p p − 2 T 1 p − 2 + T n p u 1 + n ( p − 2 ) p ( 0 , T ) .

Let ℬ M ( x , t ) be the fundamental solution of the p -Laplacian equation with L 1 mass M . Then, by Lemma 5.1, we can obtain

(5.2) ∫ { ∣ x ∣ < R } ℬ M ( x , 0 ) d x ≤ C R n + p p − 2 T 1 p − 2 + T n p ℬ M 1 + n ( p − 2 ) p ( 0 , T ) .

We are now ready to give proof for our Harnack-type inequality.

Proof of Theorem 1.3

The proof is almost the same as the one for Theorem 1.5 in [10]. For future reference, we will give the sketch of proof here.

For any M ¯ > 0 , denote by P ( M ¯ ) the class of all weak solution u ¯ = ( u ¯ 1 , … , u ¯ k ) of

( u ¯ l ) t = ∇ ⋅ ( ∣ ∇ u ¯ ∣ p − 2 ∇ u ¯ l ) , in  R n × [ 0 , ∞ ) ,

satisfying

u ¯ l ≥ 0 , ∀ 1 ≤ l ≤ k ,

and

sup t > 0 ∫ R n ∣ u ¯ ∣ ( x , t ) d x ≤ M ¯ .

Let k ∈ N and T > 0 be fixed. By Lemma 2.2, there exists a constant M ¯ u > 0 , depending on u ( ⋅ , 0 ) , such that

(5.3) ∫ R n ∣ u ∣ ( x , t ) d x ≤ M ¯ u , ∀ 0 ≤ t ≤ T .

Thus, we are going to show that (1.9) holds when u ∈ P ( M ¯ ) for some constant M ¯ > 0 . We divide the proof into two cases.

Case 1. supp ∣ u ∣ ( ⋅ , 0 ) ⊂ B 1 = { x ∈ R n : ∣ x ∣ ≤ 1 } .

If (1.9) is violated, then for each j ∈ N , there exist a constant R j > 0 , a solution u j = ( u j 1 , … , u j k ) ∈ P ( M ¯ j ) , and a number 1 ≤ i ′ ( j ) ≤ k such that

(5.4) ∫ { ∣ x ∣ < R j } u j i ′ ( x , 0 ) d x ≥ j ( μ j i ′ ) 1 + n ( p − 2 ) p R j n + p p − 2 T 1 p − 2 + T n p ( u j i ′ ) 1 + n ( p − 2 ) p ( 0 , T ) ,

where

μ j i ′ = ∫ R n u j i ′ ( x , 0 ) d x max 1 ≤ l ≤ k ∫ R n u j l ( x , 0 ) d x .

Without loss of generality, we may assume that i ′ = 1 for each j ∈ N and let

(5.5) I j = max 1 ≤ l ≤ k ∫ R n u j l ( x , 0 ) d x .

By (5.4) and (5.5),

I j → ∞ , as  j → ∞ ,

since R j n + p p − 2 T − 1 p − 2 ≥ T n p > 0 . Consider the rescaled function

v j l ( x , t ) = 1 I j p n ( p − 2 ) + p u j l I j p − 2 n ( p − 2 ) + p x , t , ∀ 1 ≤ l ≤ k .

By direct computation, one can easily check that v j = ( v j 1 , … , v j k ) is also a solution of (1.8) in R n × [ 0 , ∞ ) with

(5.6) 0 < μ 0 ≤ μ j l = ∫ R n v j l ( x , 0 ) d x ≤ 1 , ∀ 1 ≤ l ≤ k , j ∈ N .

Moreover,

supp v j l ( ⋅ , 0 ) ⊂ B 1 I j p − 2 n ( p − 2 ) + p ( 0 ) , ∀ 1 ≤ l ≤ k , j ∈ N .

By the Ascoli theorem and a diagonalization argument, the sequence { v j } j = 1 ∞ has a subsequence which we may assume without loss of generality to be the sequence itself such that

(5.7) v j l ( x , 0 ) → μ l δ , ∀ 1 ≤ l ≤ k ,

where δ is Dirac’s delta function and μ l is a constant such that 0 < μ 0 ≤ μ l ≤ 1 . Then, by Hölder’s continuity of u j l and (2.5),

(5.8) v j l → μ l μ ℬ μ , uniformly on every compact subset of  R n × ( 0 , ∞ )  as  j → ∞ ,

where μ = ∑ i = 1 k ( μ i ) 2 and ℬ μ is the fundamental solution of p -Laplacian equation with L 1 mass μ . By (5.6), (5.7), and (5.8), there exists a number j 0 ∈ N such that

(5.9) μ j 1 = μ j i ′ ≥ 1 2 μ i ′ = 1 2 μ 1 , ∀ j ≥ j 0 ,

and

(5.10) ∫ ∣ x ∣ < I j − p − 2 n p − 2 + p R j v j 1 ( x , 0 ) d x ≤ μ j 1 ≤ 2 μ 1 ≤ 2 ∫ ∣ x ∣ < I j − p − 2 n ( p − 2 ) + p R j ℬ μ ( x , 0 ) d x , ∀ j ≥ j 0 ,

and

(5.11) μ 1 μ ℬ μ ( 0 , T ) ≤ 2 v j 1 ( 0 , T ) = 2 I j p n ( p − 2 ) + p u j 1 ( 0 , T ) , ∀ j ≥ j 0 .

By (5.2), (5.9), (5.10), and (5.11),

1 I j ∫ { ∣ x ∣ < R j } u j 1 ( x , 0 ) d x = ∫ ∣ x ∣ < I j − p − 2 n ( p − 2 ) + p R j v j 1 ( x , 0 ) d x ≤ 2 ∫ ∣ x ∣ < I j − p − 2 n ( p − 2 ) + p R j ℬ μ ( x , 0 ) d x ≤ 2 C R j n + p p − 2 I j T 1 p − 2 + T n p ℬ μ 1 + n ( p − 2 ) p ( 0 , T ) ≤ 2 2 + n ( p − 2 ) p k 1 2 + n ( p − 2 ) 2 p C ( μ 1 ) 1 + n ( p − 2 ) p I j R j n + p p − 2 T 1 p − 2 + T n p ( u j 1 ) 1 + n ( p − 2 ) p ( 0 , T ) ≤ 2 3 + 2 n ( p − 2 ) p k 1 2 + n ( p − 2 ) 2 p C ( μ j 1 ) 1 + n ( p − 2 ) p I j R j n + p p − 2 T 1 p − 2 + T n p ( u j 1 ) 1 + n ( p − 2 ) p ( 0 , T ) , ∀ j ≥ j 0 .

Hence, for C 1 = 2 3 + 2 n ( p − 2 ) p k 1 2 + n ( p − 2 ) 2 p C ,

∫ { ∣ x ∣ < R j } u j 1 ( x , 0 ) d x ≤ C 1 ( μ j 1 ) 1 + n ( p − 2 ) p R j n + p p − 2 T 1 p − 2 + T n p ( u j 1 ) 1 + n ( p − 2 ) p ( 0 , T ) , ∀ j ≥ j 0 ,

which contradicts (1.9), and the case follows.

Case 2. General case ∣ u ∣ ( ⋅ , 0 ) has compact support.

Let R 0 > 1 be a constant such that

∣ u ∣ ( x , 0 ) = 0 , ∀ x ∈ R n \ B R 0 ,

and consider the rescaled functions

w l ( x , t ) = 1 R 0 p p − 2 u l ( R 0 x , t ) , ∀ 1 ≤ l ≤ k .

Then, w = ( w 1 , … , w k ) is a solution of (1.8) with

supp ∣ w ∣ ( ⋅ , 0 ) ⊂ B 1 .

By Case 1, for any 0 < R < R 0 , we can obtain

(5.12) 1 R 0 n + p p − 2 ∫ { ∣ x ∣ < R } u l ( x , 0 ) d x = ∫ ∣ x ∣ < R R 0 w l ( x , 0 ) d x ≤ C ( μ w l ) 1 + n ( p − 2 ) p 1 T 1 p − 2 R n + p p − 2 R 0 n + p p − 2 + T n p ( w l ) 1 + n ( p − 2 ) p ( 0 , T ) = C ( μ l ) 1 + n ( p − 2 ) p 1 T 1 p − 2 R n + p p − 2 R 0 n + p p − 2 + T n p R 0 n + p p − 2 ( u l ) 1 + n ( p − 2 ) p ( 0 , T ) , ∀ 1 ≤ l ≤ k ,

where

μ w l = ∫ R n w l ( x , 0 ) d x max 1 ≤ l ≤ k ∫ R n w l ( x , 0 ) d x , ∀ 1 ≤ l ≤ k .

Multiplying (5.12) by R 0 n + p p − 2 , (1.9) holds and the theorem follows.□

As mentioned in Section 1, the result of Harnack-type inequality plays an important role in the study of the initial trace theorem. We finish this article by providing the proof of Corollary 1.4.

Proof of Corollary 1.4

By the law of L 1 mass conservation (Lemma 2.2), u l ( ⋅ , t ) is uniformly bounded in L 1 ( R n ) . Thus, by an argument similar to the proof of Theorem 1.1, the convergence (1.10) can be easily proved.

We now focus on the decay rate of initial trace ρ l . By the result of Theorem 1.3, there exists a constant C 1 = C 1 ( n , m ) > 0 independent of ε ∈ 0 , T 2 such that

(5.13) ∫ { ∣ x ∣ < R } u l ( x , ε ) d x ≤ 2 1 p − 2 C 1 ( μ 0 ) 1 + n ( p − 2 ) p R n + p p − 2 T 1 p − 2 + T n p ( u l ) 1 + n ( p − 2 ) p ( 0 , T ) , ∀ R > 0 , 1 ≤ l ≤ k .

By (1.10), we can obtain

(5.14) lim ε → 0 ∫ { ∣ x ∣ < R } u l ( x , ε ) d x = ∫ { ∣ x ∣ < R } ρ l ( d x ) .

Since the constant μ 0 is independent of ε , by (5.13) and (5.14), (1.11) hold for C = 2 1 p − 2 C 1 ( μ 0 ) 1 + n ( p − 2 ) p and the corollary follows.□

  1. Funding information: Ki-Ahm Lee was supported by NRF grant funded by the Korean government (MSIP) (NRF-2020R1A2C1A01006256). Ki-Ahm Lee also held a joint appointment with the Research Institute of Mathematics of Seoul National University. This work was supported by the Research Fund, 2022 of The Catholic University of Korea.

  2. Author contributions: All authors have accepted responsibility for the entire content of this manuscript and consented to its submission to the journal, reviewed all the results, and approved the final version of the manuscript.

  3. Conflict of interest: The authors state no conflict of interest.

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Received: 2024-07-23
Revised: 2024-10-03
Accepted: 2024-10-18
Published Online: 2024-11-25

© 2024 the author(s), 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/math-2024-0094
Keywords for this article
degenerate parabolic system; entropy approach; asymptotic behaviour; Harnack-type inequality
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  60. Pullback attractors for fractional lattice systems with delays in weighted space
  61. Singularities of spherical surface in R4
  62. Variational approach to Kirchhoff-type second-order impulsive differential systems
  63. Convergence rate of the truncated Euler-Maruyama method for highly nonlinear neutral stochastic differential equations with time-dependent delay
  64. On the energy decay of a coupled nonlinear suspension bridge problem with nonlinear feedback
  65. The limit theorems on extreme order statistics and partial sums of i.i.d. random variables
  66. Hardy-type inequalities for a class of iterated operators and their application to Morrey-type spaces
  67. Solving multi-point problem for Volterra-Fredholm integro-differential equations using Dzhumabaev parameterization method
  68. Finite groups with gcd(χ(1), χc (1)) a prime
  69. Small values and functional laws of the iterated logarithm for operator fractional Brownian motion
  70. The hull-kernel topology on prime ideals in ordered semigroups
  71. ℐ-sn-metrizable spaces and the images of semi-metric spaces
  72. Strong laws for weighted sums of widely orthant dependent random variables and applications
  73. An extension of Schweitzer's inequality to Riemann-Liouville fractional integral
  74. Construction of a class of half-discrete Hilbert-type inequalities in the whole plane
  75. Analysis of two-grid method for second-order hyperbolic equation by expanded mixed finite element methods
  76. Note on stability estimation of stochastic difference equations
  77. Trigonometric integrals evaluated in terms of Riemann zeta and Dirichlet beta functions
  78. Purity and hybridness of two tensors on a real hypersurface in complex projective space
  79. Classification of positive solutions for a weighted integral system on the half-space
  80. A quasi-reversibility method for solving nonhomogeneous sideways heat equation
  81. Higher-order nonlocal multipoint q-integral boundary value problems for fractional q-difference equations with dual hybrid terms
  82. Noetherian rings of composite generalized power series
  83. On generalized shifts of the Mellin transform of the Riemann zeta-function
  84. Further results on enumeration of perfect matchings of Cartesian product graphs
  85. A new extended Mulholland's inequality involving one partial sum
  86. Power vector inequalities for operator pairs in Hilbert spaces and their applications
  87. On the common zeros of quasi-modular forms for Γ+0(N) of level N = 1, 2, 3
  88. One special kind of Kloosterman sum and its fourth-power mean
  89. The stability of high ring homomorphisms and derivations on fuzzy Banach algebras
  90. Integral mean estimates of Turán-type inequalities for the polar derivative of a polynomial with restricted zeros
  91. Commutators of multilinear fractional maximal operators with Lipschitz functions on Morrey spaces
  92. Vector optimization problems with weakened convex and weakened affine constraints in linear topological spaces
  93. The curvature entropy inequalities of convex bodies
  94. Brouwer's conjecture for the sum of the k largest Laplacian eigenvalues of some graphs
  95. High-order finite-difference ghost-point methods for elliptic problems in domains with curved boundaries
  96. Riemannian invariants for warped product submanifolds in Q ε m × R and their applications
  97. Generalized quadratic Gauss sums and their 2mth power mean
  98. Euler-α equations in a three-dimensional bounded domain with Dirichlet boundary conditions
  99. Enochs conjecture for cotorsion pairs over recollements of exact categories
  100. Zeros distribution and interlacing property for certain polynomial sequences
  101. Random attractors of Kirchhoff-type reaction–diffusion equations without uniqueness driven by nonlinear colored noise
  102. Study on solutions of the systems of complex product-type PDEs with more general forms in ℂ2
  103. Dynamics in a predator-prey model with predation-driven Allee effect and memory effect
  104. A note on orthogonal decomposition of 𝔰𝔩n over commutative rings
  105. On the δ-chromatic numbers of the Cartesian products of graphs
  106. Binomial convolution sum of divisor functions associated with Dirichlet character modulo 8
  107. Commutator of fractional integral with Lipschitz functions related to Schrödinger operator on local generalized mixed Morrey spaces
  108. System of degenerate parabolic p-Laplacian
  109. Stochastic stability and instability of rumor model
  110. Certain properties and characterizations of a novel family of bivariate 2D-q Hermite polynomials
  111. Stability of an additive-quadratic functional equation in modular spaces
  112. Monotonicity, convexity, and Maclaurin series expansion of Qi's normalized remainder of Maclaurin series expansion with relation to cosine
  113. On k-prime graphs
  114. On the existence of tripartite graphs and n-partite graphs
  115. Classifying pentavalent symmetric graphs of order 12pq
  116. Almost periodic functions on time scales and their properties
  117. Some results on uniqueness and higher order difference equations
  118. Coding of hypersurfaces in Euclidean spaces by a constant vector
  119. Cycle integrals and rational period functions for Γ0+(2) and Γ0+(3)
  120. Degrees of (L, M)-fuzzy bornologies
  121. A matrix approach to determine optimal predictors in a constrained linear mixed model
  122. On ideals of affine semigroups and affine semigroups with maximal embedding dimension
  123. Solutions of linear control systems on Lie groups
  124. A uniqueness result for the fractional Schrödinger-Poisson system with strong singularity
  125. On prime spaces of neutrosophic extended triplet groups
  126. On a generalized Krasnoselskii fixed point theorem
  127. On the relation between one-sided duoness and commutators
  128. Non-homogeneous BVPs for second-order symmetric Hamiltonian systems
  129. Erratum
  130. Erratum to “Infinitesimals via Cauchy sequences: Refining the classical equivalence”
  131. Corrigendum
  132. Corrigendum to “Matrix stretching”
  133. Corrigendum to “A comprehensive review of the recent numerical methods for solving FPDEs”
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