Double-phase parabolic equations with variable growth and nonlinear sources

1.1 Previous work

The study of the double-phase problems started in the late 80th by the works of Zhikov [47,48], where the models of strongly anisotropic materials were considered in the context of homogenization. Later on, the double-phase functionals

attracted attention of many researchers. On the one hand, the study of these functionals is a challenging mathematical problem. On the other hand, the double-phase functionals appear in a variety of physical models. We refer here to [8,46] for applications in the elasticity theory, [6] for transonic flows, [9] for quantum physics, and [13] for reaction-diffusion systems.

Equations (1.1) with p ≠ q are also referred to as the equations with the ( p , q ) -growth because of the gap between the coercivity and growth conditions: If p ≤ q and 0 ≤ a ( x ) ≤ L , then for every ξ ∈ R N ,

These equations fall into the class of equations with nonstandard growth conditions, which have been actively studied during the last decades in the cases of constant or variable exponents p and q . We refer to the works [2,14,17,18,24,26,27,28,31,34,35,39,40,45] and references therein for the results on the existence and regularity of solutions, including optimal regularity results [24].

Results on the existence of solutions to the evolution double-phase equations can be found in papers [10,43,44]. These works deal with the Dirichlet problem for systems of parabolic equations of the form

where the flux a ( x , t , ∇ u ) is assumed to satisfy the ( p , q ) -growth conditions and certain regularity assumptions. As a partial case, the class of equations (1.3) includes equation (1.1) with constant exponents p ≤ q and a nonnegative bounded coefficient a ( x , t ) . It is shown in [10, Th.1.6] that if

then problem (1.1) with F ≡ 0 has a very weak solution:

provided that u 0 ∈ W 0 1 , r ( Ω ) , r = p ( q − 1 ) p − 1 . Moreover, ∣ ∇ u ∣ is bounded on every strictly interior cylinder Q T ′ ⋐ Q T separated away from the parabolic boundary of Q T . In [43], these results were extended to the case

The paper [44] deals with weak solutions of systems of equations of the type (1.3) with ( p , q ) growth conditions. When applied to problem (1.1) with constant p and q , F ≡ f 0 ( z ) , and a ( ⋅ , t ) ∈ C α ( Ω ) with some α ∈ ( 0 , 1 ) for a.e. t ∈ ( 0 , T ) , the result of [44] guarantees the existence of a weak solution:

provided that the exponents p and q obey the inequalities:

The proofs of the existence theorems in [10,43,44] rely on the property of local higher integrability of the gradient, ∣ ∇ u ∣ p + δ ∈ L 1 ( Q T ′ ) for every sub-cylinder Q T ′ ⋐ Q T . The maximal possible value of δ > 0 indicates the admissible gap between the exponents p and q and vary in dependence on the type of the solution.

Equation (1.1) with constant exponents p and q furnishes a prototype of the equations recently studied in papers [11,21,29,36] in the context of weak or variational solutions. The proofs of existence also use the local higher integrability of the gradient, but for the existence of variational solutions, a weaker assumption on the gap q − p is required. For the solutions of the evolution p ( x , t ) -Laplace equation, global higher integrability of the gradient up to the lateral boundary of the cylinder was proven in [1] for very weak solutions and under mild restrictions on the regularity of the data. In [5], this property was established in the whole of the cylinder Q T but under stronger assumptions on the data.

Nonhomogeneous parabolic equations of the form (1.3) with the flux a ( x , t , ∇ u ) controlled by a generalized N -function were studied in [16] in the framework of Musielak-Orlicz spaces. The class of equations studied in [16] includes, as a partial case, equation (1.1) with F = f 0 ( z ) and variable exponents p , q . It is shown that problem (1.1) with bounded data u 0 and f 0 admits a unique solution u ∈ L ∞ ( 0 , T ; L 2 ( Ω ) ) ∩ L 1 ( 0 , T ; W 0 1 , 1 ( Ω ) ) with ∇ u ∈ L M ( Q T ; R N ) , where L M denotes the Musielak-Orlicz space defined by the flux a ( x , t , ξ ) . We refer here to [16,25], the survey article [14], and references therein for the issues of solvability of elliptic and parabolic equations in the Musielak-Orlicz spaces. A comprehensive review of the current state of the theory of double-phase problems is given in [38].

In the last years, the double-phase stationary problems with nonlinear convection terms were in the focus of attention of many researchers, see, e.g., papers [7,27,37], monograph [41], and references therein. However, to the best of our knowledge, there are no results on the counterpart evolution problems with nonlinear sources of the variable growth.

1.2 Description of results

In the present work, we prove the existence of strong solutions to problem (1.1). By the strong solution, we mean a solution whose time derivative is not a distribution but an element of a Lebesgue space, and the flux has better integrability properties than the properties prompted by the energy equality (the rigorous formulation is given in Definition 3.1).

• We consider first the case of the given source independent of the solution: F = f 0 ( z ) . It is shown that if the exponents p , q and the data a , u 0 , and f 0 are sufficiently smooth, and if the gap between the exponents satisfies the condition

  (1.4) 2 N N + 2 < p ( z ) ≤ q ( z ) < p ( z ) + 2 p − ( N + 2 ) p − + 2 N in Q T ,

  where p − = inf Q T p ( z ) , then problem (1.1) has a strong solution u with

  (1.5) u t ∈ L 2 ( Q T ) , ess sup ( 0 , T ) ℱ ( u ( ⋅ , t ) , t ) < ∞ , ℱ ( u ( ⋅ , t ) , t ) ≡ ∫ Ω ( ∣ ∇ u ∣ p ( z ) + a ( z ) ∣ ∇ u ∣ q ( z ) ) d x .

  Moreover, the solution possesses the property of global higher integrability of the gradient:

  (1.6) ∫ Q T ∣ ∇ u ∣ p ( z ) + r d z ≤ C for every 0 < r < 4 p − ( N + 2 ) p − + 2 N ,

  with a finite constant C depending only on ℱ ( u 0 , 0 ) , N , r , and the properties of p ( ⋅ ) and q ( ⋅ ) . The same existence result is valid for problem (1.1) with the regularized nondegenerate flux:

  ( ε 2 + ∣ ∇ u ∣ 2 ) p ( z ) − 2 2 + a ( z ) ( ε 2 + ∣ ∇ u ∣ 2 ) q ( z ) − 2 2 ∇ u , ε > 0 .

• If the source F depends on u and ∇ u , the existence of strong solutions of problem (1.1) with the property of global higher integrability (1.6) is proven for the nonlinear source function F of the form (1.2) subject to specific growth restrictions, which lead to an additional assumption on p − . We assume that

  Ψ ∈ C 0 ( Q ¯ T × R ) , Ψ ( z , 0 ) = 0 , ∣ Ψ ( z , r ) ∣ ≤ C ∣ r ∣ σ ( z ) − 1 , ∣ ∇ Ψ ( z , v ) ∣ ≤ C ( ∣ v ∣ σ ( z ) − 1 ( 1 + ∣ ln ∣ v ∣ ∣ ) + ∣ v ∣ σ ( z ) − 2 ∣ ∇ v ∣ ) ,

  with a constant C and a function σ ∈ C 0 ( Q ¯ T ) such that

  2 ≤ inf Q T σ ( z ) ≤ sup Q T σ ( z ) < min p − , 1 + p − N + 2 2 N .

  The growth of F with respect to ∇ u is subject to the following restrictions:

  Φ ∈ C 0 ( Q ¯ T × R N ) , ∣ Φ ( z , s ) ∣ ≤ L ∣ s ∣ p ( z ) − 1 if ∣ s ∣ < 1 , M 1 2 ∣ s ∣ p ( z ) 2 if ∣ s ∣ ≥ 1

  with positive constants L , and M . The functions Ψ and Φ are not assumed to be sign-definite. An example of admissible Ψ and Φ is furnished by

  Φ ( z , s ) = c ( z ) ∣ s ∣ p ( z ) − 1 1 + ∣ s ∣ p ( z ) 2 − 1 , Ψ ( z , v ) = b ( z ) ∣ v ∣ σ ( z ) − 2 v ,

  with

  b ( z ) , c ( z ) ∈ C 0 ( Q ¯ T ) , ‖ ∇ b ‖ ∞ , Q T ≤ const , ‖ ∇ σ ‖ ∞ , Q T ≤ const .

  If Ψ ≡ 0 , the restriction on p ( z ) transforms into p − ≥ 2 . For the uniqueness of strong solutions, we assume that Φ ≡ 0 and either Ψ ( z , v ) is monotone decreasing with respect to v for a.e. z ∈ Q T or Ψ ( z , v ) = b ( z ) v .

In both cases, property (1.6) of global higher integrability of the gradient turns out to be crucial for the proof of the existence of a strong solution. A traditional approach to the double-phase equations involves regularization of the equation and derivation of a local estimate (1.6) with the use of Caccioppoli-type inequalities, which is sufficient for the proof of the existence of weak solutions. As distinguished from this method, we find a solution as the limit of a sequence of finite-dimensional Galerkin’s approximations. These approximations do not solve the equation, but they are smooth up to the parabolic boundary of the cylinder Q T . The higher regularity of the approximations allows us to prove global in Q T uniform higher integrability of their gradients by means of the Gagliardo-Nirenberg inequality.

All results remain true for the multiphase equations:

provided the exponents p ( z ) and q i ( z ) satisfy the balance condition (1.4) and the source F is subject to the above-described growth conditions.

2.1 Variable Lebesgue spaces

Throughout the rest of the paper, we assume that Ω ⊂ R N , N ≥ 2 , is a bounded domain with the boundary ∂ Ω ∈ C 2 . Given r ∈ P ( Ω ) , we introduce the modular

and the set

The set L r ( ⋅ ) ( Ω ) equipped with the Luxemburg norm

becomes a Banach space. By convention, from now on, we use the notation

If r ∈ P ( Ω ) and 1 < r − ≤ r ( x ) ≤ r + < ∞ in Ω , then the following properties hold.

1. L r ( ⋅ ) ( Ω ) is a reflexive and separable Banach space.

2. For every f ∈ L r ( ⋅ ) ( Ω ) ,

   (2.2) min { ∣ f ∣ r ( ⋅ ) , Ω r − , ‖ f ‖ r ( ⋅ ) , Ω r + } ≤ A r ( ⋅ ) ( f ) ≤ max { ‖ f ‖ r ( ⋅ ) , Ω r − , ∣ f ∣ r ( ⋅ ) , Ω r + } .

3. For every f ∈ L r ( ⋅ ) ( Ω ) and g ∈ L r ′ ( ⋅ ) ( Ω ) , the generalized Hölder inequality holds:

   (2.3) ∫ Ω ∣ f g ∣ ≤ 1 r − + 1 ( r ′ ) − ‖ f ‖ r ( ⋅ ) , Ω ‖ g ‖ r ′ ( ⋅ ) , Ω ≤ 2 ‖ f ‖ r ( ⋅ ) , Ω ‖ g ‖ r ′ ( ⋅ ) , Ω ,

   where r ′ = r r − 1 is the conjugate exponent of r .

4. If p 1 , p 2 ∈ P ( Ω ) and satisfy the inequality p 1 ( x ) ≤ p 2 ( x ) a.e. in Ω , then L p 1 ( ⋅ ) ( Ω ) is continuously embedded in L p 2 ( ⋅ ) ( Ω ) and for all u ∈ L p 2 ( ⋅ ) ( Ω )

   (2.4) ‖ u ‖ p 1 ( ⋅ ) , Ω ≤ C ‖ u ‖ p 2 ( ⋅ ) , Ω , C = C ( ∣ Ω ∣ , p 1 ± , p 2 ± ) .

5. For every sequence { f k } ⊂ L r ( ⋅ ) ( Ω ) and f ∈ L r ( ⋅ ) ( Ω ) ,

   (2.5) ‖ f k − f ‖ r ( ⋅ ) , Ω → 0 iff A r ( ⋅ ) ( f k − f ) → 0 as k → ∞ .

2.2 Variable Sobolev spaces

The variable Sobolev space W 0 1 , r ( ⋅ ) ( Ω ) is the set of functions:

equipped with the norm

If r ∈ C 0 ( Ω ¯ ) , the Poincaré inequality holds: for every u ∈ W 0 1 , r ( ⋅ ) ( Ω ) ,

Inequality (2.6) means that the equivalent norm of W 0 1 , r ( ⋅ ) ( Ω ) is given by

Let us denote by C log ( Ω ¯ ) the subset of P ( Ω ) composed of the functions continuous on Ω ¯ with the logarithmic modulus of continuity:

where ω is a nonnegative function such that

If r ∈ C log ( Ω ¯ ) , then the set C c ∞ ( Ω ) of smooth functions with finite support is dense in W 0 1 , r ( ⋅ ) ( Ω ) (see Proposition 2.2). This property allows one to use the equivalent definition of the space W 0 1 , r ( ⋅ ) ( Ω ) :

2.3 Spaces of functions depending on x and t

For the study of parabolic problem (1.1), we need the spaces of functions depending on z = ( x , t ) ∈ Q T . Given a function q ∈ C log ( Q ¯ T ) , we introduce the spaces:

The norm of W q ( ⋅ ) ( Q T ) is defined by

Since q ∈ C log ( Q ¯ T ) , the space W q ( ⋅ ) ( Q T ) is the closure of C c ∞ ( Q T ) with respect to this norm.

2.4 Musielak-Sobolev spaces

Let a 0 : Ω ↦ [ 0 , ∞ ) be a given function, a 0 ∈ C 0 , 1 ( Ω ¯ ) . Assume that the exponents p , q ∈ C 0 , 1 ( Ω ¯ ) take values in the intervals ( p − , p + ) , ( q − , q + ) , and p ( x ) ≤ q ( x ) in Ω . Set

and consider the function

The set

equipped with the Luxemburg norm

becomes a Banach space. The space L ℋ ( Ω ) is separable and reflexive [25]. By W 1 , ℋ ( Ω ) , we denote the Musielak-Sobolev space

with the norm

2.5 Dense sets in W 0 1 , p ( ⋅ ) ( Ω ) and W 1 , ℋ ( Ω )

Let { ϕ i } and { λ i } be the eigenfunctions and the corresponding eigenvalues of the Dirichlet problem for the Laplacian:

The functions ϕ i form an orthonormal basis of L 2 ( Ω ) and are mutually orthogonal in H 0 1 ( Ω ) . If ∂ Ω ∈ C k , k ≥ 1 , then ϕ i ∈ C ∞ ( Ω ) ∩ H k ( Ω ) . Let us denote by H D k ( Ω ) the subspace of the Hilbert space H k ( Ω ) composed of the functions f for which

The relations

define an equivalent inner product on H D k ( Ω ) : [ f , g ] k = ∑ i = 1 ∞ λ i k f i g i , where f i , and g i are the Fourier coefficients of f , and g in the basis { ϕ i } of L 2 ( Ω ) . The corresponding equivalent norm of H D k ( Ω ) is defined by ‖ f ‖ H D k ( Ω ) 2 = [ f , f ] k . Let f ( m ) = ∑ i = 1 m f i ϕ i be the partial sums of the Fourier series of f ∈ L 2 ( Ω ) . The following assertion is well known.

Given a function u ∈ W 0 1 , p ( ⋅ ) ( Ω ) with p ∈ C log ( Ω ¯ ) , the smooth approximations of u in W 0 1 , p ( ⋅ ) ( Ω ) are obtained by means of mollification. The proof relies on the boundedness of the maximal Hardy-Littlewood operator in L p ( ⋅ ) ( Ω ) with p ∈ C log ( Ω ¯ ) .

Let us denote P m = span { ϕ 1 , … , ϕ m } , where ϕ i are the solutions of problem (2.9).

Since ∂ Ω ∈ C k with k ≥ 1 and r ( x ) , s ( x ) ≥ 2 , then W 1 , ℋ ( Ω ) ⊂ W 1 , 2 ( Ω ) and the elements of W 1 , ℋ ( Ω ) have traces on ∂ Ω . Let us denote by