Optimal global second-order regularity and improved integrability for parabolic equations with variable growth

: We consider the homogeneous Dirichlet problem for the parabolic equation

, then the problem has a 2 ( ), obtained as the limit of solutions to the regularized problems in the parabolic Hölder space.The solution possesses the following global regularity properties: T W , for any 0 4 2 , 0 ,; Ω .
We consider the homogeneous Dirichlet problem Ω 0, , , 0 , in Ω, where ) is the cylinder of a finite height T and ⊂ Ω N is a bounded domain with the C 2 -smooth or convex boundary ∂Ω.Here and throughout the text, we denote by = z x t , ( ) the points of the cylinder Q T .The nonlinear source F has the form , .
In (1.1) and (1.2), the exponents and coefficients a, → c , p, q, and s are given functions whose properties will be specified later.
The first objective of this work is to establish the global second-order spatial differentiability of solutions to Problem (1.1).We prove that if (the definition of the variable Sobolev spaces is given in Section 2), ∈ f L Q T 2 ( ), and the nonlinear source ∇ F z u u , , ( ) has a proper growth with respect to the second and the third arguments, then , with a constant depending only on the problem data, i.e., on ( ) ) for the solutions of the heat equation (see, e.g., [25,Ch.3,Sec.6]).In the case of constant p, the second-order regularity of solutions to equations or systems of parabolic equations of the type t p (1.4) were studied by many authors.We refer to [16] for the global estimates (1.3) and to [1,12,17,[21][22][23]30] for local results (see also references therein).The study of Feng et al. [22] deals with the homogeneous equation (1.4).In [23], the local second-order regularity is proven for the viscosity solutions of a more general equation under some restrictions on the values of the constant exponents p and γ.In the special case = − γ p 2, equation (1.5) coincides with (1.4) with = f 0. The results on the second-order spatial regularity of solutions to (1.4)  with some σ depending on p.In [16], inclusion (1.6) with = − σ p 2 is proven for the approximable solutions such that the solution and its gradient can be obtained as the pointwise limits of solutions and gradients of solutions to Problems (1.4) with regularized fluxes and smooth data.The proof in [16] crucially depends upon a differential inequality, which links the Hessian matrix and the Laplacian of a smooth function and furnishes a lower bound for the square of the p-Laplacian -see [14,Lemma 3.1].
As it is mentioned in [16,Remark 2.3], the global estimate (1.3) with = − σ p 2 is sharp for the solutions to Problem (1.4) because in the case = u 0 0 , the norms in (1.3) are bounded below and above by It is shown in [22] that for the weak solutions of equation (1.4) with Moreover, a counterexample shows that the inequality > − s 1 is sharp, see [22,Remark 2.4].The inclusion and estimate (1.3) proven in our present work are optimal in the following sense.On the one hand, for the solution of Problem

.3).
There is an immense literature on the second-order regularity of solutions to equations and systems of elliptic nonlinear equations with constant structure.For relevant results and an exhaustive review of the available literature, we refer to [9,14,15].Results on the local second-order regularity of solutions to elliptic equations with p x ( )-Laplacian can be found in [13,33].
Coming back to equation (1.1) with the variable exponent p z ( ), the global second-order spatial regularity is proven in [6] for ≡ F 0, and in [7,8] for equations with a double-phase flux of variable growth and nonlinear sources.It is shown in [6] that .A specific feature of equation (1.1) with variable structure is that differentiation of the flux creates first-order residual terms of higher growth, which cannot be controlled through the principal energy inequality.Nonetheless, these terms can be estimated due to the property of global higher integrability of the gradientsee [8,Theorem 2.3]: For the solutions of equation (1.1) with the regularized flux the global property (1.8) stems from the following interpolation inequality: if ∂ ∈ C Ω 2 , then for every ( ) ( ), > δ 0, an exponent ∈ q C Ω 0,1 ( ) satisfying ( ) , and any with a constant C depending on δ, N , r, q, and v L Ω 2 ‖ ‖ ( ) .Inequality (1.9) was first proven in [6] for r in the − p p min and then refined in [8] to the form used in the present work.
A solution to Problem (1.1) with the regularized flux ∇ z u , ε ( ) was constructed in [6] as the limit of the sequence of finite-dimensional Galerkin's approximations in the basis composed of the eigenfunctions of the Dirichlet problem for the Laplace operator.The regularity of eigenfunctions is defined by the regularity of the boundary ∂Ω, and the sequence of finite-dimensional approximations is a sequence of smooth functions that converges to the solution in a proper variable Sobolev space.Inclusion (1.7) follows then from the uniform estimates on the regularized fluxes.The same scheme of arguments was used in [7,8] to prove the global second-order differentiability of solutions to equations with double-phase fluxes of variable growth.A drawback of this approach is that the a priori estimates for the approximations require, in general, an extra regularity of the free term f and the initial datum u 0 .
For the solutions of the singular equation (1.4) ( ) , the second-order regularity was studied in [4,5] (see also [30] for the case of constant p).In this special range of values of p z ( ), the inclusions can be achieved without higher integrability of the gradient (1.8).
In this work, we prove the inclusion and Estimate (1.3) under the assumptions ) .The proof is based on a combination of (1.9) with the inequality where the constants and ∂Ω.An immediate by-product of (1.9)-(1.10) is the inequality with a constant C depending only on the data and r.Inequality (1.11) improves (1.8) for > p z 2 ( ) .
Inequality (1.10) holds under the same assumptions on v and ∂Ω as (1.9).However, unlike (1.9), Inequality (1.10) cannot be derived for Galerkin's approximations, which makes necessary another scheme of approximation of the solution to (1.1).This is the second objective of our work: to prove that a solution to the regularized Problem (1.1) can be obtained as the limit of classical solutions to the problem with the regularized flux ε and the data.Such an approach to construction of a solutionan approximable solution in the terminology of [16] is traditional for problems with constant nonlinearity.To the best of our knowledge, it is new for problems with the p z ( )-Laplace operator.It is an alternative to the approaches based on the finite-dimensional Galerkin's approximations or regularization of the flux by ∇ (see [2,3,19] for the existence of weak solutions to the Dirichlet problem for equation (1.4) with the exponent p z ( ) depending on t and the data . A classical solution to the problem with the regularized data is obtained as a fixed point of a mapping generated by a linear problem associated with (1.1).The existence of a fixed point follows from the Leray-Schauder principle.The cornerstone of the proof is the recent result of [20] about the Hölder continuity of the gradients of weak solutions to equation (1.1) with variable nonlinearity and nonlinear sources of general form.See also [11,29] for results on the local Hölder continuity of the spatial gradient of solutions to the evolution p x t , ( ) and p t ( )-Laplacian equation and systems without sources.The main results of the work are summarized as follows.Assume that ) is subject to proper power growth conditions in the second and third arguments, and the coefficients and exponents of nonlinearity of F belong to C Q T 0 ( ).
• For every , ∈ f L Q T 2 ( ) a solution of Problem (1.1) can be obtained as the limit of the a sequence of classical solutions to the regularized problems, which belong to the Hölder space • The constructed solution has the properties Ω , , and 0, ; Ω .
• The solution has the second-order spatial regularity in the sense (1.3) and possesses the property of global higher integrability of the gradient (1.11).• All results hold true for the convex domains Ω without assumptions on the regularity of ∂Ω, and for C 2 -domains.
In the proofs, we do not distinguish between the cases ≥ p z 2 ( ) and ≤ p z

( )
. All conclusions hold true for the variable exponent p z ( ), which may vary within the interval ⎛ ⎝ 2 Function spaces, assumptions, and main results

Variable Lebesgue and Sobolev spaces
To formulate the results, we have to introduce the function spaces the solution of Problem (1.1) belongs to.We collect the most necessary facts from the theory of these spaces, for a detailed insight we refer to the monograph [18].
Let Ω be a bounded domain with the Lipschitz continuous boundary ∂Ω and )be a measurable function.Let us define the functional (the modular) The set equipped with the Luxemburg norm is a reflexive and separable Banach space and ∞ C Ω 0 ( ) (the set of smooth functions with compact support) is dense in . The modular is lower semicontinuous.By the definition of the norm, The dual of is the space , the generalized Hölder inequality holds: Let p 1 and p 2 be two bounded measurable functions in Ω such that The variable Sobolev space is defined as the set of functions equipped with the norm Under the assumptions ∈ p C Ω 0 ( ) and ∂ ∈ Lip Ω , the Poincaré inequality holds: there is a finite constant > C 0 such that , provided ∈ p C Ω log ( ), i.e., p is continuous in Ω with the logarithmic modulus of continuity: where ⋅ ω( ) is a nonnegative function satisfying the condition For the study of parabolic Problem (1.1), we need the spaces of functions depending on

Parabolic Hölder spaces
the parabolic distance.The space of Hölder-continuous functions C Q α T , α 2 ( ), ∈ α 0, 1 ( ), is the collection of all functions with the finite norm The spaces ) are the spaces of functions with the finite norms , , , , , We will use the known properties of the Hölder spaces on smooth domains:

‖ ‖ ‖‖
• for every • the embedding • for every By C Q T 0,1 ( ), we denote the space of Lipschitz-continuous functions: a function ϕ is Lipschitz-continuous on where ⋅ ⋅ d , ( ) is the Euclidean distance in

Assumptions and main results
We assume that the exponents of nonlinearity and the coefficients in equation (1.1) satisfy the following conditions: T T T N 0,1 0 0
The notation or C 2 means that for every ∈ ∂ x Ω 0 , there exists a ball B x R 0 ( ) such that in the local coordinate system y i { } centered at x 0 with y N pointing in the direction of the exterior normal to ∂Ω at x 0 , the set ( ) can be represented as the graph of a function The symbol C stands for constants that can be evaluated through the known quantities but whose exact values are unimportant.The value of C may vary from line to line even inside the same formula.
Theorem 2.1.Assume that conditions (2.4)-(2.7)are fulfilled and , Problem (1.1) has a strong solution u.The solution satisfies the estimate with a constant C depending only on N and ∂Ω, the structural constants in conditions (2.4)-(2.7),∇ ⋅ u p 0 , Ω 0 ‖ ‖ ( ) and f Q 2, T ‖ ‖ .Moreover, the solution possesses the property of global higher integrability of the gradient: and a constant C depending on r and the same quantities as the constant in (2.9).
Theorem 2.2.Let the conditions of Theorem 2.1 be fulfilled and u be the strong solution of Problem (1.1) obtained in Theorem 2.1.Then, with a constant C depending on N, the constants in conditions Theorem 2.3.The assertions of Theorems 2.1 and 2.2 remain true for the C 2 -domains and convex domains Ω without any assumption on the regularity of ∂Ω.

Organization of the work
Section 3 is devoted to studying the problem with the regularized nondegenerate flux ∇ z u , ε ( ) and smooth data in a smooth domain Ω.The exponents of nonlinearity, the coefficients, and the data f and u 0 are approximated by sequences of smooth functions converging in the corresponding norms.A solution to the regularized problem is sought as the fixed point of the mapping generated by the linearized problem.The bulk of the section consists in checking the fulfillment of the conditions of the Leray-Schauder principle.In the Second-order regularity for parabolic equations  7 proof, we rely on the recent results on boundedness and Hölder continuity of the gradient for the weak solutions to the corresponding nonlinear problem [20].The constructed solution belongs to In Section 4, we derive the integral inequality, which allows one to estimate )‖ and the data.We use first the known integral identity, which holds for smooth functions and domains: where is the second fundamental form of the surface ∂Ω, and → ν is the unit exterior normal to ∂Ω.By straightforward computations, the integrand of the second integral is represented as the sum of and several residual terms that appear only if p depends on x.After estimating these terms and the boundary integral, we arrive at (1.10).This inequality together with another known inequality (1.9) implies higher integrability of the gradient (1.11).
In Section 5, uniform a priori estimates for the classical solutions of the regularized problems are derived.The assumptions on the growth of F allow one to estimate the nonlinear sources through the elliptic part of the equation and the lower-order norms of the solutions.In our assumptions, all terms of the regularized equation are uniformly bounded in L Q T 2 ( ).In Section 6, we prove Theorems 2.1 and 2.2.We prove first the analogs of these theorems for Problem (1.1) with the regularized flux ε .The uniform estimates allow us to choose a sequence of solutions to the regularized problems with smooth data such that the sequence itself, the derivatives of its terms, and the corresponding nonlinear terms are weakly convergent.The limits of the nonlinear terms are identified by the monotonicity of the fluxes and the pointwise convergence of the gradients together with the higher integrability property (1.11).The assertions of Theorems 2.1 and 2.2 follow after passing to the limit → + ε 0 in the family of solutions to the regularized problems.
The proof of Theorem 2.3 is given in Section 7.For a convex domain, this is done by means of approximation from the interior by a sequence of smooth convex domains, application of Theorems 2.1 and 2.2 in each of these domains, and the choice of a sequence of solutions with proper convergence properties.For the smooth convex domains, the function in (2.11) is non-positive.In this case, the first term on the right-hand side of (2.11) can be omitted, and the key estimates on the solutions to the problems in the regularized domains become independent of ∂Ω.In the case of a C 2 -domain Ω, the conclusion follows by approximation of Ω from the interior by a family of smooth domains whose boundaries are uniformly bounded in C 2 .

Regularized problem
A solution of the degenerate Problem (1.1) is obtained as the limit of the family of solutions to the following problems with the regularized flux and sources: where Throughout the text until Section 7, we assume that with some ∈ α 0, 1 ( ).

Properties of the regularized flux
Fix > ε 0 and accept the notation The function ε with > ε 0 is the regularized flux.It is straightforward to check that ε with > ε 0 possesses the following properties.
(1) Monotonicity.There is a constant For the proof, we refer to [8, Proposition 3.1].
(2) Coercivity.For every | | , we may choose = δ 1: Inequality (3.6) follows from Second-order regularity for parabolic equations  9 whence For every constant ∈ − Assume, for the sake of definiteness, that Here, we have used the inequality and Young's inequality.□

Classical solutions of the regularized problem
Let us approximate the data of Problem (3.1): We refer to [19, Sec.4] for the approximation of u 0 in the variable Sobolev space , the approximation of f is standard.The exponents and coefficients are approximated as follows: By the McShane-Whitney extension theorem [28], every continuous or Lipschitz-continuous function can be extended to the whole space in such a way that the extension preserves the modulus of continuity.The approximations for q, s, a, and → c are obtained then by mollification of the extended functions.To obtain a monotone increasing sequence p m { }, we extend p from Q T to 2 , where ϕ is the stan- dard mollifier.For all m from some m 0 on, the approximations satisfy the structural conditions (2.5)-(2.7)with the same constants except for μ 2 , which is substituted by μ.From now on, we use the abbreviate notation for the sequences of approximations satisfying (3.9): Consider Problem (3.1) with the data f m , u m 0 , and data m , and the auxiliary linear problem where ) is a given function and Proof.Let p s , m ( )be the function defined in (3.2).The assertion of the proposition will follow is we show that q v , m ( ) and ∇ s v , m ( | |) are Hölder-continuous as functions of z.For the sake of definiteness, we consider the second case.Given two points z 1 and z 2 , we denote = p z p m i im . By the Lagrange mean value theorem, The integrands are bounded by constants depending on ε, ± p m , and , .
Second-order regularity for parabolic equations  11 It follows that be the ball of radius R to be defined, and  (2) By virtue of (3.12), for every > R 0, the image of S R is bounded in ( ) ( ) and, therefore, is (3) Uniform continuity with respect to τ.Let u 1 , u 2 be the solutions of the equations = u v τ Φ , , 0 on Ω 0, .
By virtue of (3.12), ) To check that the boundary of the set S R does not contain fixed points of the mapping Φ suffices to prove that all possible fixed points belong to a ball ′ B R of a smaller radius ′ < R R. The proof amounts to deriving the a priori estimate for all possible fixed points of τ v Φ , ( ) On the one hand, u is a classical solution of the linear equation (3.11) with the smooth data On the other hand, u is a strong solution of the nonlinear Problem (3.10).
The variable nonlinearity of equation (3.10) prevents one from deriving (3.13) from the classical parabolic theory.To obtain (3.13), we make use of the recent results on the gradient regularity of weak solutions to nonlinear equations of the type (3.1).It is proven in [34] that for the solutions of equation (1.1) with a given source term in divergence form ), provided that the components of f are Hölder continuous.This local result allows one to estimate the Hölder norm of ∇u in every domain of the form ) with smooth ∂Ω and any > δ 0 - [31].This is done by means of local rectification of the boundary ∂Ω and application of the local result of [34] to the odd continuation of the solution across the lateral boundary to the exterior of the problem domain (see [31,Lemma 1.3,(2)]).
In [20], the results of [31,34] were extended to a class of equations with nonlinear sources of general form that includes (1.2) as a partial case.Let us consider the problem where , , satisfies the growth condition , , , , , const 0, the exponents , satisfy conditions 2.4 , 2.5 , and 2.6 . ) ) .
A revision of the proof of Proposition 3.
with a constant ′ C that depends on N , − p , + q , + s , T , and Ω | |, but independent of ε, τ, and m.
Second-order regularity for parabolic equations  13 Proof.Take with a constant ). Inequality (3.19) can be written as Integration in t leads to the inequality with a constant ″ C independent of u.At every point where ≥ p z 2 m ( ) , we have This observation allows one to rewrite (3.20) as To estimate the first term of t Ψ( ), we use the assumption , apply the Poincaré inequality with the constant exponent − − p μ m , and then make use of the Young's inequality: for any > δ 0, For a sufficiently small δ, after integration in t, this term is absorbed in the left-hand side of (3.22).The second term of t Ψ( ) is estimated likewise: by Young's inequality, The first integral is already estimated in (3.23).To estimate the second one, we note that the assumption and once again apply Young's inequality: , w i t h a n y 0 .
with any > δ 0 and a constant C depending on T , Ω | |, ± p , and μ, but independent of ε and m.Choosing δ sufficiently small and moving the terms containing ∇u p m | | to the left-hand side of (3.22), we arrive at (3.18).□ with a constant C depending on M, λ, N , Ω | |, and ± p and the modulus of continuity of p in Q T .
Proposition 3.5.Under the assumptions of Lemma 3.1, every weak solution of Problem (3.10) satisfies the estimate and on the same quantities as the constant ′ C in Lemma 3.1, but does not depend on ε and m.
By Lemma 3.1 and Proposition 3.4, both integrals are bounded, and the conclusion follows from Proposition 3.

□
We are now in position to derive (3.13) and estimate all possible fixed points of τ v Φ , ( ) in the norm of (3.16), by Proposition 3.5, u is bounded in Q T .Fix some > δ 0. By Proposition 3.3, there is ∈ γ 0, 1 The function w is a weak solution of the equation with a constant ′ C depending only on the data.
‖ ‖ with a constant ″ C depending only on the data.To conclude the proof, it suffices to take We summarize these arguments in the following assertion.
Proof.The existence of a classical solution u is already proven.Fix a direction = k N 1,…, , and consider the function ≡ w D u k .Differentiation of the equation for u with respect to x k shows that = w D u k satisfies the equation This is a linear uniformly parabolic equation with the coefficients and the right-hand side in Ch.III, Th.12.1], ∈ × 4 Inequalities for the regularized flux

Formulas of integration by parts
Let the conditions of Lemma 3.3 be fulfilled and u be the classical solution of Problem (3.10).For every In the local coordinates Then, by [24, Th.3.
where t is fixed and ε k ( ) denotes the kth component of ε .
Since for the smooth convex domain,

Differential inequality
The integrands of the integrals over Ω on the right-hand side of (4.2) can be transformed in the following manner.Given ∇u, we denote For every Second-order regularity for parabolic equations  17 Denote by u ( ) the Hessian matrix of u: the symmetric × N N matrix with the entries Summing up, we obtain the representation We want to show that there is a constant > C 0 such that Inequality (4.4) coincides with Inequality (3.5) in [14].Its proof can be easily adapted to our case, where . Fix a point x 0 , diagonalize the matrix u x 0 ( ( )) by means of rotation, and denote by → ζ the vector → η in the new basis: where d i are the diagonal elements of the transformed matrix.Set The function ) is continuous with respect to λ.By inequality (3.5) in [14] we know that there exists a constant C such that ), the para- meter λ is so close to one that where

( ) ( ) with the Lipschitz constant L. There exists a positive constant
where the constant K is defined in (4.1) by the second fundamental form of the surface ∂Ω.
Corollary 4.1.Let Ω be a smooth convex domain.Then, the boundary integral on the right-hand side of (4.6) can be omitted by virtue of (4.3), and the constant C in (4.6) is independent of ∂Ω.Now, we combine Inequality (4.6) with the following assertion.
Second-order regularity for parabolic equations  19 , which is equivalent to We will repeatedly use the following elementary inequality: for every > γ 0 and > σ 0 there is a constant ) such that An immediate corollary of Lemma 4.2 and Inequality (4.8) is that for every We proceed to estimate the last two terms on the right-hand side of (4.6).

Estimates on the residual terms
) are estimated as follows: by (4.7)-(4.9)and Young's inequality, and The last term on the right-hand side of (4.12) is estimated by means of the following inequality: for every > λ 0, Substituting (4.13) into (4.12), and then using (4.7) with = λ r 2 , we obtain: for every > δ 0,

Boundary integrals
with a finite constant > K 0. Thus, estimating the boundary integral in (4.2) amounts to estimating the integral of The integrals 1 and 3 are estimated as ij 6 ( ) in (4.12) and (4.13): for every ∈ δ 0, 1 ( ), By Young's inequality, where 1 is already estimated in (4.15).Gathering these estimates with a sufficiently small δ, we reformulate Lemma 4.1 as follows.
( ) ( ) with the Lipschitz constant L and and, by Corollary 4.2, Second-order regularity for parabolic equations  21 where C and ′ C depend on the same quantities as the constants in (4.16) but do not depend on ε.By Corollary 4.1, for convex domains Ω, the constants C i in (4.16) and ′ C in (4.17) are independent of ∂Ω.

A priori estimates
Throughout this section, we assume that u is the strong solution of regularized Problem (3.10) with data f m , u m 0 , and data m .The first a priori estimate is already derived in Lemma 3.1 for = τ 1.To obtain the second a priori estimate, we make use of the representation Now, we multiply (3.10 ), and apply (5.1) and the Cauchy inequality: The constant M depends on the Lipschitz constant L of p m .The last term on the right-hand side of (5.2) is estimated by virtue of (4.7), (4.9): it is sufficient to claim that ) by assumption.Using Lemma 4.3 and choosing > δ 0 sufficiently small, we obtain . By Lemma 3.1 with = τ 1, ) .By the Gagliardo-Nirenberg inequality, Moreover, we have to claim that Inequality (5.6) is stronger than (5.5).Integrating (5.4) in t and using (5.6), we obtain the inequality with an arbitrary positive δ and a constant C depending only on Proposition 5.2.Assume that the conditions of Lemma 3.1are fulfilled and

□
To estimate u t , we use (3.10) and Propositions 5.1 and 5.2: with a constant C independent of u.Gathering these estimates and recalling (3.8)-(3.9),we obtain the following assertion.
with a constant ″ C independent of ε and m.
6 Existence and regularity of strong with a constant C independent of m and ε.By virtue of (4.7), for any which yields the uniform in m and ε estimate with some positive λ.It follows from estimate (5.7) and Proposition 3.4 that with a positive constant λ and an independent of m, ε constant C.Moreover, the convergence ↗ p p m in C Q T 0,1 ( ) and (6.2) imply that for all sufficiently large m, with a constant C independent of m and ε.It follows that the sequence u m { } contains a subsequence (for which we will keep the same notation), and there exist functions u, χ, η, and ζ such that weakly in 0, ; Ω , strongly in and a.e. in , , .
Proof.Let us multiply equations (3.10) for u m and u n by = − w u u m n , integrate over Ω, and combine the results: Let us integrate this inequality over the interval T 0, ( ).Applying Hölder's inequality and (6.3) and (6.4), we estimate the last two terms of the resulting inequality by M w Q

2, T ‖ ‖
with a constant M independent of m, n.Inequality (6.7) is then continued as follows: To estimate the last term of the second inequality in (6.8), we apply the Lagrange mean value theorem (see the derivation of (3.7)) and then follow the proof of [32,Lemma 3.9].The constant C depends on which are already estimated in (6.5).
Theorem 6.1.Let conditions (2.4)-(2.7)be fulfilled, and , and ∈ ε 0, 1 ( ), Problem (3.1) has a strong solution.The solution satisfies the estimate and possesses the property of global higher integrability of the gradient: for every The constant C depends on N, ∂Ω, L, the structural constants in (2.4)- , the constant ′ C depends on the same quantities and also on r; C and C' are independent of ε.Theorem 6.2.Let the conditions of Theorem 6.1be fulfilled.If u is a strong solution of Problem (3.1), then and with C depending on the same quantities as the constant ′ C in (6.13) but independent of ε.
Proof.Let u m be the solution of Problem (3.10).For all Using (4.7), (4.9), and Lemma 5.1, we conclude that with a constant ″ C depending only on the data and independent of m and ε.It follows that there is and a subsequence of u m { } (for which we keep the same notation) such that To identify the limit Θ ij ε ( ) , we use the pointwise convergence of the sequence of gradients: for every Estimate (6.14) follows from (6.15) and (6.16).Indeed, for every with a constant ″ C independent of ε. □ Corollary 6.
with a constant C independent of ε, and there exists ) (up to a subsequence).Now, we make use of the pointwise convergence of the gradients established in the of Theorem 2.1.For every with the constant C from Theorem 6.2.□ 7 C 2 -smooth and convex domains: proof of Theorem 2.3 We begin with the case of a bounded convex domain Ω.There exists a sequence of smooth convex domains To construct such a sequence take a smooth convex exhaustion of Ω, i.e., a smooth, negative, convex function ψ such that = →∂ ψ x lim 0 x Ω ( ) , and choose for Ω k the sub-level sets of ψ: For the existence of an exhaustion function, we refer to [10].Let u k { } be the sequence of solutions of the problems , in Ω 0, , 0, on Ω 0, , , 0 , in Ω, where , and , .By Theorem 2.1, Problem (7.1) has a solution u k satisfying estimates (2.9)-(2.10).Define the functions Since each of w k satisfies the uniform estimates (2.9)-(2.10), the sequence w k { } has the convergence properties similar to (6.17): there are functions w, η, and χ such that (up to a subsequence) weakly, in 0, ; Ω , strongly, in , and a.e., in , , Moreover, by virtue of (4.17  .We now follow the proof of Lemma 6.1.Using an analog of (6.10), we find that ∇ − → ([ ] ( )) follows from the Aubin-Lions lemma.
To prove the second-order regularity, we follow the proof of Theorem 2.
and the parametrizations of ∂Ω k are uniformly bounded in C 2 , see [8,Subsec.7.4] for the choice of Ω k .Let u k { } be the sequence of solutions to Problems (7.1) in the cylinders Q T k , .The a priori estimates for u k depend on the C 2 -norms of the parame- trizations of ∂Ω k , which are independent of k.The proof is completed as in the case of the convex domain Ω.

2 ( 2 ‖
) estimate (1.3) holds with a constant C depending on f L Q T ‖ ( ) .On the other hand, by virtue of equation (1.1), it is necessary that ∈ f L Q T 2 ( ) for every solution satisfying (1

Proof.
Denote by ε i ( ) the ith component of the flux.By the mean value theorem, ∫ ∫ −

+ N 1
by a function P with the same Lipschitz constant L, take the monotone increasing sequence = in the same way.□ Since the functions f m and u m 0 have finite supports in Q T and Ω, they satisfy the first-order compatibility conditions on the set ∂ × = t Ω 0 { }.By [25, Ch.IV, Th.5.2],Problem (3.11) has a unique classical solution the boundary of the ball B R does not contain fixed points of the mapping = u Let us consider Φ as the mapping 1) Problem (3.11) with = τ 0 is the linear uniformly parabolic equation with zero initial and boundary data.By the maximum principle, the unique solution of this problem is = u 0.

2 .u u 2 1
Combining equations(3.11),we find that the difference = − w is a solution of the linear problem

β 1 2
( ) with some ∈ β 0, 1 ( ).Let u be a fixed point of the mapping τ v Φ , ( ), i.e., a strong solution of Problem (3.10) with the data τf m , τu m 0 , ∈ τ 0, 1 [ ], and data m .Since every strong solution is a weak solution in the sense of Second-order regularity for parabolic equations  15

Now, we revert, 2 (
to equation (3.11) with = v u and consider it as a linear equation with the coefficients and the right-hand side in ∕ ).The solution of Problem (3.11) belongs to

Lemma 3 . 3 .α 2 ,
If Conditions (2.4)-(2.7)are fulfilled and ∂ ∈ + C Ω then Problem (3.10) with data m has a unique classical solution ∈ the surface ∂Ω is represented by the equation = Let us denote by ξ η ; ( ) the second quadratic form of the surface ∂Ω: for every two tangent vectors → ξ and → η to ∂Ω → ν be the exterior normal to ∂Ω and → τ belong to the tangent plane to ∂Ω at the same point.Since =

10 )
By Young's inequality, this inequality is extended to the rest of the interval ⎛

Lemma 5 . 1 .α 2 .
Let conditions (2.4)-(2.7)be fulfilled and ∂ ∈ + C Ω Then, the strong solution of Problem (3.10) with the data f m , u m 0 , and data m satisfies the estimate

6. 1
Regularized problemWe begin with considering the regularized Problem (3.1).Let u m be the solution of (3.10) with the free term f m and the initial datum u m 0 satisfying(3.8),and data m .By Lemmas 3.1 and 5.1,

1 . 7 )
C independent of ∂Ω.Let us check that the limit function w is a strong solution of Problem(1.1).Let ∈ ∞ ∞ is Lipschitz-continuous in Q T , the set ∞ ∞ The sequence ∇w k {} contains a subsequence that converges to ∇w a.e. in Q T .Proof.Take an arbitrary set ⋐ ω Ω, denote = By the choice of u k 0, { } and f k , and due to (7.2), regularity for parabolic equations  31By virtue of estimates (2.9)-(2.10)for the functions u m in Q T m , and w m in Q T and the assumptions on the structure of the equation ∇ C F independent of m.Thus, Dropping the first nonnegative term on the left-hand side of (7.7), we find that for every k, using the diagonal process, we may choose a subsequence w k { } and a function Θ ij such that smooth or convex bounded domain.It is assumed ( ) is a solution of the nonlinear Problem (3.10).The existence of a fixed point will follow from the Leray-Schauder principle (see, e.g., [26, Ch.IV, Th.8.1]).Let procedure we extract a subsequence of ∇w k { } that converges to ∇w a.e. in Q T .□ By [27, Ch.1, Lemma 1.3], we conclude that = ∇ ∇