Existence of weak solutions of parabolic systems with p , q-growth

We consider evolutionary problems associated with a convex integrand f : T × R Nn → [0, ∞), which is α-Hölder continuous with respect to the x-variable and satisfies a non-standard p, q-growth condition. We prove the existence of weak solutions u : T → R N , which solve ∂t u − div ∂ζ f (x, t, Du) = 0 weakly in T . Therefore, we use the concept of variational solutions, which exist under a mild assumption on the gap q − p, namely 2n n + 2 < p ≤ q < p + 1. For 2n n + 2 < p ≤ q < p + min{2, p} α n + 2 , weprove that the spatial derivative Du of a variational solution u admits a higher integrability and is accordingly a weak solution.


Abstract.
We consider evolutionary problems associated with a convex integrand f : T × R N n → [0, ∞), which is α-Hölder continuous with respect to the x-variable and satisfies a non-standard p, q-growth condition. We prove the existence of weak solutions u : T → R N , which solve ∂ t u − div ∂ ζ f (x, t, Du) = 0 weakly in T . Therefore, we use the concept of variational solutions, which exist under a mild assumption on the gap q − p, namely 2n n + 2 < p ≤ q < p + 1.
For 2n n + 2 < p ≤ q < p + min{2, p} α n + 2 , we prove that the spatial derivative Du of a variational solution u admits a higher integrability and is accordingly a weak solution.

Introduction and statement of the results
In this paper we are interested in the existence of solutions of parabolic systems with p, q-growth of the type (1.1) W 1,q loc , that possess an uniform W 1,∞ -bound and sub-converge to a W 1,∞ loc -solution of the original p, q-growth problem. For more details we refer to [5,8,9,[18][19][20][21].
In the elliptic setting, a second approach was introduced in [12]. Therein, regularity results for functionals of the form with a convex integrand f : R N n → [0, ∞), satisfying a non-standard p, qgrowth, are established. Due to the coercivity of the integrand, the gradient of the minimizer u lies in L p . The aim is to establish, that minimizers admit a gradient in L q loc . Therefore, one tests the corresponding Euler-Lagrange system with a finite difference of u and obtains a fractional differentiability of Du. At this stage it is not clear that the minimizer is also a solution to the Euler-Lagrange system. Hence one has to perform an approximation procedure. By using fractional Sobolev embeddings and a finite iteration, the desired higher integrability of Du can be deduced. Initially, this results holds only for the regularized problem, but it is possible to show, that these minimizers sub-converge to a minimizer of the original functional.
In [13], this method was extended for functionals, where the integrand f can additionally depend on x. It is assumed that f is α-Hölder continuous with respect to the x-variable, but not differentiable. Again fractional Sobolev spaces are used, to obtain a higher integrability for the gradient of the minimizer. Although it is not possible to differentiate the integrand, the Hölder continuity of f provides a certain kind of fractional differentiability for the gradient of minimizers u. Of course, a stronger assumption, depending on α, on the difference between p and q as in [12] is needed to show the desired higher integrability of Du. For more information to this topic we refer to [7,16,24,25].
Here we are interested in existence and regularity results for parabolic systems with p, q-growth. In this setting a variational approach was developed in [4]. Therein, the notion of variational solutions, which was introduced by Lichnevsky and Temam in [15] in the context of evolutionary minimal surface equations, is adapted. The advantage of these solutions is, that the existence can be established under mild assumptions on the convex integrand, which is independent on x and t, and on p and q, namely After having the existence at hand, a parabolic version of fractional Sobolev spaces is used to achieve the higher integrability property Du ∈ L q loc in the case p ≥ 2. Moreover, higher integrability results via differentiability and interpolation in the parabolic case are obtained in [2]. Therein, Lipschitz regular integrands f (x, Du) with p(x, t)-growth are considered and the a priori estimates are proven only by using the fact that the vector fields satisfies a p, q-growth condition. The method of fractional differentiability for parabolic systems has also been used in [11].
The aim of this paper is to establish the existence of weak solutions to parabolic systems of the form (1.1), where the integrand f satisfies a non-standard p, qgrowth condition and is only Hölder continuous with respect to the x-variable. Note that besides measurability, we do not need any other assumption for the time variable. Existence results for variational solutions, can be gained in the same way as in [4]. The main effort of this works persists in proving a higher integrability result for the spatial gradient. This is accomplished by proving a suitable Caccioppoli type inequality for the regularized problem, and afterwards the higher integrability is gained by the parabolic fractional Sobolev embedding, where the condition is required. We also treat the singular case 2n n+2 < p < 2. To deal with it, we have to overcome some problems. First, the Caccioppoli inequality can not directly be applied to the spatial gradient. Here, we have to make use of the V -function, which interpolates between quadratic and p-growth. The second problem is the appearance of quadratic terms of Du. Since p < 2, it is not clear that Du ∈ L 2 loc holds. But with the help of an interpolation argument it is possible to absorb the quadratic term and to handle these problems.

The setting
We consider Cauchy-Dirichlet problems of the type where u : T ⊂ R n+1 → R N with n ≥ 2 and N ≥ 1, can be a vector valued function. Here, denotes a bounded domain in R n with n ≥ 2 and for T > 0, T := × (0, T ) denotes the space-time cylinder. The parabolic boundary of T is denoted by With Du we mean the spatial gradient, and ∂ t u stands for the differentiation with respect to the time variable. The function f : It is an easy consequence, that also holds (c.f. [18,Lemma 2.1]). For the boundary data g we assume that the following regularity assumptions hold true: where p = p p−1 denotes the Hölder conjugate of p. Note that p (q − 1) > q. In the following we will use the notation u ∈ L p (0, T ; W

The main result
Now, we state our existence result for the parabolic Cauchy-Dirichlet problem (1.2) and start with the definition of a weak solution, which has been already used in a similar way in [4]: with u(·, 0) = g(·, 0), is called a weak solution of the parabolic system (1.2) if and only if For weak solutions in the sense of Definition 1.1, we prove the following existence result: and (1.4) and further, that g is as in (1.6). Then there exists a weak solution with u(·, 0) = g(·, 0) of the parabolic system (1.2). Moreover, there exist constants χ = χ(n, p, q, α) andχ =χ(n, p, q, α) such that for any cylinder Q R (z 0 ) T the quantitative estimate holds for p ≥ 2 and for p < 2 we have and a constant c = c(n, q, p, L , ν, α, R). Remark 1.3. Now, we compare the elliptic bound for the difference between p and q with the parabolic bound for p ≥ 2. In the stationary case in [13], the assumption is needed for all p > 1, while for evolutionary problems is required for p ≥ 2. This seems to be the natural bound, since one must replace n by n + 2 and must take the parabolic deficit 2 p for p ≥ 2 into account. However, for p < 2 an interesting phenomena appears. If we take the scaling deficit 2 p/( p(n + 2) − 2n), the maximal difference would be α( pn + 2 p − 2n)/(2(n + 2)), which is smaller than αp/(n + 2). However, we can prove the better bound, which is also stable for p 2.
The first step of the proof is to show that there exists a variational solution The existence of such solutions can be established under the assumption cf. Theorem 3.4. For more details, we refer to Sect. 3.

Model examples
Here, we give some examples for integrands f , which are discussed in this paper. For instance, we can consider functions For the function h, we can take for example We could also consider functions with anisotropic growth, i.e.
we require only the weaker assumption 0 ≤ a(x) ≤ L, where a is again a α-Hölder continuous function.

Preliminaries
Here, we state some usefull tools, that will be needed throughout the paper.

Auxiliary tools and notations
With we denote the open ball in R n with centre x 0 and radius ρ, and is the parabolic standard cylinder. In order to absorb certain terms, we will use the following iteration Lemma, which can be found for instance in [14, Lemma 6.1].
We need the auxiliary function V : and from the last Lemma, we can conclude (cf. [1, Lemma 2.2]): for arbitrary ζ, η ∈ R n , not both zero if μ = 0.
Another important tool is the next interpolation inequality, which is a consequence of Gagliardo-Nirenberg's inequality (see [23,Lemma 3.2] ).

Lemma 2.4. Assume that the function
for some exponents 1 ≤ p < ∞. Then there holds, for every radius ρ ∈ (r/2, r ), with a constant c depending on n, k, p and q.

Fractional Sobolev spaces
Now we state some results for parabolic fractional Sobolev spaces. The embedding for such spaces will play a crucial part in the proof, since it provides higher integrability properties. We will only be concerned with the parabolic case, for more information for elliptic fractional Sobolev spaces see for instance [3,6].
We say that u ∈ L p (0, is finite for any multiindex β ∈ N 0 with |β| = k. Analogous to the elliptic setting we define the norm The next Lemma provides an embedding result for fractional parabolic Sobolev spaces and is proved in [4, Lemma 6.5]. ) for any 0 < ϑ < ρ and moreover, the quantitative estimate holds true with a constant c = c(n, μ, θ, r, p, s, 1/(ρ − ϑ)).

Existence of variational solutions
In this section, we prove the existence of variational solutions. In [4] such a result has already been shown for integrands, which do not depend on x or t. But the techniques are applicable in our case, too. Thus we will only describe the notion of variational solutions and give a sketch of the proof. The existence of variational solutions can be shown under much weaker assumptions, than the existence of weak solutions. Here, the integrand f must only fulfil the following growth conditions: whenever ζ, η ∈ R N n and for some 0 < ν ≤ 1 ≤ L and μ ∈ [0, 1].
To give the precise definition of variational solutions, we introduce a notion of weaker continuity with respect to time. Here ·, ·, denotes the duality pairing between X and X .
Proof. Since the proof is essentially the same as the one of Theorem 2.4 in [4], we will only give a sketch of the proof.
Step 1: First, we consider the regularized integrand . Then, f ε satisfies a standard q-growth condition and [17] ensures the existence of a unique weak solution to the Cauchy-Dirichlet problem Step 2: Next, we prove a suitable energy bound for u ε . Therefore, we take ϕ = (u ε − g) as testing function in the weak formulation (note, that this is only possible on a formal level) and with help of the growth conditions (3.1), we get the following energy bound sup t∈(0,T ) |u ε (u(·, t))| 2 dx + T |u ε | p + |Du ε | p dz ≤ c(ν, L , q, p, , g). (3.3) Step 3: Using the energy bound (3.3) and the fact, that u ε is a weak solution, we get for any 0 < t 1 < t 2 < T and ϕ ∈ C ∞ 0 ( × (t 1 , t 2 )) 1 ,t 2 )) .
This and a density argument guarantees that holds true for any s 1 , s 2 ∈ (t 1 , t 2 ) and > n+2 n . But this is the desired weak continuity property with respect to the time variable for u ε .
Step 4: The weak solutions are also variational solutions, which can be easily deduced by testing the weak formulation with ϕ = v − u ε .
Step 5: In order to prove, that there exists a variational solution, we have to pass the limit ε ↓ 0. The energy bound (3.3) and (3.4) ensure the existence of a function u ∈ L p (0, T ; W 1, p ( , R N )) such that for a (not re-labelled) subsequence. Since u ε is already a variational solution for every ε > 0, we can pass to the limit ε ↓ 0 in (3.2). Note, that the functions u ε belong to space C 0 ([0, T ]; L 2 ( , R N )), but they loose this property in the limit ε ↓ 0 and u belongs only to the space C ω ([0, T ]; L 2 ( , R N )).
Step 6: It remains to show, that there exists only one variational solution. To this end, we assume that there exist two different solutions u 1 and u 2 . If we choose v = (u 1 + u 2 )/2 as comparison map in (3.2), we get a contradiction and the desired claim follows. Note, that this choice for the comparison map is only possible on a formal level, because the functions u 1 and u 2 do not possess a time derivative in L p (0, T ; W −1, p ( , R N )). Therefore, one has to use a mollification in time to make the calculations rigorous.

A local L q -estimate for the Spatial gradient
This section contains the main effort of this work. Here, we show the higher integrability for the spatial gradient Du. To be more precisely, we first assume that Du ∈ L q loc ( T , R N ) holds, and prove that the L q loc ( T , R N )-norm of Du can be estimated only in terms of the L p ( T , R N ) of Du. This result can later on be used in an approximation scheme. For the approximating sequence, the higher integrability is known and the results from this section, ensures the higher integrability of variational solutions.
First we define and start with a Caccioppoli-type inequality.
With the Caccioppoli type inequality at hand, we can prove the desired higher integrability for the spatial gradient. To this end, we will make use of the fractional Gagliardo-Nirenberg inequality (Lemma 2.5).

Lemma 4.2. Let
be a weak solution to (1.2), where (1.3) holds. If p ≥ 2, then there exists a constant χ = χ(n, q, p, α) such that for every parabolic cylinder Q R (x 0 ) T there holds and for p < 2 there exists a constantχ =χ(n, q, p, α) such that for every parabolic cylinder Q R (x 0 ) T there holds 1 + |u| p + |Du| p dz (4.5) and N z 0 ,R := and a constant c = depending only on n, q, p, L, ν, α and R.
Since q < p + 2α n + 2 < p + 2α n + 2 − α by assumption we can choose s ∈ (q, p + 2α n+2−α ). For δ > s q > 1, where δ will be chosen later, we infer from (4.8) (4.9) Next, we want to absorb the term involving the L s -norm of Du from the right-hand side into the left, so that there remain only terms with the L p -norm of Du on the right-hand side. Therefore, we have to choose δ and s in such a way that holds, but this is equivalent to If we choose it is sufficient to show that we can find s ∈ (q, p + 2α n+2−α ) satisfying where we note that δ > s q holds. Since there exists ε = ε(α, n) > 0 such that is true. Moreover, there exists s 0 ∈ (q, p + 2α n+2−α ) such that (4.9) is true for all Hence, by (4.11), we obtain which means, we can choose δ and s such that (4.10) and s > s q holds. With this choice in (4.9), we can apply Young's inequality and obtain for some exponent χ depending on n, p, q, and α. Lemma 2.1 allows to absorb the term involving the L s -norm of Du from the left-hand side into the right and yields that and dividing the last inequality by |h| 2 leads to where we used Young's inequality with 4/( p(n+2)−2n), 2/(2− p) and 4/(2− p)n.
As in the case p ≥ 2, we want to absorb the term involving the L s -norm of Du on the right-hand side. Therefore, we have to choose δ and s in such a way that holds, but this is equivalent to If we choose δ = s− p q− p , we only need to find s ∈ (q, p + αp n+2−α ) satisfying where we note that δ > s q holds. Since there exists ε = ε(α, n, p) > 0 such that for some exponentχ depending only on n, p, q and α. Lemma 2.1 gives for some s ∈ [s 0 , p + αp n+2−α ). This finishes the proof of the Lemma.

Proof of Theorem 1.2
In this section we complete the proof of Theorem 1.2. Therefore, we regularize the functional and obtain variational solutions u ε , which also solve the associated parabolic system. Lemma 4.2 guarantees an L q loc -bound for the spatial gradient of u ε . In the limit ε ↓ 0, this property can be transferred to the variational solution u and hence it also a weak solution.
Proof of Theorem 1.2. The procedure will be the same as in Sect. 7 of [4], so we will only give a sketch of the proof.
For every fixed ε, ∂ ζ f ε satisfies a standard q-growth condition and we obtain a unique weak solution u ε ∈ L q 0, T ; W 1,q ( , R N ) ∩ C 0 [0, T ], L 2 ( , R N ) to the parabolic Cauchy-Dirichlet problem Step 2: In the following we want to pass to the limit ε ↓ 0. Since we perform the same approximation schema as in Sect. 3  As in Sect. 3, we can only show that u ∈ C ω ([0, T ]; L 2 ( , R N )) holds, although u ε ∈ C 0 ([0, T ]; L 2 ( , R N )) is true for every ε > 0.
Step 3: With help of [26,Theorem 6], we obtain ⎧ ⎨ ⎩ u ε → u strongly in L 2 ( T , R N ) and L q (Q 0 , R N ) for any Q 0 T , Du ε → Du strongly in L p (Q 0 , R N n ) for any Q 0 T , u ε (·, t) → u(·, t) strongly in L 2 (O, R N ) for any O and any t ∈ (t 1 , t 2 ).
Step 4: The convergence results allow us to pass to the limit ε ↓ 0 in the weak formulation for u ε , which implies that u is also a weak solution to the Cauchy-Dirichlet problem (1.2). The bounds of Lemma 4.2 can be transferred from u ε to u, which completes the proof.