Existence of weak solutions to a certain homogeneous parabolic Neumann problem involving variable exponents and cross-diffusion

This paper deals with a homogeneous Neumann problem of a nonlinear diffusion system involving variable exponents dependent on spatial and time variables and cross-diffusion terms. We prove the existence of weak solutions using Galerkin’s approximation and we derive suitable energy estimates. To this end, we establish the needed Poincaré type inequality for variable exponents related to the Neumann boundary problem. Furthermore, we show that the investigated problem possesses a unique weak solution and satisfies a stability estimate, provided some additional assumptions are fulfilled. In addition, we show under which conditions the solution is nonnegative.

problem was considered, see also [40,49]. Furthermore, in [25] an Aubin-Lions type theorem was established, which we will also use in this paper. In addition, the author of [25] considered more general vector-fields, which are related to the parabolic p(x, t)-Laplacian, and inhomogeneities. Moreover, in [26] the existence, uniqueness and stability of a weak solution to the equation where ≥ 0 and the vector-field a(x, t, ⋅) satisfies certain p(x, t)-growth and monotonicity conditions, cf. [25], was shown, see also [14] for p =constant. Additionally, in [8] it is shown that the solutions of a similar problem may vanish in finite time even if the equation combines the directions of slow and fast diffusion, and the extinction moment is estimated in terms of the data. Further, very recently the existence of weak solutions to a homogeneous Dirichlet problem of a nonlinear diffusion equation involving anisotropic variable exponents and convection was studied in [39].

Plan of the paper
The paper is organised as follows: The rest of this sections is focused on the formulations of the problem, which we are going to study. Furthermore, we will refer some known results on nonstandard Lebesgue and Sobolev spaces, before we state some preliminary results and tools, which are needed to established our existence result. In Sect. 2, we will state our main result. Then, in Sect. 3, we prove the existence of weak solutions to the considered parabolic Neumann boundary problem using Galerkin's approximation and we derive suitable energy estimates. Moreover, in Sect. 4, we will establish under which conditions the weak solution is unique. Finally, in Sect. 5, we will prove that the solution in nonnegative, provided certain assumptions are fulfilled.

Notation and formulation of the problem
In this paper, Ω ⊂ ℝ n denotes a bounded Lipschitz domain of dimension n ≥ 2 and we write Q T ∶= Ω × (0, T) for the space-time cylinder over Ω of height T > 0 . Here, u t or t u respectively denote the partial derivative with respect to the time variable t and ∇u denotes the one with respect to the space variable x. Moreover, we denote by P Q T ∶= (Ω × {0}) ∪ ( Ω × (0, T)) the parabolic boundary of Q T and we write z = (x, t) for points in ℝ n+1 . The aim of this paper is the investigation of the following Neumann problem: u t − div(a(x, t, ∇u)) = − |u| p(x,t)−2 u, with u 0 , v 0 ∈ L 2 (Ω) , where S T ∶= Ω × (0, T) , denotes the exterior normal to the boundary Ω and d i > 0 , i = 1, 2 , ≥ 0 with which is the case if Furthermore, the vector-fields A i (x, t, ⋅) are assumed to be Carathéodory functions and satisfy the following coercivity, growth and monotonicity conditions: p(x,t)−1 , which we will define later. In addition, the functions k (⋅) , k = 1, … , 4 are measurable functions satisfying Moreover, the growth exponent p ∶ Q T → [2, ∞) satisfies the following conditions: There exist constants 1 and 2 , such that hold for any choice of z 1 , z 2 ∈ Q T , where ∶ [0, ∞) → [0, 1] denotes a modulus of continuity. More precisely, we assume that (⋅) is a concave, non-decreasing function with lim ↓0 ( ) = 0 = (0). Moreover, the parabolic distance is given by In addition, for the modulus of continuity (⋅) we assume the following weak logarithmic continuity condition Similarly, the exponent q(x, t) is assumed to fulfil the conditions: (1.10) 1 < q(z) ≤ 2 and |q(z 1 ) − q(z 2 )| ≤ (d P (z 1 , z 2 ))

Function spaces
The spaces L p (Ω) , W 1,p (Ω) and W 1,p 0 (Ω) denote the usual Lebesgue and Sobolev spaces, while the nonstandard p(z)-Lebesgue space L p(z) (Q T , ℝ k ) is defined as the set of those measurable functions v: The set L p(z) (Q T , ℝ k ) equipped with the Luxemburg norm becomes a Banach space. This space is separable and reflexive, see [4,19]. For elements of L p(z) (Q T , ℝ k ) the generalised Hölder's inequality holds in the following form: we have see also [4]. Moreover, the norm ‖ ⋅ ‖ L p(z) (Q T ) can be estimated as follows Notice that we will use also the abbreviation p(⋅) for the exponent p(z). Next, we introduce nonstandard Sobolev spaces for fixed t ∈ (0, T) . From assumption (1.8) we know that p(⋅, t) satisfy |p(x 1 , t) − p(x 2 , t)| ≤ (|x 1 − x 2 |) for any choice of x 1 , x 2 ∈ Ω and for every t ∈ (0, T) . Then, we define for every fixed t ∈ (0, T) the Banach space W 1,p(⋅,t) (Ω) as equipped with the norm In addition, we define W Here, it is worth to mention that the notion on the lateral boundary of the cylinder Q T , i.e. u ∈ W p(⋅) g (Q T ) . In addition, we denote by W p(⋅) (Q T ) � the dual of the space W p(⋅) where ∇ i v i has to be interpreted as a distributional derivate. By we mean that there exists w t ∈ W p(⋅) (Q T ) � , such that see also [19]. The previous equality makes sense due to the inclusions which allow us to identify w as an element of W p(⋅) (Q T ) � . Finally, we are in the situation to give the definition of a weak solution to the parabolic problem (1.1): is called a weak solution of (1.1) if and only if (u, v) ∈ (L ∞ (0, T;L 2 (Ω)) ∩ W p(⋅) (Q T )) 2 and for every test function i ∈ C ∞ 0 (Ω × [0, T)) , i = 1, 2 , the following equalities hold: where (1.3) and the initial value conditions u( (1.14) are fulfilled.

Preliminary results and tools
To derive our existence result, we will need the following Poincaré type estimate, which is a modification of the Poincaré type estimate from [22,Lemma 3.9].
After proving the energy estimate for the (weak) solutions, we will derive from Lemma 1.2 the needed L p(⋅) (Q T )−bounds for the approximate solution to (1.1). This together with the following Aubin-Lions type Theorem [25, Theorem 1.3] will guarantee the strong convergence of the approximate solution to the solution in L p(⋅) (Q T ) . The Aubin-Lions type Theorem reads as follows: Moreover, the next two lemmas, which are useful tools when dealing with p-growth problems, we will need to prove the uniqueness of the weak solution to system (1.1). Therefore, we define a function by Moreover, we cite the following lemma from [33, Lemma 2.1], which is established for the case ≥ 0 in [31] and in the case 0 > > −1 in [33]. We are using Lemma 1.6 only for 1 < q(⋅) ≤ 2 . However, Lemma 1.6 holds true for all 1 < q(⋅).

Statement of results
In this section we state the main results of this paper. The existence result reads as follows: where the initial values are given. Furthermore, suppose that growth exponent , to the homogeneous Neumann problem (1.1), which satisfies the following energy estimate: iii) and in case that we have for all k = 1, … , 4 and (x, t) ∈ Q T , and additionally is satisfied, then system (1.1) possesses a unique weak solution.
Please compare the uniqueness result from [10], here a similar restriction occurs due to the term |u| q(⋅)−2 u . In addition, one can conclude from the proof of the Theorem 2.2 immediately the following stability result:

Proof of the existence result
In this section, we prove our existence result utilising Galerkin's approximations, cf. [9,25,50].

Proof of Theorem 2.1
The construction of a sequence of Galerkin's approximations is as follows: First of all, we want to recall that Ω ⊆ ℝ n is an open, bounded Lipschitz domain and due to the dense embeddings W 1,s (Ω) ⊂ L 2 (Ω) and (L 2 (Ω)) � ⊂ W −1,s � (Ω) one has the inclusions where the injections are compact. Note that W 1,s 0 (Ω) ⊂ W 1,s (Ω) also holds true. Furthermore, it is known that for 1 < 1 ≤ s ≤ 2 < ∞ the space L s (Ω) is a separable and reflexive Banach space, and similarly, W 1,s (Ω) is a separable and reflexive Banach space. In the case of Dirichlet boundary values one would consider , which is an orthonormal basis in L 2 (Ω) , while here one can follow the approach from [13], i.e. one considers the spectral problem: Find f ∈ W 1,2 (Ω) and ∈ ℝ such that where ̂ is the unit outward normal. Then, problem (3.1) possesses a sequence of nondecreasing eigenvalues { i } ∞ i=1 and a sequence of corresponding eigenfunctions forming an orthogonal basis in W 1,2 (Ω) and an orthonormal basis in L 2 (Ω) ), see also [35]. Next, fix a positive integer m and define the approximate solution to problem (1.1) in the following way: where the coefficients c

Then, we choose test functions
Note that t and t̃ exist, since the coefficients i (t) and ̃i(t) lie in C 1 ([0, T]) . Thus, we have and From this we can conclude that with a constant c 1 = c 1 (a 0 , a 1 , d 1 , d 2 , 1 , 2 , 1 , 2 , ‖u 0 ‖ L 2 , ‖v 0 ‖ L 2 , �Q T �, L 1 , ‖h 1 ‖ L p � (⋅) ) , where we applied the generalised Hölder's inequality (1.11), the growth condition (1.5), the condition (1.7), (1.12), the fact p � (x, t) ≤ 2 ≤ p(x, t) and the energy estimate (3.5). Similarly, we can deduce that with a constant where we also used the Poincaré type estimate (1.17) with u (m) Summarising, we have the weak convergences for the sequences u (m) and v (m) (up to a subsequence): Moreover, by Theorem 1.4 we can conclude that the sequences u (m) and v (m) (up to a subsequence) converges strongly in L p(⋅) (Q T ) to some function u, v ∈ W(Q T ) with u Ω = v Ω = 0 . Thus, we have the desired convergences In addition, the growth assumption on A i (z, ⋅) and the estimate (3.5) imply that the sequences A 1 (z, ∇u (m) ) m∈ℕ and A 2 (z, ∇v (m) ) m∈ℕ are bounded in L p � (⋅) (Q T , ℝ n ) . Consequently, after passing to a subsequence once more, we can find limit maps The next aim is to show that By the method of construction [7], we have from (3.2) and (3.3) for all test function i ∈ W s (Q T ) , i = 1, 2 with s ≤ m for an arbitrary fixed m ∈ ℕ that and Then, passing to the limit m → ∞ we get and ⎧ ⎪ ⎨ ⎪ ⎩ u (m) ⇀ * u and v (m) ⇀ * v weakly* in L ∞ (0, T;L 2 (Ω)), ∇u (m) ⇀ ∇u and ∇v (m) ⇀ ∇v weakly in L p (⋅) for every i ∈ W s (Q T ) , i = 1, 2 . According to the monotonicity assumption (1.6), we also know that we can deduce by subtracting (3.9) and (3.10) from (3.12) and (3.13), respectively, and passing to the limit m → ∞ that for all i ∈ W s (Q T ) , i = 1, 2 . Then, choosing 1 = u ± 1 and 2 = v ± 2 with arbitrary i ∈ W p(⋅) (Q T ) finally yields Passing to the limit ↓ 0 , then implies The last step in our existence proof is to check if the initial value condition is satisfied, which is similar to [9,13]. Consider functions . Then, choose test functions from (3.14) with i (⋅, T) = 0 . Thus, we can conclude from (3.7) and (3.8), integrating by parts and passing to the limit m → ∞ the following: where we used u (m) (⋅, 0) → u 0 as m → ∞ , cf. [25], and similarly On the other side hand we know from (3.9) and (3.10) that Finally, since i are arbitrary we have that u(⋅, 0) = u 0 and v(⋅, 0) = v 0 , which completes the proof. ◻ Remark 3.1 There are some additional assumptions one can make to weaken the condition on the exponent q(x, t): i) For a modified version of problem (1.1), i.e.
we do not need Gronwall's inequality to derive the energy estimate (3.5), since we would have for any 1 < q(x, t) < ∞ , cf. Lemma 1.6, instead of Thus, we don't need the restriction 1 < q(x, t) ≤ 2. ii) An other approach would be the following: Assume that there exist constants q − and q + , such that 1 < q − ≤ q(x, t) ≤ q + ≤ p(x, t) , then we can conclude that with constants c 1 , c 2 and c 3 dependent on (n, q − , q + , diam(Ω), 1 , 2 ) , where we used Hölder's, Poincaré's and Young's inequality to derive this estimate. To be able to absorb these terms on the left-hand side the structure constants of system (1.1) have to satisfy where Thus, we don't need again the restriction 1 < q(x, t) ≤ 2 , but other restrictions on the system coefficients.

Proof of the uniqueness result
Now, we are in the situation to prove the uniqueness of the weak solution to problem (1.1) according to Theorem 2.2.

Proof of Theorem 2.2
For the proof of uniqueness, we assume that there exist two pairs of solutions (u, v) and (u 1 , v 1 ) with the same initial value (u 0 , v 0 ) . Therefore, we choose 1 = u − u 1 and 2 = v − v 1 as admissible test functions. i) Subtracting the weak formulation for (u 1 , v 1 ) from the weak formulation for (u, v), [cf. (1.14) & (1.15)] and using integration by parts, we get where we applied the monotonicity condition (2.2) and (1.7). Similarly, we have Combining these estimates, using Cauchy's and Hölder's inequality, we get Proof of Lemma 2. 3 We assume that there exist two pairs of solutions (u, v) and (u 1 , v 1 ) with the different initial values (u 0 , v 0 ) ∈ (L 2 (Ω)) 2 and (u 1 0 , v 1 0 ) ∈ (L 2 (Ω)) 2 . Then, following the proof of Theorem 2.2, we can conclude for = 0 that which implies for a.e. t ∈ [0, T) . ◻

Proof of the nonnegativity of the weak solutions
Our finally aim is to prove of the nonnegativity of the weak solutions.