On the Boundary Value Condition of an Isotropic Parabolic Equation

The well-posedness problem of anisotropic parabolic equation with variable exponents is studied in this paper. The weak solutions and the strong solutions are introduced, respectively. By a generalized Gronwall inequality, the stability of strong solutions to this equation is established, and the uniqueness of weak solutions is proved. Compared with the related works, a new boundary value condition, QN i=1aiðx, tÞ = 0, ðx, tÞ ∈ ∂Ω × 1⁄20, T , is introduced the first time and has been proved that it can take place of the Dirichlet boundary value condition in some way.


Introduction
In this paper, we mainly pay attention on the stability of solutions to the following anisotropic parabolic equation with variable exponents: with the initial value and the boundary value condition Here, a i ðx, tÞ ≥ 0, p i ðx, tÞ > 1, a i ðx, tÞ ∈ Cð Q T Þ, p i ðx, tÞ ∈ Cð Q T Þ, and Ω ⊂ ℝ N is a bounded domain with the smooth boundary ∂Ω, Q T = Ω × ð0, TÞ.
When p i ðx, tÞ = p is a constant, i = 1, 2, ⋯, N, equation (1) arises in the mathematical modelling of various physical processes such as flows of incompressible turbulent fluids, gases in pipes, and processes of filtration in glaciology. The equation in this case has been studied widely [1][2][3][4][5]. When p i ðx, tÞ = pðx, tÞ is a measurable function, i = 1, 2, ⋯, N, equation (1) is similar with the equation with the type which arises in the phenomena of electrorheological fluids [6,7]. The existence of solutions of the initial-boundary value problem to this equation can be found in [8][9][10][11]. Also, one can refer to [12][13][14][15][16][17][18] for some other related works. If a i ðx, tÞ = a i ðxÞ and satisfies the well-posedness problem of the following equations has been studied by the second author in recent years [19][20][21]. Instead of boundary value condition (3), only a partial boundary value condition is imposed, where ∑ p ∈ ∂Ω is a relatively open subset which has different expression according to different kinds of f ðx, t, u,∇uÞ and sometimes is just an empty set [19][20][21]. Compared with [19,20] and [21], since the diffusion coefficient a i ðx, tÞ and the variable exponent p i ðx, tÞ both depend on the time variable t, equation (1) has a wider applications than equation (6), and in mathematical theory, there are some essential difficulties to be overcome. More than that, instead of (5), we only assume that and do not require that which is similar as (5) in [19][20][21].
To see the essential difference between (9) and (10), let us give a special case of equation when N = 2, Ω ⊂ ℝ 2 , ∂Ω = Γ 1 ∪ Γ 2 , and Γ 1 and Γ 2 are relatively open subset of ∂Ω, Γ 1 ∩ Γ 2 = ∅. Consider the equation where then (9) is true, i.e., More precisely, for example, However, in (12) and (13), This fact makes us feel that only under the boundary value condition (3), the uniqueness (or the stability) of weak solutions to equation (11) can be true. The following works seem to supply more evidences. One is [22] in which the equation is studied. The others are the equations arising from the double phase obstacle problems where aðxÞ + bðxÞ > 0, which have gained a wide attention in recent years, one can refer to [23,24] and the references therein. In these papers, the boundary value condition (3) is imposed without exception.
The main dedication of this paper is that the stability of weak solutions to equation (11)(in general, (1)) can be established independent of boundary value condition (3). Such a conclusion totally overthrows our imagination. In theory, condition (9) can take place of boundary value condition (3) is found the first time. In applications, condition (9) reflects a synthesized effect of an anisotropic diffusion process.
This paper is arranged as follows. In Section 1, we have given a simple introduction. In Section 2, we will introduce the definitions of weak solution and strong weak solution, respectively, quote some basic lemmas, and give the main results. In Section 3, we will study the stability of weak solutions to equation (1) with the new boundary value condition (9). In Section 4, we will study the uniqueness of weak solution to equation (1) independent of the boundary value condition (7). In Section 5, we will give the outline of the proof on the existence of strong solutions.

Definitions and Main Results
We denote First of all, let us introduce the definition of solutions.
and for any function φ ∈ C 1 0 ðQ T Þ, This definition of weak solution is similar as that defined in [20], where a i ðx, tÞ = a i ðxÞ, p i ðx, tÞ = p i is a constant. Also, we can prove the existence of weak solutions similar as that defined in [20], so we do not repeat the details in this paper. As an improvement from the existing result in [20], we introduce the following definition.

Definition 2.
A function uðx, tÞ is said to be a strong solution of equation (1) with the initial value (2), if and for any function φ ∈ C 1 0 ðQ T Þ, u satisfies (21) and the initial value is satisfied in the sense The proof of the existence of strong solution will be given at Section 5 of this paper. Since a i ðx, tÞ is positive when x ∈ Ω, (22) means that u t and ∇u exist almost everywhere in Q T . This is the reason that we call uðx, tÞ as a strong solution of equation (1). Moreover, from Definition 2, for all φðx, tÞ ∈ L p + ð0, T ; W 1,p + 0 ðΩÞÞ, we still have the integral equality (21), which implies that Thus, if uðx, tÞ is a strong solution of equation (1), then it is a weak solution.

Journal of Function Spaces
Basing on Lemma 3, by generalizing the Gronwall inequality, we will prove the following stability theorems, in which the initial values satisfy Theorem 4. Let uðx, tÞ and vðx, tÞ be two strong solutions of (1) with the initial values u 0 ðxÞ and v 0 ðxÞ, respectively; p − > 1 and a i ðx, tÞ ∈ Cð Q T Þ satisfy (8) and (9) and If for η small enough, where for any t ∈ ½0, TÞ, Ω ηt = fx ∈ Ω : ð Q N j=1 a j ðx, tÞÞ > ηg. In this paper, the constant cðTÞ represents that c depends on T. If we only want to prove the uniqueness of weak solutions, condition (34) is not necessary; we have the following result. (8) and (9) and If uðx, tÞ and vðx, tÞ are two strong solutions of equation (1) with the initial values u 0 ðxÞ and v 0 ðxÞ, respectively, then for any Ω 1 ⊂ ⊂Ω, which implies that the uniqueness of weak solution is true.
One can see that both Theorem 4 and Theorem 5 imply the uniqueness of solution is true. However, in Theorem 5, the convection function b i ð·, x, tÞ is independent of the diffusion coefficient a i ðx, tÞ, so as a uniqueness theorem, it is a better than Theorem 4.

The Stability of Strong Solutions Independent of the Boundary Value Condition
For small η > 0, let Obviously, h η ðsÞ ∈ CðℝÞ, and At first, we give a generalization of the Gronwall inequality.
Secondly, we give the proof of Theorem 4.
Proof of Theorem 4. Let uðx, tÞ and vðx, tÞ be two strong solutions of equation (1) with the initial values u 0 ðxÞ and v 0 ðxÞ, respectively.
Using Lemma 6, we have Journal of Function Spaces Then by the arbitrary of τ, This lemma can be generalized from of Lemma 2.2 in [28] simply; we do not give the details here.
By a process of limitation, we may choose φ = χ ½τ,s Q N j=1 a β j j ðu − vÞ as a test function. Denoting that Q τs = Ω × ½τ, s, then, At first, we have Here, we have used the fact that ja x i j ≤ c and p 1i = p + i or p ′ 1i has a similar meaning. Now, if we denote that Q 1 = fðx, tÞ ∈ Q T : 1 < pðx, tÞ < 2g and Q 2 = fðx, tÞ ∈ Q T : pðx, tÞ ≥ 2g, by that β i ≥ 2, we have and by the Hölder inequality, where k < 1.
By this theorem, Theorem 5 is true clearly.

The Strong Solutions Dependent on the Initial Value
For the completeness of the paper, we will give a basic theorem about the existence of strong solutions.
Here, b is ðs, x, tÞ = ð∂b i ðs, x, tÞÞ/∂s. Before we give the proof of Theorem 9, we would like to point out that condition (86) is just a sufficient condition; we also can use other conditions to replace them. For example, when a i ðx, tÞ ≡ aðxÞ, p i ðx, tÞ = p i , by the conditions Theorem 9 had been obtained in [19].
Proof of Theorem 9. Consider the following regularized problem Here, u ε0 ∈ C ∞ 0 ðΩÞ, ju ε0 j L ∞ ðΩÞ ≤ ju 0 j L ∞ ðΩÞ , and j∇u ε0 j converges to j∇u 0 ðxÞj in L p + ðΩÞ. It is well-known that the above problem has a unique weak solution u ε ∈ L ∞ ðQ T Þ and a i ðx, tÞju x i j p i ðx,tÞ ∈ L 1 ðQ T Þ [8], and where Similar as with Theorem 2.1 of Chapter 2 in [1] (also, one can refer to [19] in which p i ðx, tÞ = p i is just a constant), we are able to show that for any φ ∈ C 1 0 ðQ T Þ. At last, by a process of limit, we can choose the test function φðx, tÞ = χ ½t 1 ,t 2 ϕðxÞ in which ϕðxÞ ∈ C ∞ 0 ðΩÞ and χ ½t 1 ,t 2 is the characteristic function of ½t 1 , t 2 ⊂ ð0, TÞ. Then, Let t = t 2 and t 1 → 0. Then, we have (23) and u is a strong solution of equation (1) with the initial value (2) in the sense of Definition 2.
At last, we give a simple comment. The condition a i u x i ∈ L 1 ðΩÞ can not assure the boundary value condition is imposed in the sense of the trace. In fact, we have the following proposition. Then, Thus, if a i ðx, tÞ satisfies (85), the partial boundary value condition (105) can be imposed in the sense of trace. However, in this paper, we pay our attention on the studying the stability (or the uniqueness) of solutions independent of the boundary value condition (105), so Proposition 10 is not important.

Conclusion
It is clear of that Lemma 6 has a wider applications than the classical Gronwall inequality. Moreover, compared with reference [18], there is at least the essential difference in two aspects. The first one is that condition (9), i.e., is much weaker than condition (5) appearing in [19], i.e.
a i x ð Þ = 0, x ∈ ∂Ω, i = 1, 2, ⋯, N: Such a degeneracy is the special nature of the anisotropic equation. The second one is that we have not used boundary value condition (3) throughout this paper; in other words, condition (9) may replace boundary value condition (3) in some way. Moreover, using some techniques developed by the second author in his work [10], in which the wellposedness of weak solutions to equation, v j j β−1 v t = div b x, t ð Þ ∇v j j p x,t ð Þ−2 ∇v has been discussed; the method used in this paper can be applied to study a more general equation in the future.

Data Availability
There is not any data in the paper.

Conflicts of Interest
The authors declare that they have no competing interests.