The nonnegative weak solution of a degenerate parabolic equation with variable exponent growth order

<jats:p>A degenerate parabolic equation of the form <jats:disp-formula><jats:alternatives><jats:tex-math>$$\bigl( \vert v \vert ^{\beta-1}v\bigr)_{t}= \operatorname{div} \bigl(b(x,t) \vert \nabla v \vert ^{p(x,t)-2}\nabla v \bigr)+\nabla\vec{g}\cdot\nabla\vec{\gamma}(v) $$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mo>|</mml:mo><mml:mi>v</mml:mi><mml:msup><mml:mo>|</mml:mo><mml:mrow><mml:mi>β</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mi>v</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mi>t</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mo>div</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mi>b</mml:mi><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo><mml:mo>|</mml:mo><mml:mi>∇</mml:mi><mml:mi>v</mml:mi><mml:msup><mml:mo>|</mml:mo><mml:mrow><mml:mi>p</mml:mi><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo><mml:mo>−</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mi>∇</mml:mi><mml:mi>v</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:mi>∇</mml:mi><mml:mover><mml:mi>g</mml:mi><mml:mo>→</mml:mo></mml:mover><mml:mo>⋅</mml:mo><mml:mi>∇</mml:mi><mml:mover><mml:mi>γ</mml:mi><mml:mo>→</mml:mo></mml:mover><mml:mo>(</mml:mo><mml:mi>v</mml:mi><mml:mo>)</mml:mo></mml:math></jats:alternatives></jats:disp-formula> is considered, where <jats:inline-formula><jats:alternatives><jats:tex-math>$\vec{g}=\{g^{i}(x,t)\}$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mover><mml:mi>g</mml:mi><mml:mo>→</mml:mo></mml:mover><mml:mo>=</mml:mo><mml:mo>{</mml:mo><mml:msup><mml:mi>g</mml:mi><mml:mi>i</mml:mi></mml:msup><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo><mml:mo>}</mml:mo></mml:math></jats:alternatives></jats:inline-formula>, <jats:inline-formula><jats:alternatives><jats:tex-math>$\vec{\gamma}(v)=\{ \gamma_{i}(v)\}$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mover><mml:mi>γ</mml:mi><mml:mo>→</mml:mo></mml:mover><mml:mo>(</mml:mo><mml:mi>v</mml:mi><mml:mo>)</mml:mo><mml:mo>=</mml:mo><mml:mo>{</mml:mo><mml:msub><mml:mi>γ</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>(</mml:mo><mml:mi>v</mml:mi><mml:mo>)</mml:mo><mml:mo>}</mml:mo></mml:math></jats:alternatives></jats:inline-formula>. If the diffusion coefficient <jats:inline-formula><jats:alternatives><jats:tex-math>$b(x,t)\geq0$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>b</mml:mi><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo><mml:mo>≥</mml:mo><mml:mn>0</mml:mn></mml:math></jats:alternatives></jats:inline-formula> is degenerate on the boundary, by adding some restrictions on <jats:inline-formula><jats:alternatives><jats:tex-math>$b(x,t)$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>b</mml:mi><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:math></jats:alternatives></jats:inline-formula> and <jats:italic>g⃗</jats:italic>, the existence and uniqueness of weak solutions are proved. Based on the uniqueness, the stability of weak solutions can be proved without any boundary condition.</jats:p>


Introduction and the main results
Consider the degenerate parabolic problem with exponent variable growth order |v| β-1 v t = div b(x, t)|∇v| p(x,t)-2 ∇v + ∇ g · ∇ γ (v), (x, t) ∈ Q T = Ω × (0, T), (1.1) where b(x, t) and p(x, t) are C(Q T ) nonnegative functions, g = {g i (x, t)}, γ (v) = {γ i (v)}, β > 0. We denote that p -= min (x,t)∈Q T p(x, t) > 1 and p + = max (x,t)∈Q T p(x, t) in this paper. The initial value matching up to equation (1.1) is is dispensable. If g = 0, equation (1.1) arises from the branches of flows of electro-rheological or thermo-rheological fluids (see [1][2][3]), and the processing of digital images [4][5][6][7][8][9][10][11][12][13][14][15]. If the variable exponent p(x, t) is replaced by a constant p, equation (1.1) becomes the well-known non-Newtonian polytropic filtration equation with orientated convection [16], as well as the convection-diffusion-reaction equation in which the variable can be interpreted as temperature for heat transfer problems, concentration for dispersion problems, etc. [17]. Now, let us give some details in part of the above references. Ye and Yin studied the propagation profile for the equation in which the orientation of the convection was specified to be either the convection with counteracting diffusion or the convection with promoting diffusion, that is, β(x) · (-x) ≥ 0 or β(x) · x ≥ 0, respectively [16]. Guo, Li, and Gao considered the following evolutionary p(x)-Laplacian equation: v t = div |∇v| p(x)-2 ∇v + |v| r-2 v, (x, t) ∈ Q T , subject to homogeneous Dirichlet boundary condition, where r > 1 is a constant. By using the energy estimate method, the regularity of weak solutions and blow-up in finite time were revealed in [7]. Antontsev and Shmarev have published a series of papers [8][9][10][11][12][13] on the homogeneous Dirichlet problem for the doubly nonlinear parabolic equation provided that a(x, t, v) ≥ a > 0. They established conditions on the data that guarantee the comparison principle and uniqueness of bounded weak solutions in suitable Orlicz-Sobolev spaces subject to some additional restrictions [12]. Gao, Chu, and Gao in [14] studied the nonlinear diffusion equation with the homogeneous Dirichlet boundary condition (1.3), where f is a continuous function satisfying with a 0 > 0 and α > 1. They constructed suitable function spaces and used Galerkin's method to obtain the existence of weak solutions. It is worth pointing out that the requirement on p t (x, t) is only negative and integrable, which is a weaker condition than the corresponding conditions appearing in other papers. Recently, Liu and Dong [15] generalized [14]'s result to a more general equation and gave a classification of the weak solutions. In addition, the equation arising from the double phase obstacle problems of the type has gained a wide attention [18,19] etc., where a(x) + b(x) > 0.
In this paper, for any t ∈ [0, T], we assume and with a more complicate convection term ∇ g · ∇ γ (v). These nonlinearities not only bring some essential changes to the proof of the existence, but also add difficulties to proving the stability of weak solutions. The readers will see that, in order to overcome these difficulties, a new technique based on the mean value theorem is posed to prove the uniqueness of weak solution, another new technique based on the proof by contradiction is introduced. Both of them supply a new method to prove uniqueness of weak solution for the nonlinear degenerate parabolic equations.
and for ∀φ(x, t) ∈ C 1 0 (Q T ), Initial value (1.2) is true in the sense The main results are the following theorems.   (1.12) and one of the following conditions is true: (1.14) Conditions (1.9), (1.10), and (1.12) all reflect the internal mutually dependent relationships between the diffusion coefficient b(x, t) and the convective coefficients g i (x, t). Such an internal mutually dependent relationship that can affect the finite propagation has been studied in [16], while the internal mutually dependent relationships between the diffusion coefficient and the convection term arise in mathematics finance model for studying the agent's decision under the risk [25].
At the end of introduction, it might be advisable to summarize briefly. First, as a classical work on the well-posed results of the solution of a nonlinear parabolic equation, there are many papers devoted to this problem (one can refer to [26][27][28] and the references therein). Secondly, the model studied in this paper is a parabolic equation with variable exponential term; we would like to point out that more details on the structural characteristics and the physical background of the variable exponential term have been described in [29][30][31][32][33], etc. Thirdly, one can see that the new method to prove uniqueness of weak solution can be generalized to study the double phase obstacle problems.

The existence of weak solutions
Let us consider the approximate initial-boundary value problem and for any φ(x, t) ∈ C 1 0 (Q T ), there holds Then we say that v(x, t) is said to be the weak solution of problem (2.1)-(2.3).
For any k > 0, let is an even function and is defined as 3), we now consider the following problem: is nonnegative, similar to the process subject to the existence of weak solutions in [12](also [34]), one can prove that there is a nonnegative weak solution v ε ∈ Proof of Theorem 1.2 Let us choose v ε as a test function. Then we have Moreover, let us multiply (2.1) with v εt , and obtain (2.14) Since and by |γ i (s)| 2 |s| 1-β ≤ c, p -≥ 2 and by (1.9), using the Young inequality, we have (2.16) From (2.14)-(2.16), we extrapolate that and Let ε → 0 in (2.10). Similar to that in [35], which is subject to the evolutionary p- Also, we can show that initial value (1.2) is true in the sense of (1.8) as in [12]. Theorem 1.2 is proved.
Last but not least, by the mean value theorem, where ζ ∈ (v, u).
One of the possibilities of (3.15) is that, for any s ≥ τ , is clear. Another possibility of (3.15) is that there is s 0 ≥ τ such that Combining (3.14)-(3.15) with (3.18), we can extrapolate that which contradicts assumption (3.17). In other words, (3.17) is impossible. This fact implies that, for any s, τ ∈ [0, T), inequality (3.16) is always true. By the arbitrariness of τ , we have