key: cord-0738824-mvbrvkfe
authors: Zhan, Huashui; Feng, Zhaosheng
title: Existence and stability of the doubly nonlinear anisotropic parabolic equation
date: 2020-12-13
journal: J Math Anal Appl
DOI: 10.1016/j.jmaa.2020.124850
sha: dc962810fc07e235050cd5e6cefed9dc8b7150c6
doc_id: 738824
cord_uid: mvbrvkfe

In this paper, we are concerned with a doubly nonlinear anisotropic parabolic equation, in which the diffusion coefficient and the variable exponent depend on the time variable t. Under certain conditions, the existence of weak solution is proved by applying the parabolically regularized method. Based on a partial boundary value condition, the stability of weak solution is also investigated.

Since the significant disruption that is being caused by the coronavirus pandemic, we are aware that all communities must resolutely work together to battle the pandemic amid globalization. A growing number of mathematical models have been developed by health care systems, academic institutions and others to help forecast coronavirus spread, deaths, and medical supply needs, including ventilators, hospital beds and intensive care units, timing of patient surges and more. Mathematically, a model of infectious disease can be regarded as a special reaction-diffusion process. Motivated by this fact, in this study we consider a kind of reaction-diffusion equation, namely, a doubly nonlinear parabolic anisotropic equation:

where Q T = Ω × (0, T ), Ω ⊂ R N is a bounded domain with a C 2 boundary ∂Ω, 0 ≤ a i (x, t) ∈ C 1 (Q T ), 1 < p i (x, t) ∈ C 1 (Q T ), g i (x, t) ∈ C 1 (Q T ), B (u) = b(u) ≥ 0 and B(0) = 0. Equation (1) arises from physics, fluid mechanics, as well as from the epidemic model of diseases in biology and ecology [27] . Compared with the isotropic-type equations, equation (1) is much closer to a diffusion process such as the epidemic of coronavirus disease. If 

we conjecture that it inevitably leads to u(x, t) = 0, (x, t) ∈ ∂Ω × [0, T ), which was partially proved in [25] .

The biological explanation of condition (2) lies in the fact that if u(x) represents the velocity of spreading progress of an infectious disease such as coronavirus disease, condition (2) implies that the virus (or disease) can not transmit across ∂Ω, when the region remains under lockdown.

A special case of equation (1) is the so-called evolutionary p(x)−Laplacian equation, which takes the form:

u t = div(|∇u| p(x,t) ∇u), (x, t) ∈ Q T , and has been extensively studied in the past decades [1, 2, 5, 6, 9, 18, 20, 26] etc. Equation (1) can also be regarded as a generalized version of the polytropic infiltration equation:

where m > 0 and p > 1. For more details and recent results on equation (3), we refer the reader to [4, 11, 13, 22, 23, 32] and the references therein.

Recently, a number of issues considered the anisotropic equation [7, 8] :

with the initial-boundary value conditions

A more general anisotropic equation [10, 17, 24, 28, 29, 30] :

was studied on the stability and the well-posedness. Here what interests us most is that if we take the diffusion coefficient a i (x) | x∈∂Ω = 0 and consider a partial boundary value condition

can the stability of equation (1) be achieved too? From [28, 29, 30] , we know that when Σ p = Σ 1 × (0, T ), where Σ 1 is a submanifold of ∂Ω (or Σ 1 = ∅), the stability of weak solution of equation (5) can be true. For equation (1) , the diffusion coefficient a i (x, t), the variable exponent p i (x, t) and the convection coefficient g i (x, t) are all dependent on the time variable t. Distinguished from [28, 29, 30] , we will show that Σ p is a submanifold of ∂Ω × (0, T ) and is generally not a cylinder as Σ 1 × (0, T ). Assume that B(u) is a strictly increasing function. For examples, B(u) can been chosen as u m , e u − 1, ln(1 + u) and

For convenience, we denote (1), if

and for any function ϕ ∈ C(0, T ; W 1,p+ 0

The initial value condition (4) is satisfied in the sense of

and the partial boundary value condition (6) is true in the sense of trace.

Let us summarize our main results as follows. For convenience, we use c to represent a constant that may change from line to line throughout the whole paper,

Suppose that a i (x, t) satisfies condition (2) and one of the following conditions:

Suppose that u 0 (x) ≥ 0 satisfies

Then there is a nonnegative solution of equation (1) under condition (4) .

Suppose that for i = 1, 2, · · · , N, a i (x, t) ≡ a(x, t) satisfies condition (2) and for the large n there holds

and

Suppose that u(x, t) and v(x, t) are two weak solutions of equation (1) with the initial values u 0 (x) and v 0 (x) respectively, and with a partial homogeneous boundary value condition

Then we have

It is remarkable that Theorem 3 can be generalized to the case of a i (x, t) = a j (x, t) as i = j. The proof can be processed in an analogous manner.

The rest of the paper is organized as follows. Proofs of Theorems 2 and 3 are presented in Sections 2 and 3, respectively. The characteristic function method is introduced in Section 4. We show that this method can also be applied to study the stability for other degenerate parabolic equations. A brief conclusion is given in Section 5.

For simplicity, we assume that B(u) is a C 1 strictly monotone increasing function. We prove Theorem 2 by starting to consider a parabolically regularized system:

Proof of Theorem 2. Since u 0 (x) ≥ 0 satisfies (13) , similar to the evolutionary p−Laplacian equation [27] , by using the monotone convergence method, we can prove that there exists a constant M such that the solution u ε ∈ L 1 (0, T : W 1,p(x) (Ω)) of the initial-boundary value problem (18)- (20) satisfies

For more results on the existence of weak solutions to the initial-boundary value problem (18)- (20) , we refer to [5, 6] . Multiplying both sides of (18) 

Note that

To derive (26) from (10)-(12), we have used the following facts. From condition (i), i.e. ∂ai(x,t)

In view of ∂pi ∂t ≤ 0, we deduce

where the small constant δ satisfies δb(M ) ≤ 1 2 .

and

From inequalities (21), (23) and (28), it implies

In order to prove u to be the solution of equation (1), we shall prove that

This further leads to

Let 0 ψ ∈ C ∞ 0 (Q T ) and ψ = 1 on suppϕ 1 

It follows from (33)-(34) that

Letting ε → 0, we have

Taking ϕ = ψB(u) in (32), we get

By combing (36) and (37), we have

In particular, taking v = B −1 (B(u) − λϕ) and λ > 0, we find

When λ goes to zero, we have

Similarly, when λ < 0, we get

Since ψ = 1 on suppϕ, we arrive at (30) . The initial value condition (4) in the sense of (9) can be derived from (29) . We omit the details here. Consequently, u(x) satisfies equation (1) in the sense of Definition 1.

To discuss the stability of solutions of equation (1), we need to introduce the following technical lemma. Let p(x) ∈ C 1 (Ω), and denote p + = max x∈Ω p(x) and p − = max x∈Ω p(x).

(Ω) are reflexive Banach spaces.

(II) Let p 1 (x) and p 2 (x) be real functions with 1 p1(x) + 1 p2(x) = 1 and p 1 (x) > 1. Then the conjugate

For any large integer n, we define an odd function S n (s) by Meanwhile, since a i (x, t) ≡ a(x, t) ≥ 0, for any λ > 0 we define

where Ω λt = {x ∈ Ω : a(x, t) > λ}. Proof of Theorem 3. Supposed that u(x, t) and v(x, t) are two weak solutions of equation (1). After a process of limit, we can choose ϕ n S n (B(u) − B(v)) as a test function. In view of a i (x, t) ≡ a(x, t), we have

Note that the second term in the left hand side of (40) satisfies

To evaluate the third term on the left hand side of (40), we use

In view of condition (14) , by the straightforward calculations we can deduce that

→0, as n → 0.

It follows from Hölder's inequality and (15) that

Recall the partial boundary value condition (16), i.e.

Then we have

and

Since B(r) ≥ 0 is monotone, it follows that

By (41)-(46), letting n → ∞ in (40) yields

Using Gronwall's inequality, we obtain

We can generalize the method described in the preceding section to prove the stability of weak solutions. Let χ(x, t) be a nonnegative C 1 (Q T ) function as

If we denote

for t ∈ [0, T ), then χ t is the weak characteristic function of Ω as defined in [29] . Likewise, we can simply call χ(x, t) a weak characteristic function of Q T . For λ > 0, we define

where q i (x, t), p + it and q + it are the same as given in Theorem 3. Suppose that u(x, t) and v(x, t) are two weak solutions of equation (1) with the initial values u 0 (x) and v 0 (x) respectively, with a partial homogeneous boundary value condition

Then we have

Proof of Theorem 5. Choose ϕ n S n (B(u) − B(v)) as a test function. Then we have

As discussing in the proof of Theorem 3, we have

In view of condition (47), we can deduce

Similar to the derivation of (43), using Hölder's inequality and (48), we obtain

qi(x,t)

Note that the right hand side of (54) goes to 0 as n → 0. Here,

In view of (49), we get

Similarly, we can deduce that both (45) and (46) hold too. From (51)-(55), we arrive at the desired result (50).

We can see that by choosing different appropriate characteristic function of Q T , we can obtain the corresponding stability results under various conditions. For example, i) If we take χ(

where a kxi = ∂a k (x,t) ∂xi , k = 1, 2, ..., N . By virtue of Theorem 5, we obtain

Suppose that u(x, t) and v(x, t) are two weak solutions of equation (1) with the initial values u 0 (x) and v 0 (x) respectively, with a partial homogeneous boundary value condition

Then we have the stability of weak solution in the sense of (17) .

ii) If we take χ(x, t) = χ [τ,s] (t)d α (x), where d(x) = dist(x, ∂Ω) is the distance function from the boundary ∂Ω and α ≥ 1 is a constant, then

According to Theorem 5, we can also obtain

Suppose that u(x, t) and v(x, t) are two weak solutions of equation (1) with the initial values u 0 (x) and v 0 (x) respectively, with a partial homogeneous boundary value condition u(x, t) = v(x, t) = 0, (x, t) ∈ x ∈ ∂Ω × (0, T ) : 

In this study, we applied an analytical method to study the stability of weak solution for a doubly nonlinear anisotropic parabolic equation, where the diffusion coefficient and the variable exponent depend on the time variable t. Under certain parametric choices, it includes the heat equation, reaction-diffusion equations, non-Newtonian fluid equation and electrorheological fluid equation and the epidemic model of diseases as particular cases.

When a i (x, t)| x∈Ω > 0 and B(u) is a strictly monotone increasing function, it excludes the strongly degenerate hyperbolic-parabolic equation, for which only under the entropy conditions, the uniqueness of weak solution can be guaranteed [3, 14, 31] . However, only under the condition B (u) = b(u) ≥ 0 or a i (x, t) is degenerate in the interior of Ω, how to prove the uniqueness of weak solution to equation (1) is still an interesting and challenging problem. In addition, if there is an external forcing term f (u) ≥ 0 in equation (1), i.e.

we conjecture that weak solutions may blow-up in finite time. How to show such a blow-up behavior and the long time behavior of solutions to equation (56) seems more interesting and helpful from the physical and biological point of view. We will continue to work on this problem in a subsequent work.

Regularity results for stationary electrorheological fluids

Nonlinear flow through double porosity media in variable exponent Sobolev spaces

Well-posedness results for triply nonlinear degenerate parabolic equations

Large time behavior of solutions to the Neumann problem for a quasilinear second order degenerate parabolic equation in domains with noncompact boundary

Anisotropic parabolic equations with variable nonlinearity

Parabolic equations with double variable nonlinearlities

A weighted anisotropic variant of the Caffarelli-Kohn-Nirenberg inequality and applications

Liouville-type theorems and symmetry results for a general class of quasilinear anisotropic equations

Renormalized solutions for a nonlinear parabolic equation with variable exponents and L 1 −data

Nonlocal in time problem for anisotropic parabolic equations with variable exponents of nonlinearities

Convergence in C

L 2 (Ω)) of weak solutions to perturbed doubly degenerate parabolic equations

On the spaces L p(x) (Ω) and W m,p(x)

Asymptotic expansion of solutions to the Cauchy problem for doubly degenerate parabolic c equations with measurable coefficients

Uniqueness and existence for anisotropic degenerate parabolic equations with boundary conditions on a bounded rectangle

On spaces L p(x) and W k,p(x)

Large time geometric properties of solutions of the evolution p-Laplacian equation

Blow-up analyses in parabolic equations with anisotropic nonstandard damping source

Solutions for a singular elliptic problem involving the p(x)-Laplacian, Filomat

Large time behavior of solutions to a class of doubly nonlinear parabolic equations

Existence of solutions of the parabolic variational inequality with variable exponent of nonlinearity

Mixed Morrey spaces and their applications to partial differential equations

Cauchy problem for doubly degenerate parabolic equation with gradient source

The interface blow-up phenomenon and local estimates for doubly degenerate parabolic equations

Existence of Lipschitz continuous solutions to the Cauchy-Dirichlet problem for anisotropic parabolic equations

Smoothing and decay estimates for nonlinear diffusion equation

Intrinsic Harnack inequalityies for parabolic equations with variable exponents

Nonlinear Diffusion Equations

The stability of the solutions of an anisotropic diffusion equation

The well-posedness problem of an anisotropic parabolic equation

Solutions of evolutionary equation based on the anisotropic variable exponent Sobolev space

Partial boundary value condition for a nonlinear degenerate parabolic equation

Existence and uniqueness of solutions for a class of doubly degenerate parabolic equations