The fundamental solution and blow-up problem of an anisotropic parabolic equation

This paper is devoted to the study of anisotropic parabolic equation related to the pi\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$p_{i}$\end{document}-Laplacian with a source term f(u)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$f(u)$\end{document}. If f(u)=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$f(u)=0$\end{document}, then the fundamental solution of the equation is constructed. If there are some restrictions on the growth order of u in the source term, the initial energy E(0)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$E(0)$\end{document} is positive and has a super boundedness, which depends on the Sobolev imbedding index, then the local solution may blow up in finite time.


Introduction
An anisotropic parabolic equation with the initial value condition and the Dirichlet boundary value condition is considered in this paper, where ⊂ R N is a bounded domain with smooth boundary ∂ , T is any given positive constant, p i is a positive constant, i = 1, . . ., N , and the function f (s) is continuous.As usual, we call problem (1)-( 2)-( 3) the (first) initial boundary value problem.We denote p + = max{p 1 , p 2 , . . ., p N }, p -= min{p 1 , p 2 , . . ., p N }, Equation (1) appears in several places in the literature.For instance, in biology, it acts as a model describing the spread of an epidemic disease in heterogeneous environments, and in fluid mechanics, it emerges as a mathematical description of the dynamics of fluids with different conductivities in different direction; one can refer to [1,4,5] for details.
People are more familiar with the evolution p-Laplacian equation with a source term As is well known, equation ( 4) arises from non-Newtonian fluids as a fast diffusion equation if p < 2, and a slow diffusion equation if p > 2. When p = 2, equation ( 4) becomes the classical heat conduction equation.When f (u) < 0, it is called an absorption term, equation (4) may have a global solution.While f (u) ≥ 0, it is called a source term, and equation ( 4) generally only has a local weak solution.This kind of equation has important physical significance and has received much attention for a long time.Sometimes, the divergence div(|∇u| p-2 ∇u) is replaced by N i=1 ∂ ∂x i (|u x i | p-2 u x i ).Some details are given in what follows.Consider the slow diffusion equation When f (u, x, t) = f (x, t), Nakao studied the existence of periodic solutions in [16].If f (u, x, t) = |u| α u, then the energy functional is set as the potential well is defined as where which is called the depth of potential well.By the potential well theory it means that we can study the existence of the global solution and the blow-up of local solutions to equation ( 5) by means of analyzing the relationship between I(u) and J(u).Under the assumption and u 0 (x) ∈ W , that is, J(u 0 ) < d and I(u 0 ) > 0, Tsutsumi [23] showed that equation ( 5) admits a global solution.While, for the case of negative initial energy J(u 0 ) < 0, the solutions of equation ( 5) can blow up in finite time [23].If I(u 0 ) > 0 or I(u 0 ) = 0, 0 < J(u 0 ) ≤ d, Liu and Zhao [13] showed that problem ( 5)-( 2)-( 3) admits a global solution.If J(u 0 ) > d, then using the comparison principle and by the variational method, the global existence and finite time blow-up of solutions were approached by Xu in [25].Along this way, i.e., by the potential well theory, Liu, Yu, and Li made a new progress in the study of the finite time blow-up phenomena to equation ( 5) with the subcritical initial energy and the critical initial energy in [12] recently.The excellent insights into the theory of blow-up of solutions to parabolic equations with constant nonlinearity can be found in the monographs [9,14,19].In these monographs, instead of the energy function, the proofs of the main results are based on the reduction of the problem to the study of a nonlinear ordinary differential inequality for a suitably chosen function associated with the solution.It happens so that every function satisfying such an inequality becomes unbounded in a finite time, which yields the finite time blow-up of the corresponding weak solution.
Let us come back to the anisotropic parabolic equation (1).For its stationary case, i.e., if u(x, t) = u(x) satisfies the following degenerate elliptic equation: then there are many papers devoted to its multiple solutions under different conditions of f (u, x, t), one can refer to [7,8,10,17,18] for details.Here, λ > 0 is a constant.Naturally, there have been some important results of the anisotropic parabolic equation (1) itself in recent years.For example, if is a Banach space; one can refer to [20] for details.At the same time, we have noted with pleasure that the well-posedness of the initial boundary value problem of an anisotropic parabolic equation with the exponent variables was studied by Antontsev and Shmarev in [2,3] and by Alkis S. Tersenov and Aris S. Tersenov in [21].
However, different from the evolution p-Laplacian equation, the fundamental solution of the anisotropic parabolic equation cannot be found in the references at our hand.So in this paper, we first discuss the local solutions of initial boundary value problem ( 1)-( 3) and give a fundamental solution of equation ( 8), and then we pay our main attention to the blow-up phenomena of weak solutions to equation (1).
To describe the methods used in this paper, we prefer to give more information about the existence and nonexistence of global weak solutions to equation (4).If then the weak solution may blow up in finite time, this is an innovative work by Zhao [28] in 1993.Here and in what follows, After 2000, by setting the energy function and Liu and Wang [11] showed that the weak solution of problem ( 4)-( 2)-(3) blows up in finite time.Here, α 1 , E 1 are two constants defined as B 1 is the usual Sobolev embedding constant, and The latest progress can be found in [6] by Chung and Choi, in which f (u) satisfies the condition and λ 1,p is the first eigenvalue of p-Laplacian.In this paper, we give a new energy functional matching up with an anisotropic equation, employ some different techniques to study the blow-up phenomena of weak solution to equation (1).Since p i = p j when i = j, although we are using some ideas of [3,11], how to constructed a suitable functional becomes more difficult, how to obtain the corresponding inequalities to obtain the maximum value becomes a very complicated analysis.
The paper is arranged as follows.In Sect.2, the definition of weak solution and the main results are given.In Sect.3, the existence of the local solution to equation ( 1) is described and the fundamental solution to equation ( 8) is given.In Sect.4, some lemmas related to the initial energy are provided.In Sects. 4 and 5, the theorems on blow-up phenomena are proved.

The main results
For simplicity, sometimes we denote that u x i L p i ( ) = u x i p i , . . ., and so on.
1 p i when N -q > 0, and it is true for all r ≥ 1 when q ≥ N .Then W This lemma can be found in [15,22,27], and W As [3,11], let Then, combining Lemma 1 with ( 14), similar to the Sobolev embedding inequality, we can show that where r ∈ (1, N p N-p ) and 1 p = 1 N N i=1 1 p i , and B is the optimal constant satisfying The weak solution is defined as follows.From this definition, let Then, obviously, we have Now, for a given u being a solution of the initial-boundary problem ( 1)-(3), we define We set the constant α 1 by and define here and in what follows By these definitions, we have the following.

Theorem 3 Let u be a solution of initial boundary value problem (1)-(3)
. Suppose that f (s) in C(R) satisfies ( 14)-( 15), and there exist positive constants σ , r, and α 2 such that α 2 > α 1 , and If the initial energy satisfies as well as then the solution u(x, t) of problem (1)-(3) blows up in finite time.
If α 1 ≤ 1, then we define and let α 10 be a constant satisfying Theorem 4 Let u be a solution of initial boundary value problem (1)- (3).Suppose that f (s) in C(R) satisfies ( 14)- (15), and there exist positive constants σ , r, and α 20 such that and the initial energy satisfies as well as then the solution u(x, t) of problem (1)-(3) blows up in finite time.
One can see that if p + = p -and σ = 1, then the growth order condition ( 22) is just the same as (13).However, in our paper, σ can be less than 1.So our paper improves the results of [11].Actually, the blow-up phenomena shown in [13,16,19,24,25,28] also arise in the anisotropic equation (1).If the readers are interested in these problems, to generalize the results in [13,16,19,24,25,28], they will find that the complexity mainly reflects in the index relationships just like our theorems obtained in this paper.

The fundamental solution and the existence of weak solution
About two years ago, my graduate student Miss Yang first constructed the fundamental solution of equation ( 8) in her master thesis [26].Since it has not been published in any magazine, we extract a part of [26] here.
Proposition 5 There is a fundamental solution of equation (8).
Theorem 7 Let f (s) ∈ C(R) and there exist a function g(s) , where T 1 is a small positive constant.If f (s) is a Lipschitz function, then the local solution is unique.
For simplicity of the paper, we omit the details of the proof here.
Lemma 11 Suppose that u(x, t) is a solution of problem (1)-( 3), and the initial value satisfies (29) as well as N i=1 u 0x i p i (x) > α 10 .Then, for any t ≥ 0, we have  Once more, we want to give an estimate of G r/2 (t).By inequality (22) and the Hölder inequality, we can extrapolate that 2 ).

M
* = inf m, the solvability of the initial boundary value problem (1)-(3) was approached in [20] by Starovoitov and Tersenov, provided that the function f : R → R is continuous, |f (ξ )| ≤ f (η) for all ξ and η such that |ξ | ≤ η, and m the initial value condition (2) is true in the sense lim t→0 u(x, t)u 0 (x) dx = 0, and the boundary value condition (3) is true in the sense of trace.