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

Abstract

Abstract This paper is devoted to the study of anisotropic parabolic equation related to the p i $p_{i}$ -Laplacian with a source term f ( u ) $f(u)$ . If f ( u ) = 0 $f(u)=0$ , 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 ) $E(0)$ is positive and has a super boundedness, which depends on the Sobolev imbedding index, then the local solution may blow up in finite time.

Keywords

• Fundamental solution
• Anisotropic parabolic equation
• Initial energy
• The growth order
• Blow-up