Initial Value And Initial Boundary Value Problem For Two Classes Of Nonlinear Evolution Equations

The present paper studies the initial value problem of semilinear parabolic equation and the initial boundary value problem for equation of nerve conduction type in arbitrary dimensions. Outwardly they look like two different types of equations. In fact, the derivations of the semilinear parabolic equations wtih respect to t is the nerve conduction equationThis paper studies the initial value problem of semilinear parabolic equation by a family of potential wells. Firstly it introduces a family of potential wells and its corresponding sets and gives a series of their properties. Secondly it proves the invariance of some sets and vacuum isolating of solutions. Thirdly it obtains a threshold result for the global existence and nonexistence of solutions. Fourthly it proves finite time blow-up of solutions. Finally it discusses the asymptotic behavior of the solutions.This paper also studies the existence of global strong solutions to the initial boundary value problem of nerve conduction equation in arbitrary dimensions by Galerkin method together with energy estimation. It proves that when n≤3, the problem admits a global in time L~∞strong solution if nonlinear terms satisfy some conditions. When n≥4, the problem admits a global in time L~2 strong solution if f∈C,g∈C~1, f and g are bounded below, f and g satisfy certain growth condition. The paper introduces a new concept of global strong solutions when n≥4.