On Blow-up set of a Semilinear Heat Equation with Neumann Boundary Condition

Abstract

This paper deals with the blow-up set of a semilinear heat equation defined on a ball, with nonlinear boundary condition, where the reaction term and the boundary term are powers of exponential types. In [9], under some restricted assumption, it has been proved that, in case of, the nonlinear terms are of exponential types without powers, the blow-up occurs only on the boundary. Our aim is to extend that result, for some regains of the powers those appear on the reaction and boundary terms. 1 Introduction In this paper, we consider the initial –boundary problem: ……..(1)where , is a ball in , is the outward normal, is nonnegative symmetric, nondecreasing, smooth function satisfies the conditions ………………………(2) …. (3)where It is known that, the existence and uniqueness of local classical solutions to this problem are guaranteed by the standard theory see [9],[5]. On the other hand, the nontrivial solutions of this problem blow-up in finite time and the blow-up set contains , and that due to comparison principle,[7], and the known blow-up results of problem where (see[2]).In [9], it has been proved that, the lower blow-up rate is obtained as follows …….(4)