On the Blow up Problem for Semilinear Heat Equations

Abstract

This paper is concerned with the instability behaviour of a differential system arising from the theory of channel and cylindrical flow of viscous fluids with high heat generation. The spatial domain under consideration is the whole space $R^n $, since usually in this type of problem a bounded domain can be transformed into an unbounded domain. It is shown that if the physical parameter $g(x,t)$, which corresponds to the stress on the fluid is such that $g(x,t) \geq \lambda t^{n + \alpha - 1} H(x,t)$, where $H(x,t)$ is the fundamental solution of the heat equation and $\lambda $ and $\alpha $ are positive constants, then for certain classes of initial conditions the corresponding solution of the initial value problem grows unbounded in a finite time. We obtain an upper bound for the blow up time. We also explain how the method can be applied to a typical physical problem.