Report 1999-23

Optimal sub- or supersolutions in reaction-diffusion problems

R. Sperb

Abstract: The type of problem under consideration is $$ \left\{ \begin{array}{lll} u_t = \Delta u + f(u) & {\rm in} & \Omega \times (0,T) \\[1ex] \displaystyle\frac{\partial u}{\partial n} + g(u) = 0 & {\rm on} & \partial \Omega \times (0,T) \\[2ex] u(x,0) = u_0(x)\,. \end{array}\right. \leqno(*) $$ Here $\Omega$ is a finite domain of $\R^N$. The solution of (*) is compared with a corresponding solution of the $N$-ball or a finite interval whose size depends on different quantities of an associated linear elliptic problem for $\Omega$, such as e.g. the fixed membrane problem. Possible applications include estimates for the blow-up or finite vanishing time.

Paper: Available as PDF (322 KB) or as hardcopy to order reports@sam.math.ethz.ch.