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Optimal sub- or supersolutions in reaction-diffusion problems

by R. Sperb

(Report number 1999-23)

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.

Keywords:

@Techreport{S99_257,
  author = {R. Sperb},
  title = {Optimal sub- or supersolutions in reaction-diffusion problems},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {1999-23},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports1999/1999-23.pdf },
  year = {1999}
}

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