Solutions of Diffusion Equations in Channel Domains

Abstract

We will discuss the existence and the qualitative properties of solutions of the problem

$$\begin{gathered} \Delta u + f(u)\; = \;0{\text{ }}in{\text{ }}\Omega \hfill \\ {\text{ }}u = \;0{\text{ }}on{\text{ }}\partial \Omega ; \hfill \\ \end{gathered}$$

here Ω is a channel of the form

$$\Omega \; = \;\{ {\mkern 1mu} (x,{\mkern 1mu} y)\;:\;x\; \in \;{\mathbb{R}^1},\quad 0 < y < \;\emptyset (x)\;\} .$$

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