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Reaction-diffusion problems in infinite cylinders with no invariance by translation. (Problèmes de réaction-diffusion sans invariance par translation dans des cylindres infinis.) (French. Abridged English version) Zbl 0839.35045
Summary: This note deals with existence results and a priori estimates of solutions \((c, u)\) of reaction-diffusion in infinite cylinders \(\Sigma= \{(x_1, y)\in \mathbb{R}^N, x_1\in \mathbb{R}, y\in \omega\}\) with outward unit normal \(\nu\). These equations can be written: \[ \Delta u- (c+ \alpha(y)+ \gamma(x_1)) \partial_1 u+ f(x_1, u)= 0\quad\text{in }\Sigma \] with Neumann boundary conditions on \(\partial\Sigma\), and limits 0 and 1 as \(x_1\to \pm\infty\). When the term \(f\) depends only on \(u\) and has an “ignition temperature” profile, the existence of solutions \((c, u)\) is proved for small functions \(\gamma\), and one enounces a continuity theorem as \(\gamma\to 0\). In a second part, one assumes that \(\gamma\) and \(f\) are monotone with respect to \(x_1\); the existence of solutions is related to two asymptotic problems obtained by taking the limits \(x_1\to \pm\infty\), and one shows that the solutions are necessarily increasing in \(x_1\).

35J65 Nonlinear boundary value problems for linear elliptic equations
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
35B45 A priori estimates in context of PDEs