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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 $$x_1$$.

##### MSC:
 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