Large time monotonicity of solutions of reaction-diffusion equations in $R^N$

International audienceIn this paper, we consider nonnegative solutions of spatially heterogeneous Fisher-KPP type reaction-diffusion equations in the whole space. Under some assumptions on the initial conditions, including in particular the case of compactly supported initial conditions, we show that, above any arbitrary positive value, the solution is increasing in time at large times. Furthermore, in the one-dimensional case, we prove that, if the equation is homogeneous outside a bounded interval and the reaction is linear around the zero state, then the solution is time-increasing in the whole line at large times. The question of the monotonicity in time is motivated by a medical imagery issue.Dans cet article nous étudions les solutions positives d'équations de réaction diffusion dans l'espace entier. Sous certaines conditions sur la donnée initiale, nous démontrons que, au dessus d'une certaine valeur arbitrairement petite, la solution est croissante en temps pour des temps assez grands