The semiflow of a reaction diffusion equation with a singular potential

Abstract

We study the semiflow $\mathcal{S}(t)$ defined by a semilinear parabolic equation with a singular square potential $V(x)=\frac{\mu }{|x{|}^2}$. It is known that the Hardy-Poincaré inequality and its improved versions, have a prominent role on the definition of the natural phase space. Our study concerns the case $0<\mu \le {\mu }^{\ast }$, where ${\mu }^{\ast }$ is the optimal constant for the Hardy-Poincaré inequality. On a bounded domain of ${\mathbb{R}}^N$, we justify the global bifurcation of nontrivial equilibrium solutions for a reaction term $f(s)=\lambda s-|s{|}^{2\gamma }s$, with $\lambda$ as a bifurcation parameter. The global bifurcation result is used to show that any solution $\varphi (t)=\mathcal{S}(t){\varphi }_0$, initiating form initial data ${\varphi }_0\ge 0$ (${\varphi }_0\le 0$), ${\varphi }_0\not\equiv 0$, tends to the unique nonnegative (nonpositive) equilibrium.

20 pages, 3 figures