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-Poincar\'{e} inequality and its improved versions, have a prominent role on the definition of the natural phase space. Our study concerns the case $0<\mu\leq\mu^*$, where $\mu^*$ is the optimal constant for the Hardy-Poincar\'{e} 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 $\phi(t)=\mathcal{S}(t)\phi_0$, initiating form initial data $\phi_0\geq 0$ ($\phi_0\leq 0$), $\phi_0\not\equiv 0$, tends to the unique nonnegative (nonpositive) equilibrium.