The semiflow of a reaction diffusion equation with a singular potential

Karachalios, Nikos I.; Zographopoulos, Nikos B.

The semiflow of a reaction diffusion equation with a singular potential

Authors

Description

We study the semiflow $\mathcal{S}(t)$ defined by a semilinear parabolic equation with a singular square potential $V(x)=\fracμ{|x|^2}$. 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 $0<μ\leqμ^*$, where $μ^*$ 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 $f(s)=λs-|s|^{2γ}s$, with $λ$ as a bifurcation parameter. The global bifurcation result is used to show that any solution $ϕ(t)=\mathcal{S}(t)ϕ_0$, initiating form initial data $ϕ_0\geq 0$ ($ϕ_0\leq 0$), $ϕ_0\not\equiv 0$, tends to the unique nonnegative (nonpositive) equilibrium.
20 pages, 3 figures