, Volume 130, Issue 1, pp 63-91
Date: 03 Jun 2009

The semiflow of a reaction diffusion equation with a singular potential

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Abstract

We study the semiflow \({\mathcal{S}(t)_{t\geq 0}}\) 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 < μ ≤ μ*, 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. We remark some qualitative differences of the branches in the subcritical case μ < μ* and the critical case μ = μ*. The global bifurcation result is used to show that any solution \({\phi(t)}\) , initiating form initial data \({\phi_0\geq 0}\) tends to the unique nonnegative equilibrium.