The semiflow of a reaction diffusion equation with a singular potential

We study the semiflow defined by a semilinear parabolic equation with a singular square potential . 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 , 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 , initiating form initial data tends to the unique nonnegative equilibrium.