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The semiflow of a reaction diffusion equation with a singular potential. (English) Zbl 1178.35221

Let ${\Omega }$ be an open set in ${ℝ}^{N}$ and assume that $\lambda ,\mu ,\gamma$ are positive parameters. This paper deals with the study of dynamics of the nonlinear parabolic equation

${\partial }_{t}\varphi -{\Delta }\varphi -\frac{\mu }{{|x|}^{2}}\phantom{\rule{0.166667em}{0ex}}\varphi =\lambda \varphi -{|\varphi |}^{2\gamma }\varphi ,\phantom{\rule{1.em}{0ex}}x\in {\Omega },\phantom{\rule{4pt}{0ex}}t>0,$

subject to the initial condition

$\varphi \left(x,0\right)={\varphi }_{0}\left(x\right),\phantom{\rule{2.em}{0ex}}x\in {\Omega }$

and the boundary condition

$\varphi =0,\phantom{\rule{2.em}{0ex}}x\in \partial {\Omega },\phantom{\rule{4pt}{0ex}}t>0·$

The authors establish bifurcation results and they study the asymptotic behaviour of solutions as $\mu ↗{\mu }^{*}$, where ${\mu }^{*}$ denotes the optimal constant for the Hardy-Poincaré inequality. The proofs combine variational methods, elliptic and parabolic estimates and related inequalities involving singular terms.

##### MSC:
 35K67 Singular parabolic equations 35B40 Asymptotic behavior of solutions of PDE 26D10 Inequalities involving derivatives, differential and integral operators 35B41 Attractors (PDE) 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems 35K58 Semilinear parabolic equations 35K20 Second order parabolic equations, initial boundary value problems 35B32 Bifurcation (PDE)
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