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

Let Ω be an open set in N and assume that λ,μ,γ are positive parameters. This paper deals with the study of dynamics of the nonlinear parabolic equation

t φ-Δφ-μ |x| 2 φ=λφ-|φ| 2γ φ,xΩ,t>0,

subject to the initial condition

φ(x,0)=φ 0 (x),xΩ

and the boundary condition

φ=0,xΩ,t>0·

The authors establish bifurcation results and they study the asymptotic behaviour of solutions as μμ * , where μ * 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:
35K67Singular parabolic equations
35B40Asymptotic behavior of solutions of PDE
26D10Inequalities involving derivatives, differential and integral operators
35B41Attractors (PDE)
46E35Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
35K58Semilinear parabolic equations
35K20Second order parabolic equations, initial boundary value problems
35B32Bifurcation (PDE)
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