manuscripta mathematica

, Volume 130, Issue 1, pp 63–91

The semiflow of a reaction diffusion equation with a singular potential

  • Nikos I. Karachalios
  • Nikolaos B. Zographopoulos
Article

DOI: 10.1007/s00229-009-0284-1

Cite this article as:
Karachalios, N.I. & Zographopoulos, N.B. manuscripta math. (2009) 130: 63. doi:10.1007/s00229-009-0284-1

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|2γs, 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.

Mathematics Subject Classification (2000)

35B40 35B41 26D10 46E35 

Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  • Nikos I. Karachalios
    • 1
  • Nikolaos B. Zographopoulos
    • 2
  1. 1.Department of MathematicsUniversity of the AegeanKarlovassiGreece
  2. 2.Division of Mathematics, Department of SciencesTechnical University of CreteChaniaGreece