Locally uniform convergence to an equilibrium for nonlinear parabolic equations on $R^N$

Abstract

We consider bounded solutions of the Cauchy problem { ut −∆u = f(u), x ∈ R , t > 0, u(0, x) = u0(x), x ∈ R , where u0 is a nonnegative function with compact support and f is a C1 function on R with f(0) = 0. Assuming that f ′ is locally Holder continuous and f satisfies a minor nondegeneracy condition, we prove that, as t → ∞, the solution u(·, t) converges to an equilibrium φ locally uniformly in RN . Moreover, the limit function φ is either a constant equilibrium, or there is a point x0 ∈ RN such that φ is radially symmetric and radially decreasing about x0, and it approaches a constant equilibrium as |x−x0| → ∞. The nondegeneracy condition ∗Supported in part by the Australian Research Council. †Supported in part by NSF Grant DMS-1161923. ‡This work was partly carried out during a visit of P. Polacik to the University of New England and a visit of Y. Du to the Institute of Mathematics and its Applications, University of Minnesota. §AMS Classification: 35K15, 35B40