Critical exponents for the decay rate of solutions in a semilinear parabolic equation

This paper is concerned with the Cauchy problem ut = uxx - |u|p-1u u in R x (0, ∞), u(x, 0) =u0(x) in R. A solution u is said to decay fast if t1/(p-1)u → 0 as t → ∞ uniformly in R, and is said to decay slowly otherwise. For each nonnegative integer k, let Λk be the set of uniformly bounded functions on R which change sign k times, and let Pk > 1 be defined by Pk = 1 + 2/(k + 1). It is shown that any nontrivial bounded solution with u0 ∈ Λk decays slowly if 1 < p < Pk, whereas there exists a nontrivial fast decaying solution with u0 ∈ Λk if p ≥ Pk.