Critical Exponents for the Decay Rate of Solutions in a Semilinear Parabolic Equation

This paper is concerned with the Cauchy problem A solution u is said to decay fast if as uniformly in R, and is said to decay slowly otherwise. For each nonnegative integer k, let be the set of uniformly bounded functions on R which change sign k times, and let be defined by . It is shown that any nontrivial bounded solution with decays slowly if , whereas there exists a nontrivial fast decaying solution with if . (Accepted April 24, 1998)