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

This paper is concerned with the Cauchy problem u(t) = u(xx) - \u\p(-1)u in R x (0, infinity), u(x, 0) = u(0)(x) in R. A solution It is said to decay fast if t(1/(p-1))u --> 0 as t --> infinity uniformly in R, and is said to decay slowly otherwise. For each nonnegative integer k, let Lambda(k) be the set of uniformly bounded functions on R which change sign k times, and let p(k) > 1 be defined by p(k) = 1 + 2/(k + 1). It is shown that any nontrivial bounded solution with u(0) is an element of Lambda(k) decays slowly if 1 < p < p(k), whereas there exists a nontrivial fast decaying solution with u(0) is an element of Lambda(k) if p greater than or equal to p(k).