Slow decay of solutions in a semilinear dissipative parabolic equation

This paper is concerned with a Cauchy problem [GRAPHICS] where p > 1 and u(0) is an element of L-infinity(R). A solution u of (P) is said to decay Fast as t --> infinity if lim(t --> infinity) t(1/(p-1))u(x, t) = 0 uniformly in R and to decay slowly as t --> infinity otherwise. We prove that if u(0)(x) does not decay faster than \x\(-q) with some q < 2/(p - 1) as x --> infinity or x --> -infinity, then u decays slowly as t --> i nfinity. In a process of the proof, we give an estimate of solutions near t he spatial infinity for general initial data, which implies that none of ze ros of u(t) goes to +/- infinity at each t > 0. (C) 1999 Academic Press.