Slow decay of solutions in a semilinear dissipative parabolic equation

This paper is concerned with a Cauchy problem , where p > 1 and u0 ε L∞(R). A solution u of (P) is said to decay fast as t → ∞ if limt → ∞ tl/p −1u(x, t) = 0 uniformly in R and to decay slowly as t → ∞ otherwise. We prove that if u0(x) does not decay faster than ¦x¦−q with some q < 2/(p −1) as x → ∞ or x → −∞, then u decays slowly as t → ∞. In a process of the proof, we give an estimate of solutions near the spatial infinity for general initial data, which implies that none of zeros of u(t) goes to ±∞ at each t > 0.