Convection-diffusion equation with absorption and non-decaying initial data

In this paper we give the precise description of the large time behavior of the solution u of the Cauchy problem by using the ordinary differential equation ζ ^'=-ζ ^β and a linear parabolic equation. Here N≥1, a∈R ^N, α>1, β>1, λ>0, and φ is a bounded continuous function such that φ∈L ^p(R ^N) for some 1≤p<∞. Furthermore, we study the large time behavior of the hot spots for the solution u.