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,{∂tu=Δu+a⋅∇uα−uβin RN×(0,∞),u(x,0)=λ+φ(x)⩾0in RN, by using the ordinary differential equation ζ′=−ζβζ′=−ζβ and a linear parabolic equation. Here N⩾1N⩾1, a∈RNa∈RN, α>1α>1, β>1β>1, λ>0λ>0, and φ   is a bounded continuous function such that φ∈Lp(RN)φ∈Lp(RN) for some 1⩽p<∞1⩽p<∞. Furthermore, we study the large time behavior of the hot spots for the solution u.