A class of semilinear parabolic equations with singular initial data

We consider the initial-boundary value problem for the semilinear parabolic equation on a smooth domain Ω⊂RNΩ⊂RN, equation(1.1){ut=Δu+|∇u|p|u|q−1uin(0,∞)×Ω,u(t,x)=0in(0,∞)×∂Ω,u(0,x)=u0(x)inΩ, where 1≤p≤21≤p≤2 and q≥1q≥1. In this paper, we are concerned with the existence of solutions with singular initial data u0⁄∈L∞u0⁄∈L∞. We study the problem (1.1) on several singular spaces of initial data. More precisely, we investigate the subquadratic case p<2p<2 in the Lebesgue class {Lr}1≤r<∞{Lr}1≤r<∞ and in the singular Sobolev class {W01,r}1≤r