Critical exponent for the parabolic equation u t = .u m + h (t)u p in a cone

In this paper, we study the initial-boundary value problem of porous medium equation u t = .u m + h (t)u p in a cone D = (0,.) . ., where h(t) . t .. Let . 1 denote the smallest Dirichlet eigenvalue for the Laplace-Beltrami operator on . and let l denote the positive root of l 2+(n.2)l = . 1. We prove that if m<p.m+2(.+1)n+l+.(m.1) , then the problem has no global nonnegative solutions for any nonnegative u 0 unless u 0 = 0; if p>m+2(.+1)n+l+.(m.1) , then the problem has global solutions for some u 0 . 0.