Some blow-up problems for a semilinear parabolic equation with a potential

The blow-up rate estimate for the solution to a semilinear parabolic equation u(t) = Delta u + V(x)vertical bar u vertical bar(p-1) u in Omega x (0, T) with 0-Dirichlet boundary condition is obtained. As an application, it is shown that the asymptotic behavior of blow-up time and blow-up set of the problem with nonnegative initial data u (x, 0) = M phi(x) as M goes to infinity, which have been found in [C. Cortazar, M. Elgueta, J.D. Rossi, The blow-up problem for a semilinear parabolic equation with a potential, preprint, arXiv: math.AP/0607055, July 2006], is improved under some reasonable and weaker conditions compared with [C. Cortazar, M. Elgueta, J.D. Rossi, The blow-up problem for a semilinear parabolic equation with a potential, preprint, arXiv: math.AP/0607055, July 2006]. (C) 2007 Elsevier Inc. All rights reserved.