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    Some Blow-Up Problems for a Semilinear Parabolic Equation with a Potential

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    The blow-up rate estimate for the solution to a semilinear parabolic equation ut=Δu+V(x)up1uu_t=\Delta u+V(x) |u|^{p-1}u in Ω×(0,T)\Omega \times (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\vf (x) as MM goes to infinity, which have been found in \cite{cer}, are improved under some reasonable and weaker conditions compared with \cite{cer}.Comment: 29 page