On the analysis of blowup phenomena for a nonlinear parabolic equation

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In this research, we investigated a blowup problem for a semilinear diffusion equation with power nonlinerity. The blowup rate for the equation under the Neumann boundary condition or the Dirichlet boundary condition in a non-convex domain is considered. A well-lnown result due to Giga Kohn cannot be applied to these cases, so we showed the blowup is of type I making use of Liouville type theorem. Here a solution of the equation is said to exhibit the type I blowup if the blowup rate is as same as that of solutions to the corresponding ordinary equation. Next, the 1 location of the blowup set of solutions to the Cauchy-Neumann problem is described by the property of the domain in the case of large diffusion. We also studied the continuation after blowup for supercritical exponent in the Sobolev sense. Solutions blowing up in finite time and being a classical solution for all time after blowup with various behaviors as time tends to infinity. Moreover, we got a solution which blows up in finite time and becomes regular immediately after the blowup time and blows up again.