THE BLOWUP PROPERTY OF SOLUTIONS TO SOME DIFFUSION-EQUATIONS WITH LOCALIZED NONLINEAR REACTIONS

In this paper we investigate the blowup property of solutions to the equation ut = Δu + f{hook}(u(x0, t)), where x0 is a fixed point in the domain. We show that under certain conditions the solution blows up in finite time. Moreover, we prove that the set of all blowup points is the whole region. Furthermore, the growth rate of solutions near the blowup time is also derived. Finally, the results are generalized to the following nonlocal reaction-diffusion equation ut = Δu + ∝Ω f{hook}(u) dx. © 1992.