Semilinear parabolic equations with prescribed energy
In this paper we study the reaction-diffusion equationut=Δu+f(u, k(t)) subject to appropriate initial and boundary conditions, wheref(u, k(t))=up−k(t) ork(t)up, withp>1 andk(t) an unknown function. An additional energy type condition is imposed in order to find the solution pairu(x, t) andk(t). This type of problem is frequently encountered in nuclear reaction processes, where the reaction is known to be very strong, but the total energy is controlled. It is shown that the solution blows up in finite time for the first class of functionsf, for some initial data. For the second class of functionsf, the solution blows up in finite time ifp>n/(n−2) while it exists globally in time if 1<p<n/(n−2), no matter how large the initial value is. Partial generalizations are given for the case wheref(u, k(t)) appears in the boundary conditions.
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