Semilinear Parabolic Equations with Prescribed Energy
نویسنده
چکیده
In this paper we study the following reaction-di usion equation ut = u+ f(u; k(t)) subject to appropriate initial and boundary conditions, where f(u; k(t)) = u k(t) or k(t)u with p > 1 and k(t) is an unknown function. An additional energy type condition is imposed in order to nd the solution u(x; t) and k(t). This type of problem is frequently encountered in nuclear reaction process, where the reaction is known to be very strong, but the total energy is controlled. It is shown that the solution blows up in nite time for the rst class of function f for some initial data. For the second class of function f , the solution blows up in nite time if p > n=(n 2) while it exists globally in time if 1 < p < n=(n 2), no matter how large the initial value is. Partial results are generalized into the case where f(u; k(t)) appears on the boundary.
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