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Semilinear parabolic equations with prescribed energy. (English) Zbl 0856.35063
Summary: We study the reaction-diffusion equation ${u}_{t}={\Delta }u+f\left(u,k\left(t\right)\right)$ subject to appropriate initial and boundary conditions, where $f\left(u,k\left(t\right)\right)={u}^{p}-k\left(t\right)$ or $k\left(t\right){u}^{p}$, with $p>1$ and $k\left(t\right)$ an unknown function. An additional energy type condition is imposed in order to find the solution pair $u\left(x,t\right)$ and $k\left(t\right)$. 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 functions $f$, for some initial data. For the second class of functions $f$, the solution blows up in finite time if $p>n/\left(n-2\right)$ while it exists globally in time if $1, no matter how large the initial value is. Partial generalizations are given for the case where $f\left(u,k\left(t\right)\right)$ appears in the boundary conditions.
##### MSC:
 35K57 Reaction-diffusion equations 35B40 Asymptotic behavior of solutions of PDE
##### Keywords:
blow up; energy type condition
##### References:
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