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Boundedness of global solutions for a heat equation with exponential gradient source. (English) Zbl 1254.35122

Summary: We consider a one-dimensional semilinear parabolic equation with exponential gradient source and provide a complete classification of large time behavior of the classical solutions: either the space derivative of the solution blows up in finite time with the solution itself remaining bounded, or the solution is global and converges in \(C^1\) norm to the unique steady state. The main difficulty is to prove \(C^1\) boundedness of all global solutions. To do so, we explicitly compute a nontrivial Lyapunov’s functional by carrying out the method of Zelenyak.

MSC:

35K58 Semilinear parabolic equations
35B40 Asymptotic behavior of solutions to PDEs
35B44 Blow-up in context of PDEs
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