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Periodic solutions of the boundary value problem for the nonlinear heat equation. (English) Zbl 0555.35062

The authors deal with the periodic Dirichlet problem for the nonlinear heat equation \[ u_ t(t,x) - u_{xx}(t,x) = g(t,x,u(t,x)) + h(t,x), \]
\[ u(t,0) = u(t,\pi) = 0 \text{ for }t\in [0,2\pi], \quad u(0,x) - u(2\pi,x) = 0 \text{ for }x\in [0,\pi], \] where g satisfies Carathéodory type conditions. Extending previous results of Fuchik, Mawhin, Vejvoda, the authors prove the existence of a generalized periodic solution: a typical condition to this end is the existence, for some \(m>0\), of functions \(\gamma\), \(\Gamma\) (satisfying certain additional properties) such that \(m^ 2\leq \gamma(t,x) \leq \lim \inf_{| u| \to \infty} g(t,x,u) / u \leq \limsup_{| u| \to \infty} g(t,x,u) / u \leq \Gamma(t,x) \leq (m+1)^ 2\). The proof is based on Leray-Schauder techniques as those, involving as well conditions of the same type, used by other authors in case of ODE’s, and elliptic and wave PDE’s.
Reviewer: P.de Mottoni

MSC:

35K55 Nonlinear parabolic equations
35B10 Periodic solutions to PDEs
47H10 Fixed-point theorems
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