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Periodic parabolic problems with nonlinearities indefinite in sign. (English) Zbl 1146.35051

Authors’ abstract: Let \(\Omega\subset\mathbb{R}^{N}\) be a smooth bounded domain. We give sufficient conditions (which are also necessary in many cases) on two nonnegative functions \(a\), \(b\), that are possibly discontinuous and unbounded, for the existence of nonnegative solutions for semilinear Dirichlet periodic parabolic problems of the form \(Lu=\lambda a\left( x,t\right) u^{p}-b\left( x,t\right) u^{q}\) in \(\Omega\times\mathbb{R}\), where \(0 < p, q < 1\) and \(\lambda > 0 \). In some cases we also show the existence of solutions \(u_{\lambda}\) in the interior of the positive cone and that \(u_{\lambda}\) can be chosen such that \(\lambda\rightarrow u_{\lambda}\) is differentiable and increasing. A uniqueness theorem is also given in the case \(p\leq q\). All results remain valid for the corresponding elliptic problems.

MSC:

35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
35K55 Nonlinear parabolic equations
35B10 Periodic solutions to PDEs
35J65 Nonlinear boundary value problems for linear elliptic equations
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