## 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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