A remark on the existence of positive periodic solutions of superlinear parabolic problems

Esteban, Maria J.

doi:10.1090/S0002-9939-1988-0915730-7

A remark on the existence of positive periodic solutions of superlinear parabolic problems
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by Maria J. Esteban PDF

Proc. Amer. Math. Soc. 102 (1988), 131-136 Request permission

Abstract:

We prove the existence of a solution for the following problem: \[ {\partial _t}u - \Delta u = f(t,x,u){\text { in (}}0,T) \times \Omega ,\;\quad u{\text { > }}0{\text { in (}}0,T) \times \Omega ,u(T) = u(0){\text { in }}\Omega ,\;\quad u = 0{\text { on (}}0,T) \times \partial \Omega ,\] where $\Omega$ is a bounded domain of ${R^N}$ and the function $f(t,x, \cdot )$ grows more slowly than ${u^\alpha }$ at $+ \infty$, with $\alpha {\text { < }}N/(N - 2)$. On démontre ici l’existence d’une solution positive pour le problème parabolique périodique suivant \[ \begin {gathered} {\partial _t}u - \Delta u = f(t,x,u){\text { in (0,}}T{\text {)}} \times \Omega {\text {,}}\quad \;u{\text { > }}0{\text { in (0,}}T) \times \Omega , u(T) = u(0){\text { in }}\Omega ,\quad \;u = 0{\text { on (}}0,T) \times \partial \Omega ,\end {gathered} \] où $\Omega$ est un domaine borné de ${R^N}$ et la fonction $f(t,x, \cdot )$ croit plus lentement que ${u^\alpha }$ à l’infini, avec $\alpha {\text { < }}N/(N - 2)$.

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A remark on a priori bounds for global solutions of semilinear equations

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