Positive unstable periodic solutions for superlinear parabolic equations. (English) Zbl 0828.35063
The paper is concerned with stable and unstable \(T\)-periodic solutions for a semilinear parabolic problem
\[
{\partial u \over \partial t} - \Delta u = u^p + h(t,x) \quad \text{in } \mathbb{R}_+ \times \Omega
\]
\[ u = 0 \quad \text{in } \mathbb{R}_+ \times \partial \Omega, \qquad u(t) = u(t + T) \quad \text{in } \overline \Omega_1, \quad u > 0 \text{ in } \mathbb{R}_+ \times \Omega, \] where \(1 < p < (N + 2)/(N - 2)\) if \(N \geq 3\) and \(1 < p < \infty\) if \(N \leq 2\). It is proved that there exists a stable and an unstable positive \(T\)-periodic solution for this problem if \(h\) is sufficiently small in \(L^\infty\).
\[ u = 0 \quad \text{in } \mathbb{R}_+ \times \partial \Omega, \qquad u(t) = u(t + T) \quad \text{in } \overline \Omega_1, \quad u > 0 \text{ in } \mathbb{R}_+ \times \Omega, \] where \(1 < p < (N + 2)/(N - 2)\) if \(N \geq 3\) and \(1 < p < \infty\) if \(N \leq 2\). It is proved that there exists a stable and an unstable positive \(T\)-periodic solution for this problem if \(h\) is sufficiently small in \(L^\infty\).
Reviewer: G.Aniculaesei (Iaşi)
MSC:
| 35K60 | Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations |
| 35K20 | Initial-boundary value problems for second-order parabolic equations |
| 35B10 | Periodic solutions to PDEs |
Keywords:
semilinear parabolic problemReferences:
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