On positive solutions of semilinear periodic-parabolic problems. (English) Zbl 0556.35066

Hess, P.

Infinite-dimensional systems, Proc. Conf., Retzhof/Austria 1983, Lect. Notes Math. 1076, 101-114 (1984).

[For the entire collection see Zbl 0535.00013.]
The author discusses the eigenvalue problem \(Lu=\lambda m(x,t)u,\) \(u(x,t)=0\) if \(x\in \partial \Omega\), u is T periodic in t, where L is a uniformly parabolic second order linear operator and m is a weight function which changes sign. He discusses analogues of his and T. Kato’s earlier work [Commun. Partial Differ. Equations 5, 999-1030 (1980; Zbl 0477.35075)] for the elliptic problem and sketches the proofs. Full details appear in a paper of A. Beltramo and the author [ibid. 9, 919-941 (1984)]. He also discusses estimates for the smallest real eigenvalue and applications to nonlinear equations.

Reviewer: E.Dancer

MSC:

35K20	Initial-boundary value problems for second-order parabolic equations
35P05	General topics in linear spectral theory for PDEs
35P15	Estimates of eigenvalues in context of PDEs
35K60	Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
35P30	Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs

Keywords:

semilinear parabolic problem; uniformly parabolic second order linear operator; weight function; smallest real eigenvalue

Citations:

Zbl 0535.00013; Zbl 0477.35075