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Some remarks on parabolic solutions of parabolic differential equations. (English) Zbl 0551.35044
Dynamical systems II, Proc. int. Symp., Gainesville/Fla. 1981, 227-246 (1982).
[For the entire collection see Zbl 0532.00014.]
The author studies the existence and stability of time-periodic solutions of semilinear and linear second order parabolic differential equations under Dirichlet boundary conditions in a bounded domain. Several results, analogous to known ones for elliptic problems are obtained; in particular, the result of J.-L. Kazdan and F. W. Warner [Commun. Pure Appl. Math. 28, 567-597 (1975; Zbl 0325.35038)] is generalized to this setting. The main tools are the notion (developed here) of principal eigenvalue and eigenfunction for linear periodic Dirichlet problems, and the Kolesov method of upper and lower solutions for proving the existence of solutions.
Reviewer: P.Fife

35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
35B10 Periodic solutions to PDEs
35B35 Stability in context of PDEs