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Semilinear Periodic-Parabolic Equations with Nonlinear Boundary Conditions

✍ Scribed by M.N. Nkashama

Publisher
Elsevier Science
Year
1996
Tongue
English
Weight
839 KB
Volume
130
Category
Article
ISSN
0022-0396

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✦ Synopsis

Recently much work has been devoted to periodic-parabolic equations with linear homogeneous boundary conditions. However, very little has been accomplished in the literature for periodic-parabolic problems with nonlinear boundary conditions. It is the purpose of this paper to prove existence and regularity results for (classical) periodic solutions to semilinear second order parabolic partial differential equations with nonlinear boundary conditions provided ordered upper and lower solutions are given. Fractional order function spaces, Ehrling Gagliardo Nirenberg and Lions Peetre Caldero n type interpolation inequalities for functions in (anisotropic) Sobolev Slobodecki@$ spaces play an important role in the obtainment of a priori boundary and interior estimates. In proving our existence results we make use of topological degree techniques and regularity results for linear parabolic partial differential equations under linear nonhomogeneous boundary conditions. We also indicate how one can obtain minimal and maximal timeperiodic solutions to parabolic problems with nonlinear boundary conditions.

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## Abstract We study a semilinear parabolic partial differential equation of second order in a bounded domain Ξ© βŠ‚ ℝ^__N__^, with nonstandard boundary conditions (BCs) on a part Ξ“~non~ of the boundary βˆ‚Ξ©. Here, neither the solution nor the flux are prescribed pointwise. Instead, the total flux throu