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Keywords:
parabolic initial-boundary value problem; inhomogeneous Robin boundary conditions; existence of weak solution; continuity up to the boundary; asymptotic behavior; asymptotically almost periodic solution
parabolic initial-boundary value problem; inhomogeneous Robin boundary conditions; existence of weak solution; continuity up to the boundary; asymptotic behavior; asymptotically almost periodic solution
Summary:
Second order parabolic equations on Lipschitz domains subject to inhomogeneous Neumann (or, more generally, Robin) boundary conditions are studied. Existence and uniqueness of weak solutions and their continuity up to the boundary of the parabolic cylinder are proved using methods from the theory of integrated semigroups, showing in particular the well-posedness of the abstract Cauchy problem in spaces of continuous functions. Under natural assumptions on the coefficients and the inhomogeneity the solutions are shown to converge to an equilibrium or to be asymptotically almost periodic.
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