Inhomogeneous parabolic Neumann problems. (English) Zbl 1349.35154

Nittka, Robin

Czech. Math. J. 64, No. 3, 703-742 (2014).

The author studies second order parabolic equations on Lipschitz domains subject to inhomogeneous Neumann (or, more generally Robin) boundary conditions \[ \begin{cases} u_t(t,x)-Au(t,x)= f(t,x) & t>0, \;x\in\Omega\\ {{\partial u(t,z)}\over{\partial \nu}}= g(t,z) & t>0, z\in \partial \Omega\\ u(0,x)=u_0(x) & x\in\Omega\,.\end{cases} \] 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.

Reviewer: Lubomira Softova (Aversa)

MSC:

35K20	Initial-boundary value problems for second-order parabolic equations
35B15	Almost and pseudo-almost periodic solutions to PDEs
35B65	Smoothness and regularity of solutions to PDEs
35D30	Weak solutions to PDEs

Keywords:

parabolic initial-boundary value problem; inhomogeneous Neumann boundary conditions; Robin boundary conditions; existence of weak solutions; continuity up to the boundary; asymptotic behavior; asymptotically almost periodic solutions

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