Inhomogeneous parabolic Neumann problems. (English) Zbl 1349.35154

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.


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
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