Almost periodicity of parabolic evolution equations with inhomogeneous boundary values
Under consideration is the existence and uniqueness of the almost periodic solution to the inhomogeneous evolution boundary problem in Banach spaces \[ u'(t)=A_m(t) u(t) + g(t), \;B(t) u(t)=h(t), \quad t\in{\mathbb R}, \] where typically \(A_m(t)\) is an elliptic partial differential operator acting in \(L^p(\Omega)\) and \(B(t)\) is a boundary operator mapping \(W^2_p(\partial\Omega)\) into a space like \(W^{1-1/p}_p(\partial\Omega)\). The problem is to show that the solution \(u\) inherits the almost periodicity of the inhomogeneities \(g\) and \(h\). The main results concern the case that the evolution family of the problem has exponential dichotomies on time intervals \((-\infty, -T]\) and \([T, \infty)\). The main theorem gives a Fredholm alternative for mild solution to the problem in a space of continuous functions asymptotically almost periodic on \({\mathbb R}_-\) and \({\mathbb R}_+\). These results are obtained by the technique of evolution equations in extrapolation spaces and periodic equations in the context of exponential dichotomies.
scientific article; zbMATH DE number 5636884