Boundary value problems for parabolic systems (Q1069035)

Existence and uniqueness theorems are proved for weak solutions of a specified class of second order parabolic boundary value problems. The case of bounded domains \(Q=\Omega \times (0,T)\subset R^{n+1}\) is considered first. Subject to various technical hypotheses, an existence and uniqueness theorem is proved, and estimates for the solution are given in the space \(L^ 2(0,T;V)\cap L^{2,\infty}(Q),\) where V contains \(H^ 1_ 0(\Omega)\). Unlike previous results of this type, the estimates obtained are independent of T. Similar results, depending on the measure of \(\Omega\), and on T, were obtained by M. Marino. In the final section these results are generalized to unbounded domains. The results are related to those of O. Arena, though with different hypotheses.