Non-variational parabolic systems with solutions satisfying a condition of generalized periodicity (Q1313214)

The author studies a class of nonlinear parabolic systems in non- divergent form of the type \[ a \bigl( x,t,H(u) \bigr) - {\partial u \over \partial t} = f(x,t) \tag{1} \] on a domain \(Q_T = \Omega \times (\tau, \tau + T)\), where \(\Omega \subset \mathbb{R}_n\) is bounded, open and convex, \(u : \Omega \to \mathbb{R}_N\), \(H(u) = \{D_{ij} u\}_{i,j = 1, \ldots, n}\) and \(a\) is a Caratheodory function satisfying the condition (A) there exist \(\alpha, \gamma, \delta > 0\) with \(\gamma + \delta < 1\) so that \[ \Bigl |\sum_{i = 1, \ldots, n} \tau_{ii} - \alpha \bigl \{a(x,t, \tau + \eta) - a(x,t, \eta) \bigr\} \Bigr |_N \leq \gamma |\tau |_{n^2N} + \delta |\sum_i \tau_{ii} |_N, \] \(\forall (x,t) \in Q_T, \forall \tau, \eta \in\mathbb{R}_{n^2N}\). For a bounded linear operator \(B\) on \(L_2 (\Omega)\) (which commutes with derivatives and with \(a)\) there is proved existence and uniqueness of a weak solution \(u \in W^{1,2}_T\) of (1) satisfying generalized periodicity condition \[ u(x,t) = Bu(x,t + T) \] a.e. in \(\Omega\). The standard periodic problem corresponds to the case \(B = I\). The proof is based on the analogous assertion for the heat equation and a perturbation argument using condition \((A)\).