On time periodic solutions of Dirichlet problem for degenerate parabolic equations of nondivergence type (Q1922930)

The authors study the degenerate parabolic equation \(u_t=u^p(\Delta u+u+f)\) under Dirichlet conditions on a bounded domain. They consider the time-periodic problem \(u(t+T)= u(t)\) for a given \(T\)-periodic force \(f\). They prove for positive \(f\) and all \(p\) the existence of a positive periodic solution \(u\), provided the domain is small (precisely for \(\lambda_1>1\), with \(\lambda_1\) the first eigenvalue of the Laplacian), while for negative \(f\) and \(\lambda_1<1\) there is a nonnegative solution provided \(1\leq p<3\). It is interesting to note again, that \(\lambda_1=1\) and \(p=3\) are borderlines for these problems.