Strong Maximum Principles for Time Degenerate Parabolic Operators
Margaret C. Waid · SIAM Journal on Applied Mathematics · 1974
We study the degenerate parabolic operator $Lu = \sum olimits_{i,j = 1}^n {a^{ij} u_{x_i x_j } } + \sum olimits_{i = 1}^n {b^i u_{x_i } - cu_t + du} $, where $u,a^{ij} ,b^i ,c,d$ are bounded, real-valued functions defined on a bounded domain $D \subset R^{n + 1} $. Assume that L is parabolic. Though classically $c \equiv 1$, we assume only that c is nonnegative. Using proofs substantially different from classical ones, which rely heavily on the minimum of c in $\bar D$ being positive, we prove the usual strong maximum principles as well as uniqueness of solutions to the first and second initial-boundary value problems.