A unique continuation theorem involving a degenerate parabolic operator

Abstract:

We consider the degenerate parabolic operator $L[u] = \gamma B[u] - {u_t}$ on a domain $D = \Omega \times (0,T]$ where $B[u] = \Sigma _{i,j = 1}^n{({a_{ij}}(x){u_{{x_j}}})_{{x_i}}}$ and $\gamma$ is an arbitrary complex number. Classically, $\gamma = 1$ and the real-valued matrix $({a_{ij}})$ is positive definite. We assume $({a_{ij}})$ is a real-valued symmetric matrix but not necessarily definite. We prove that any complex-valued function u which satisfies the inequality $|L[u]| \leqslant c|u|$ for some nonnegative constant c and vanishes initially as well as on the boundary of $\Omega$ must vanish on all of D. The theorem is particularly useful in studying uniqueness for many systems which are not parabolic.