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A unique continuation theorem involving a degenerate parabolic operator

Author: Alan V. Lair
Journal: Proc. Amer. Math. Soc. 66 (1977), 41-45
MSC: Primary 35K10
MathSciNet review: 0466968
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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 $ \vert L[u]\vert \leqslant c\vert u\vert$ 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.

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Keywords: Degenerate parabolic operator, unique continuation
Article copyright: © Copyright 1977 American Mathematical Society

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