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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



The first initial-boundary value problem for some nonlinear time degenerate parabolic equations

Author: Margaret C. Waid
Journal: Proc. Amer. Math. Soc. 42 (1974), 487-494
MSC: Primary 35K55
MathSciNet review: 0336083
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Abstract: Consider the nonuniformly parabolic operator

$\displaystyle Lu = \sum\limits_{i,j = 1}^n {{a^{ij}}(x,t){u_{{x_i}{x_j}}} + \sum\limits_{i = 1}^n {{b^i}(x,t){u_{{x_i}}} - c(x,t,u){u_t} + d(x,t)u,} } $

where u, $ {a^{ij}},{b^i}$ c, d are bounded, real-valued functions defined on a domain $ D = \Omega \times [0,T] \subset {R^{n + 1}}$. Assume that $ c(x,t,u)$ is Lipschitz continuous in $ \vert\bar \cdot \vert _\alpha ^D$ of $ {C_\alpha }(D)$, and that $ c(x,t,u) \geqq 0$ on D. Sufficient conditions on c are found which guarantee existence of a unique solution $ u \in {\bar C_{2 + \alpha }}$ to the first initial-boundary value problem $ Lu = f(x,t), u = \psi $, on the normal boundary of D, where $ \psi \in {\bar C_{2 + \alpha }}$. Existence is proved by direct application of a fixed point theorem due to Schauder using existence of a solution to the linear problem as well as a priori estimates.

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Keywords: Degenerate parabolic equation, nonlinear parabolic operators, nonuniformly parabolic operators, existence theorems for nonlinear equations, applications of fixed point theorems
Article copyright: © Copyright 1974 American Mathematical Society