The first initial-boundary value problem for some nonlinear time degenerate parabolic equations
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- by Margaret C. Waid
- Proc. Amer. Math. Soc. 42 (1974), 487-494
- DOI: https://doi.org/10.1090/S0002-9939-1974-0336083-1
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Abstract:
Consider the nonuniformly parabolic operator \[ Lu = \sum \limits _{i,j = 1}^n {{a^{ij}}(x,t){u_{{x_i}{x_j}}} + \sum \limits _{i = 1}^n {{b^i}(x,t){u_{{x_i}}} - c(x,t,u){u_t} + d(x,t)u,} } \] 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 $|\bar \cdot |_\alpha ^D$ 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.References
- A.-P. Calderón, Uniqueness in the Cauchy problem for partial differential equations, Amer. J. Math. 80 (1958), 16–36. MR 104925, DOI 10.2307/2372819
- Avner Friedman, Partial differential equations of parabolic type, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964. MR 0181836 S. Middleman, Transport phenomena in the cardiovascular system, Wiley-Interscience Series on Biomedical Engineering, Wiley, New York, 1972.
- Margaret C. Waid, Second-order time degenerate parabolic equations, Trans. Amer. Math. Soc. 170 (1972), 31–55. MR 304860, DOI 10.1090/S0002-9947-1972-0304860-1
Bibliographic Information
- © Copyright 1974 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 42 (1974), 487-494
- MSC: Primary 35K55
- DOI: https://doi.org/10.1090/S0002-9939-1974-0336083-1
- MathSciNet review: 0336083