Weak solutions of the porous medium equation in a cylinder
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- by Björn E. J. Dahlberg and Carlos E. Kenig PDF
- Trans. Amer. Math. Soc. 336 (1993), 701-709 Request permission
Abstract:
We show that if $D \subset {{\mathbf {R}}^n}$ is a bounded domain with smooth boundary, and $u \in {L^m}(D \times (\varepsilon ,T))$, $u \geq 0$, solves $\frac {{\partial u}} {{\partial t}} = \Delta {u^m}$, $m > 1$, in the sense of distributions on $D \times (0,T)$, and vanishes on $\partial D \times (0,T)$ in a suitable weak sense, then $u$ is Hölder continuous in $\overline D \times (0,T)$.References
- D. G. Aronson and L. A. Caffarelli, The initial trace of a solution of the porous medium equation, Trans. Amer. Math. Soc. 280 (1983), no. 1, 351–366. MR 712265, DOI 10.1090/S0002-9947-1983-0712265-1
- Philippe Bénilan, Michael G. Crandall, and Michel Pierre, Solutions of the porous medium equation in $\textbf {R}^{N}$ under optimal conditions on initial values, Indiana Univ. Math. J. 33 (1984), no. 1, 51–87. MR 726106, DOI 10.1512/iumj.1984.33.33003
- Björn E. J. Dahlberg and Carlos E. Kenig, Nonnegative solutions of the porous medium equation, Comm. Partial Differential Equations 9 (1984), no. 5, 409–437. MR 741215, DOI 10.1080/03605308408820336
- Björn E. J. Dahlberg and Carlos E. Kenig, Nonnegative solutions of generalized porous medium equations, Rev. Mat. Iberoamericana 2 (1986), no. 3, 267–305. MR 908054
- Björn E. J. Dahlberg and Carlos E. Kenig, Nonnegative solutions of the initial-Dirichlet problem for generalized porous medium equations in cylinders, J. Amer. Math. Soc. 1 (1988), no. 2, 401–412. MR 928264, DOI 10.1090/S0894-0347-1988-0928264-9 —, Weak solutions of the porous medium equation, preprint.
- L. L. Helms, Introduction to potential theory, Pure and Applied Mathematics, Vol. XXII, Wiley-Interscience [A division of John Wiley & Sons, Inc.], New York-London-Sydney, 1969. MR 0261018 M. Pierre, Uniqueness of the solutions of ${u_t} - \Delta \varphi (u) = 0$ with initial datum a measure, Nonlinear Analysis 7 (1983), 387-409.
- Paul E. Sacks, Continuity of solutions of a singular parabolic equation, Nonlinear Anal. 7 (1983), no. 4, 387–409. MR 696738, DOI 10.1016/0362-546X(83)90092-5
Additional Information
- © Copyright 1993 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 336 (1993), 701-709
- MSC: Primary 35D05; Secondary 35K55, 76S05
- DOI: https://doi.org/10.1090/S0002-9947-1993-1085940-6
- MathSciNet review: 1085940