Solutions with compact support of the porous medium equation in arbitrary dimensions
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- by Michiaki Watanabe PDF
- Proc. Amer. Math. Soc. 103 (1988), 149-152 Request permission
Abstract:
Compactness of the support is discussed of a solution $u$ to the Cauchy problem for the porous medium equation ${u_t} = \Delta \phi (u),t > 0$, in ${R^N}$ of arbitrary dimension $N \geq 1$, where $\phi$ is a nondecreasing function on ${R^1}$. It is shown that if $u(0,x) = 0$ for $\left | x \right | \geq R,R > 0$, then for all $t \geq 0$ \[ u(t,x) = 0\quad {\text {a}}{\text {.e}}{\text {.}}\left | x \right | \geq R + C{t^{1/2}}\] with a constant $C$ depending on $\phi$ and $u(0, \cdot )$. The result is well known when $N = 1$, but the study for $N > 1$ has somehow been neglected.References
- Philippe Benilan, Haim Brezis, and Michael G. Crandall, A semilinear equation in $L^{1}(R^{N})$, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 2 (1975), no. 4, 523–555. MR 390473
- Haïm Brézis and Michael G. Crandall, Uniqueness of solutions of the initial-value problem for $u_{t}-\Delta \varphi (u)=0$, J. Math. Pures Appl. (9) 58 (1979), no. 2, 153–163. MR 539218
- J. Ildefonso Díaz Díaz, Solutions with compact support for some degenerate parabolic problems, Nonlinear Anal. 3 (1979), no. 6, 831–847. MR 548955, DOI 10.1016/0362-546X(79)90051-8
- Barry F. Knerr, The porous medium equation in one dimension, Trans. Amer. Math. Soc. 234 (1977), no. 2, 381–415. MR 492856, DOI 10.1090/S0002-9947-1977-0492856-3
- L. A. Peletier, The porous media equation, Applications of nonlinear analysis in the physical sciences (Bielefeld, 1979), Surveys Reference Works Math., vol. 6, Pitman, Boston, Mass.-London, 1981, pp. 229–241. MR 659697
- M. Schatzman, Stationary solutions and asymptotic behavior of a quasilinear degenerate parabolic equation, Indiana Univ. Math. J. 33 (1984), no. 1, 1–29. MR 726104, DOI 10.1512/iumj.1984.33.33001
- Michiaki Watanabe, Trotter’s product formula for semigroups generated by quasilinear elliptic operators, Proc. Amer. Math. Soc. 92 (1984), no. 4, 509–514. MR 760935, DOI 10.1090/S0002-9939-1984-0760935-8
Additional Information
- © Copyright 1988 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 103 (1988), 149-152
- MSC: Primary 35K55; Secondary 35K65
- DOI: https://doi.org/10.1090/S0002-9939-1988-0938660-3
- MathSciNet review: 938660