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Solutions with compact support of the porous medium equation in arbitrary dimensions


Author: Michiaki Watanabe
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
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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\vert x \right\vert \geq R,R > 0$, then for all $ t \geq 0$

$\displaystyle u(t,x) = 0\quad {\text{a}}{\text{.e}}{\text{.}}\left\vert x \right\vert \geq R + C{t^{1/2}}$

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 [Enhancements On Off] (What's this?)

  • [1] Ph. Bénilan, H. Brézis, and M. G. Crandall, A semilinear elliptic equation in $ {L^1}({R^N})$, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 2 (1975), 523-555. MR 0390473 (52:11299)
  • [2] H. Brézis and M. G. Crandall, Uniqueness of solutions of the initial value problem for $ {u_t} - \Delta \phi (u) = 0$, J. Math. Pures Appl. 58 (1979), 153-163. MR 539218 (80e:35029)
  • [3] J. I. Diaz Diaz, Solutions with compact support for some degenerate parabolic problems, Nonlinear Anal., Theory, Methods and Appl. 3 (1979), 831-847. MR 548955 (80i:35107)
  • [4] B. F. Knerr, The porous medium equation in one dimension, Trans. Amer. Math. Soc. 234 (1977), 381-415. MR 0492856 (58:11917)
  • [5] L. A. Peletier, The porous media equation, Applications of Nonlinear Analysis in the Physical Sciences (H. Amann, N. Bazley, and K. Kirchgässner, eds.), Pitman, 1981, pp. 229-241. MR 659697 (83k:76076)
  • [6] M. Schatzman, Stationary solutions and asymptotic behavior of a quasilinear degenerate parabolic equation, Indiana Univ. Math. J. 33 (1984), 1-29. MR 726104 (85i:35081)
  • [7] M. Watanabe, Trotter's product formula for semigroups generated by quasilinear elliptic operators, Proc. Amer. Math. Soc. 92 (1984), 509-514. MR 760935 (86g:35103)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1988-0938660-3
Keywords: Cauchy problem for porous medium equation, solution with compact support
Article copyright: © Copyright 1988 American Mathematical Society

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