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

MathSciNet review:
938660

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Compactness of the support is discussed of a solution to the Cauchy problem for the porous medium equation , in of arbitrary dimension , where is a nondecreasing function on . It is shown that if for , then for all

The result is well known when , but the study for has somehow been neglected.

**[1]**Philippe Benilan, Haim Brezis, and Michael G. Crandall,*A semilinear equation in 𝐿¹(𝑅^{𝑁})*, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4)**2**(1975), no. 4, 523–555. MR**0390473****[2]**Haïm Brézis and Michael G. Crandall,*Uniqueness of solutions of the initial-value problem for 𝑢_{𝑡}-Δ𝜙(𝑢)=0*, J. Math. Pures Appl. (9)**58**(1979), no. 2, 153–163. MR**539218****[3]**J. Ildefonso Díaz Díaz,*Solutions with compact support for some degenerate parabolic problems*, Nonlinear Anal.**3**(1979), no. 6, 831–847. MR**548955**, 10.1016/0362-546X(79)90051-8**[4]**Barry F. Knerr,*The porous medium equation in one dimension*, Trans. Amer. Math. Soc.**234**(1977), no. 2, 381–415. MR**0492856**, 10.1090/S0002-9947-1977-0492856-3**[5]**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****[6]**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**, 10.1512/iumj.1984.33.33001**[7]**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**, 10.1090/S0002-9939-1984-0760935-8

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
35K55,
35K65

Retrieve articles in all journals with MSC: 35K55, 35K65

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