Singular limit of solutions of as

Author:
Kin Ming Hui

Journal:
Trans. Amer. Math. Soc. **347** (1995), 1687-1712

MSC:
Primary 35K55; Secondary 35B40

DOI:
https://doi.org/10.1090/S0002-9947-1995-1290718-6

MathSciNet review:
1290718

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We will show that the solutions of in converge weakly in for any compact subset of as to the solution of the porous medium equation in with where , satisfies for some function such that whenever a.e. . The convergence is uniform on compact subsets of .

**[A]**J. R. Anderson,*Local existence and uniqueness of solutions of degenerate parabolic equations*, Comm. Partial Differential Equations**16**(1991), 105-143. MR**1096835 (92d:35163)****[Ar]**D. G. Aronson,*The porous medium equation*, CIME lectures in Some Problems in Nonlinear Diffusion, Lecture Notes in Math., vol. 1224, Springer-Verlag, New York, 1986. MR**877986 (88a:35130)****[B]**G. I. Barenblatt,*On self-similar motions of compressible fluid in porous media*, Prikl. Mat. Mech.**16**(1952), 679-698. (Russian) MR**0052948 (14:699h)****[BBH]**P. Bénilan, L. Boccardo and M. Herrero,*On the limit of solutions of**as*, Some Topics in Nonlinear PDE's, Proceedings Int. Conf. Torino 1989, M.Bertsch et al., ed.**[CF]**L. A. Caffarelli and A. Friedman,*Asymptotic behaviour of solutions of**as*, Indiana Univ. Math. J.**36**(1987), 711-728. MR**916741 (88m:35075)****[DFK]**B. E. J. Dahlberg, E. B. Fabes and C. Kenig,*A Fatou theorem for solutions of the porous medium equation*, Proc. Amer. Math. Soc.**91**(1984), 205-212. MR**740172 (85e:35064)****[DK]**B. E. J. Dahlberg and C. Kenig,*Nonnegative solutions of generalized porous medium equations*, Rev. Mat. Iberoamericana**2**(1986), 267-305. MR**908054 (88k:35223)****[DiK]**J. I. Diaz and R. Kersner,*On a nonlinear degenerate parabolic equation in infiltration or evaporation through a porous medium*, J. Differential Equations**69**(1987), 368-403. MR**903393 (88i:35088)****[EZ]**M. Escobedo and E. Zuazua,*Large time behaviour for solutions of a convection diffusion equation in*, J. Funct. Anal.**102**(1991), 119-161. MR**1124296 (92i:35063)****[ERV]**J. R. Esteban, A. Rodriguez and J. L. Vazquez,*A nonlinear heat equation with singular diffusivity*, Comm. Partial Differential Equations**13**(1988), 985-1039. MR**944437 (89h:35167)****[G1]**B. H. Gilding,*Properties of solutions of an equation in the theory of infiltration*, Arch. Rational Mech. Anal.**65**(1977), 203-225. MR**0447847 (56:6157)****[G2]**-,*A nonlinear degenerate parabolic problem*, Ann. Scuola Norm. Sup. Pisa**4**(1977), 393-432. MR**0509720 (58:23077)****[HP]**M. A. Herrero amd M. Pierre,*The Cauchy problem for**when*, Trans. Amer. Math. Soc.**291(1)**(1985), 145-158. MR**797051 (86i:35065)****[H1]**K. M. Hui,*Asymptotic behaviour of solutions of**as*, Nonlinear Anal., TMA**21**(1993), 191-195. MR**1233960 (94h:35103)****[H2]**K. M. Hui,*Singular limit of solutions of the generalized**-Laplacian equation*, Nonlinear Anal., TMA (to appear).**[Ke]**R. Kersner,*Degenerate parabolic equations with general nonlinearities*, Nonlinear Anal., TMA**4**(1980), 1043-1062. MR**591298 (82h:35056)****[LSU]**O. A. Ladyzenskaya, V. A. Solonnikov and N. N. Uraltceva,*Linear and quasilinear equations of parabolic type*, Transl. Math. Monos., Vol. 23, Amer. Math. Soc., Providence, RI, 1968.**[PV]**A. De Pablo and J. L. Vazquez,*Travelling waves and finite propagation in a reaction-diffusion equation*, J. Differential Equations**93**(1991), 19-61. MR**1122305 (92h:35108)****[P]**L. A. Peletier,*The porous medium equation*, Applications of Nonlinear Analysis in the Physical Sciences, (H. Amann, N. Bazley, and K. Kirchgassner, eds.), Pitman, Boston, 1981.**[S1]**P. E. Sacks,*Continuity of solutions of a singular parabolic equation*, Nonlinear Anal., TMA**7**(1983), 387-409. MR**696738 (84d:35081)****[S2]**P. E. Sacks,*A singular limit problem for the porous medium equation*, J. Math. Anal. Appl. (1989), 456-466. MR**1001869 (90f:35099)****[SX]**R. E. Showalter and X. Xu,*An approximate scalar conservation law from dynamics of gas absorption*, J. Differential Equations**83**(1990), 145-165. MR**1031381 (90k:35170)****[S]**E. M. Stein,*Singular integral and differentiability properties of functions*, Princeton Univ. Press, Princeton, NJ, 1971. MR**0290095 (44:7280)****[X]**X. Xu,*Asymptotics behaviour of solutions of hyperbolic conservation laws**as**with inconsistent values*, Proc. Royal Soc. Edinburgh**113A**(1989), 61-71. MR**1025454 (90k:35043)**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
35K55,
35B40

Retrieve articles in all journals with MSC: 35K55, 35B40

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1995-1290718-6

Keywords:
Asymptotic behaviour,
porous medium equation with convection term,
existence,
uniqueness,
nonnegative solutions

Article copyright:
© Copyright 1995
American Mathematical Society