Admissible conditions for parabolic equations degenerating at infinity

Kamin, Sh.; Pozio, M.; Tesei, A.

doi:10.1090/S1061-0022-08-00996-5

Admissible conditions for parabolic equations degenerating at infinity
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by Sh. Kamin, M. A. Pozio and A. Tesei

St. Petersburg Math. J. 19 (2008), 239-251

DOI: https://doi.org/10.1090/S1061-0022-08-00996-5

Published electronically: February 7, 2008

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Abstract:

Well-posedness in $L^\infty (\mathbb {R}^n)$ $(n \ge 3)$ of the Cauchy problem is studied for a class of linear parabolic equations with variable density. In view of degeneracy at infinity, some conditions at infinity are possibly needed to make the problem well-posed. Existence and uniqueness results are proved for bounded solutions that satisfy either Dirichlet or Neumann conditions at infinity.

References

Haïm Brezis and Shoshana Kamin, Sublinear elliptic equations in $\textbf {R}^n$, Manuscripta Math. 74 (1992), no. 1, 87–106. MR 1141779, DOI 10.1007/BF02567660
S. D. Eidelman and S. Kamin, On stabilization of solutions of the Cauchy problem for parabolic equations degenerating at infinity, Asymptot. Anal. 45 (2005), no. 1-2, 55–71. MR 2181259
S. Eidelman, S. Kamin, and F. Porper, Uniqueness of solutions of the Cauchy problem for parabolic equations degenerating at infinity, Asymptot. Anal. 22 (2000), no. 3-4, 349–358. MR 1753769
D. Eidus, The Cauchy problem for the nonlinear filtration equation in an inhomogeneous medium, J. Differential Equations 84 (1990), no. 2, 309–318. MR 1047572, DOI 10.1016/0022-0396(90)90081-Y
D. Eidus and S. Kamin, The filtration equation in a class of functions decreasing at infinity, Proc. Amer. Math. Soc. 120 (1994), no. 3, 825–830. MR 1169025, DOI 10.1090/S0002-9939-1994-1169025-2
M. Guedda, D. Hilhorst, and M. A. Peletier, Disappearing interfaces in nonlinear diffusion, Adv. Math. Sci. Appl. 7 (1997), no. 2, 695–710. MR 1476273
Kazuhiro Ishige and Minoru Murata, An intrinsic metric approach to uniqueness of the positive Cauchy problem for parabolic equations, Math. Z. 227 (1998), no. 2, 313–335. MR 1609065, DOI 10.1007/PL00004378
P. Rosenau and S. Kamin, Nonlinear diffusion in a finite mass medium, Comm. Pure Appl. Math. 35 (1982), no. 1, 113–127. MR 637497, DOI 10.1002/cpa.3160350106
Shoshana Kamin, Robert Kersner, and Alberto Tesei, On the Cauchy problem for a class of parabolic equations with variable density, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 9 (1998), no. 4, 279–298 (1999) (English, with English and Italian summaries). MR 1722787
Robert Kersner and Alberto Tesei, Well-posedness of initial value problems for singular parabolic equations, J. Differential Equations 199 (2004), no. 1, 47–76. MR 2041511, DOI 10.1016/j.jde.2003.10.015
Minoru Murata, Nonuniqueness of the positive Dirichlet problem for parabolic equations in cylinders, J. Funct. Anal. 135 (1996), no. 2, 456–487. MR 1370610, DOI 10.1006/jfan.1996.0016
O. A. Ladyženskaja, V. A. Solonnikov, and N. N. Ural′ceva, Linear and quasilinear equations of parabolic type, Translations of Mathematical Monographs, Vol. 23, American Mathematical Society, Providence, R.I., 1968 (Russian). Translated from the Russian by S. Smith. MR 0241822
O. A. Oleĭnik, A. S. Kalašinkov, and Yuĭ-Lin′Čžou, The Cauchy problem and boundary problems for equations of the type of non-stationary filtration, Izv. Akad. Nauk SSSR Ser. Mat. 22 (1958), 667–704 (Russian). MR 0099834
Yehuda Pinchover, On uniqueness and nonuniqueness of the positive Cauchy problem for parabolic equations with unbounded coefficients, Math. Z. 223 (1996), no. 4, 569–586. MR 1421956, DOI 10.1007/PL00004275
Maria Assunta Pozio and Alberto Tesei, On the uniqueness of bounded solutions to singular parabolic problems, Discrete Contin. Dyn. Syst. 13 (2005), no. 1, 117–137. MR 2128795, DOI 10.3934/dcds.2005.13.117
Guillermo Reyes and Juan Luis Vázquez, A weighted symmetrization for nonlinear elliptic and parabolic equations in inhomogeneous media, J. Eur. Math. Soc. (JEMS) 8 (2006), no. 3, 531–554. MR 2250171, DOI 10.4171/jems/66
A. Tesei, On uniqueness of the positive Cauchy problem for a class of parabolic equations, Current problems of analysis and mathematical physics (Italian) (Taormina, 1998) Aracne, Rome, 2000, pp. 145–160. MR 1809023