Admissible conditions for parabolic equations degenerating at infinity

Sh. Kamin ; M. A. Pozio ; A. Tesei

Algebra i analiz, Tome 19 (2007) no. 2, pp. 105-121.

Résumé

Well-posedness in $L^\infty(\mathbb{R}^n)$ $(n\ge3)$ 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.

Détail

Keywords: Parabolic Cauchy problem, linear parabolic equations with variable density, bounded solutions.

Bibliographie

[1] Brezis H., Kamin S., “Sublinear elliptic equations in $\mathbb{R}^n$”, Manuscripta Math., 74 (1992), 87–106 | DOI | MR | Zbl

[2] Eidelman S. D., Kamin S., “On stabilization of solutions of the Cauchy problem for parabolic equations degenerating at infinity”, Asymptot. Anal., 45 (2005), 55–71 | MR | Zbl

[3] Eidelman S. D., Kamin S., Porper F., “Uniqueness of solutions of the Cauchy problem for parabolic equations degenerating at infinity”, Asymptot. Anal., 22 (2000), 349–358 | MR | Zbl

[4] Eidus D., “The Cauchy problem for the nonlinear filtration equation in an inhomogeneous medium”, J. Differential Equations, 84 (1990), 309–318 | DOI | MR | Zbl

[5] Eidus D., Kamin S., “The filtration equation in a class of functions decreasing at infinity”, Proc. Amer. Math. Soc., 120 (1994), 825–830 | DOI | MR | Zbl

[6] Guedda M., Hilhorst D., Peletier M. A., “Disappearing interfaces in nonlinear diffusion”, Adv. Math. Sci. Appl., 7 (1997), 695–710 | MR | Zbl

[7] Ishige K., Murata M., “An intrinsic metric approach to uniqueness of the positive Cauchy problem for parabolic equations”, Math. Z., 227 (1998), 313–335 | DOI | MR | Zbl

[8] Kamin S., Rosenau P., “Nonlinear diffusion in finite mass medium”, Comm. Pure Appl. Math., 35 (1982), 113–127 | DOI | MR | Zbl

[9] Kamin S., Kersner R., Tesei A., “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), 279–298 | MR | Zbl

[10] Kersner R., Tesei A., “Well-posedness of initial value problems for singular parabolic equations”, J. Differential Equations, 199 (2004), 47–76 | DOI | MR | Zbl

[11] Murata M., “Nonuniqueness of the positive Dirichlet problem for parabolic equations in cylinders”, J. Funct. Anal., 135 (1996), 456–487 | DOI | MR | Zbl

[12] Ladyzhenskaya O. A., Solonnikov V. A., Uraltseva N. N., Lineinye i kvazilineinye uravneniya parabolicheskogo tipa, Nauka, M., 1967

[13] Oleinik O. A., Kalashnikov A. S., Chzhou Yui-Lin, “Zadacha Koshi i kraevye zadachi dlya uravnenii tipa nestatsionarnoi filtratsii”, Izv. AN SSSR. Ser. mat., 22:5 (1958), 667–704 | MR | Zbl

[14] Pinchover Y., “On uniqueness and nonuniqueness of the positive Cauchy problem for parabolic equations with unbounded coefficients”, Math. Z., 223 (1996), 569–586 | MR | Zbl

[15] Pozio M. A., Tesei A., “On the uniqueness of bounded solutions to singular parabolic problems”, Discrete Contin. Dyn. Syst., 13 (2005), 117–137 | DOI | MR | Zbl

[16] Reyes G., Vazquez J. L., A weighted symmetrization for nonlinear elliptic and parabolic equations in inhomogeneous media, Preprint, 2005 | MR

[17] Tesei A., “On uniqueness of the positive Cauchy problem for a class of parabolic equations”, Current Problems of Analysis and Mathematical Physics (Taormina, 1998), Aracne, Rome, 2000, 145–160 (Italian) | MR