References
A. S. Kalashnikov, “Some questions of the qualitative theory of nonlinear degenerate second order parabolic equations,” Uspekhi Mat. Nauk,42, No. 2, 135–176 (1987).
A. A. Samarskiî, V. A. Galaktionov, S. P. Kurdyumov, and A. P. Mikhaîlov, Processes with Strain for Quasilinear Parabolic Problems [in Russian], Nauka, Moscow (1987).
R. Kersner, “Degenerate parabolic equations with general nonlinearities,” Nonlinear Anal.,4, No. 6, 1043–1062 (1980).
A. S. Kalashnikov, “On the influence of absorption on heat propagation in a medium with heat conductivity depending on temperature,” Zh. Vychisl. Mat. i Mat. Fiz.,16, No. 3, 689–696 (1976).
S. Kamin, L. A. Peletier, and J. L. Vazquez, “Classification of singular solutions of a nonlinear heat equation,” Duke Math. J.,58, No. 3, 601–615 (1989).
H. Brezis, L. A. Peletier, and D. Terman, “A very singular solutions of the heat equations with absorption,” Arch. Rational Mech. Anal.,95, No. 3, 185–209 (1986).
S. Kamin and L. A. Peletier, “Source-type solution of degenerate diffusion equations with absorption,” Israel J. Math.,50, No. 3, 219–230 (1985).
L. A. Peletier and D. Terman, “A very singular solutions of the porous media equation with absorption,” J. Differential Equations,65, 396–410 (1986).
S. Kamin and L. Veron, “Existence and uniqueness of the very singular solution for the porous medium equation with absorption,” J. Analyse Math.,51, 245–258 (1968).
P. Benilan, M. Grandall, and M. Pierre, “Solutions of the porous medium equation in 53-1 under optimal conditions on initial values,” Indiana Univ. Math. J.,33, No. 1, 51–87 (1984).
D. G. Aronson and L. A. Caffarelli, “The initial trace of a solution of the porous media equation,” Trans. Amer. Math. Soc.,280, No. 1, 51–366 (1983).
B. E. G. Dahlberg and C. E. Kenig, “Non-negative solutions of the porous medium equation,” Comm. Partial Differential Equations,9, No. 5, 409–437 (1984).
M. A. Herrero and M. Pierre, “The Cauchy problem foru t=Δu m when 0<m<1,” Trans. Amer. Math. Soc.,291, No. 1, 145–158 (1985).
N. O. Maksimova, “On uniqueness of solution to the Cauchy and boundary value problems in unbounded domains for certain classes of quasilinear degenerate parabolic equations,” Trudy Sem. Petrovsk.,11, 12–31 (1985).
M. Bertsch, “A class of degenerate diffusion equations with a singular nonlinear term,” Nonlinear Anal.,7, No. 1, 117–127 (1983).
A. Friedman, Partial Differential Equations of Parabolic Type [Russian translation], Mir, Moscow (1968).
G. N. Watson, A Treatise on Theory of Bessel Functions. Part 1 [Russian translation], Izdat. Inostr. Lit., Moscow (1949).
O. A. Ladyzhenskaya and N. N. Ural'tseva, Linear and Quasilinear Equations of Elliptic Type [in Russian], Nauka, Moscow (1973).
T. Kato, “Schrödinger operators with singular potentials,” Israel J. Math.,13, 135–148 (1972).
S. Kamin, L. A. Peletier, and J. L. Vazquez, “A nonlinear diffusion-absorption equation with unbounded initial data,” Nonlinear Diffusion Equations and their Equilibrium States,3, 243–263 (1992).
J. B. McLeod, L. A. Peletier, and J. L. Vazquez, “Solutions of a nonlinear ODE appearing in the theory of diffusion with absorption,” Differential Integral Equations,4, No. 1, 1–14 (1991).
Additional information
Vitebsk. Translated fromSibirskiî Matematicheskiî Zhurnal, Vol. 34, No. 1, pp. 47–64, January–February, 1993.
Translated by G. V. Dyatlov
Rights and permissions
About this article
Cite this article
Gladkov, A.L. The Cauchy problem for certain degenerate quasilinear parabolic equations with absorption. Sib Math J 34, 37–54 (1993). https://doi.org/10.1007/BF00971239
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00971239