July  2013, 12(4): 1783-1812. doi: 10.3934/cpaa.2013.12.1783

On existence and uniqueness classes for the Cauchy problem for parabolic equations of the p-Laplace type

1. 

Computational Aeroacoustics Laboratory, Keldysh Institute of Applied Mathematics, Moscow 125047, Russian Federation

2. 

Department of Mathematics, Vladimir State University, Vladimir 600000, Russian Federation

Received  April 2012 Revised  June 2012 Published  November 2012

We prove the existence and uniqueness of global solutions to the Cauchy problem for a class of parabolic equations of the p-Laplace type. In the singular case $p<2$ there are no restrictions on the behaviour of solutions and initial data at infinity. In the degenerate case $p>2$ we impose a restriction on growth of solutions at infinity to obtain global existence and uniqueness. This restriction is given in terms of weighted energy classes with power-like weights.
Citation: Mikhail D. Surnachev, Vasily V. Zhikov. On existence and uniqueness classes for the Cauchy problem for parabolic equations of the p-Laplace type. Communications on Pure and Applied Analysis, 2013, 12 (4) : 1783-1812. doi: 10.3934/cpaa.2013.12.1783
References:
[1]

Yu. A. Alkhutov, S. N. Antontsev and V. V. Zhikov, Parabolic equations with variable order of nonlinearity, Zb. Prats' Inst. Mat. NAN Ukr., 6 (2009), 23-50.

[2]

Yu. A. Alkhutov and V. V. Zhikov, Existence theorems for solutions of parabolic equations with variable order of nonlinearity, Proceedings of the Steklov Institute of Mathematics, 270 (2010), 15-26. doi: 10.1134/S0081543810030028.

[3]

D. G. Aronson and L. A. Caffarelli, The initial trace of a solution of the porous medium equation, Trans. Amer. Math. Soc., 380 (1983), 351-366. doi: 10.1090/S0002-9947-1983-0712265-1.

[4]

Philippe Bénilan, Michael Crandall and Michel Pierre, Solutions of the Porous Medium Equation in $\mathbb{R}^{N}$ under optimal conditions on initial values, Indiana Univ. Math. J., 33 (1984), 51-87. doi: 10.1512/iumj.1984.33.33003.

[5]

B. E. J. Dahlberg and C. E. Kenig, Nonnegative solutions of the porous medium equation, Comm. Partial Differential Equations, 9 (1984), 409-437. doi: 10.1080/03605308408820336.

[6]

E. DiBenedetto, "Degenerate Parabolic Equations,'' Springer, 1993.

[7]

E. DiBenedetto and M. A. Herrero, On the Cauchy problem and initial traces for a degenerate parabolic equation, Trans. AMS, 314 (1989), 187-224. doi: 10.1090/S0002-9947-1989-0962278-5.

[8]

E. DiBenedetto and M. A. Herrero, Non negative solutions of the evolution p-Laplacian equation. Initial traces and Cauchy problem when $1, Arch. Rational Mech. Anal., 111 (1990), 225-290. doi: 10.1007/BF00400111.

[9]

A. S. Kalashnikov, The Cauchy problem in classes of increasing functions for certain quasi-linear degenerate parabolic equations of the second order, Differencial'nye Uravnenija, 9 (1973), 682-691.

[10]

A. S. Kalashnikov, Uniqueness conditions for the generalized solutions of the Cauchy problem for a class of quasi-linear degenerate parabolic equations, Differencial'nye Uravnenija, 9 (1973), 2207-2212.

[11]

O. A. Ladyzhenskaya, V. A. Solonnikov and N. N. Uraltseva, "Linear and Quasilinear Equations of Parabolic Type,'' Translations of Mathematical Monographs 23, American Mathematical Society, Providence, R.I. 1968.

[12]

J.-L. Lions, "Quelques méthodes de résolution des problémes aux limites non linéires,'' Dunod, Paris, 1969.

[13]

V. V. Zhikov and S. E. Pastukhova, Parabolic lemmas on compensated compactness and their applications, Dokl. Math., 81 (2010), 227-232. doi: 10.1134/S1064562410020171.

[14]

V. V. Zhikov and S. E. Pastukhova, Lemmas on compensated compactness in elliptic and parabolic equations, Proceedings of the Steklov Institute of Mathematics, 270 (2010), 104-131. doi: 10.1134/S0081543810030089.

show all references

References:
[1]

Yu. A. Alkhutov, S. N. Antontsev and V. V. Zhikov, Parabolic equations with variable order of nonlinearity, Zb. Prats' Inst. Mat. NAN Ukr., 6 (2009), 23-50.

[2]

Yu. A. Alkhutov and V. V. Zhikov, Existence theorems for solutions of parabolic equations with variable order of nonlinearity, Proceedings of the Steklov Institute of Mathematics, 270 (2010), 15-26. doi: 10.1134/S0081543810030028.

[3]

D. G. Aronson and L. A. Caffarelli, The initial trace of a solution of the porous medium equation, Trans. Amer. Math. Soc., 380 (1983), 351-366. doi: 10.1090/S0002-9947-1983-0712265-1.

[4]

Philippe Bénilan, Michael Crandall and Michel Pierre, Solutions of the Porous Medium Equation in $\mathbb{R}^{N}$ under optimal conditions on initial values, Indiana Univ. Math. J., 33 (1984), 51-87. doi: 10.1512/iumj.1984.33.33003.

[5]

B. E. J. Dahlberg and C. E. Kenig, Nonnegative solutions of the porous medium equation, Comm. Partial Differential Equations, 9 (1984), 409-437. doi: 10.1080/03605308408820336.

[6]

E. DiBenedetto, "Degenerate Parabolic Equations,'' Springer, 1993.

[7]

E. DiBenedetto and M. A. Herrero, On the Cauchy problem and initial traces for a degenerate parabolic equation, Trans. AMS, 314 (1989), 187-224. doi: 10.1090/S0002-9947-1989-0962278-5.

[8]

E. DiBenedetto and M. A. Herrero, Non negative solutions of the evolution p-Laplacian equation. Initial traces and Cauchy problem when $1, Arch. Rational Mech. Anal., 111 (1990), 225-290. doi: 10.1007/BF00400111.

[9]

A. S. Kalashnikov, The Cauchy problem in classes of increasing functions for certain quasi-linear degenerate parabolic equations of the second order, Differencial'nye Uravnenija, 9 (1973), 682-691.

[10]

A. S. Kalashnikov, Uniqueness conditions for the generalized solutions of the Cauchy problem for a class of quasi-linear degenerate parabolic equations, Differencial'nye Uravnenija, 9 (1973), 2207-2212.

[11]

O. A. Ladyzhenskaya, V. A. Solonnikov and N. N. Uraltseva, "Linear and Quasilinear Equations of Parabolic Type,'' Translations of Mathematical Monographs 23, American Mathematical Society, Providence, R.I. 1968.

[12]

J.-L. Lions, "Quelques méthodes de résolution des problémes aux limites non linéires,'' Dunod, Paris, 1969.

[13]

V. V. Zhikov and S. E. Pastukhova, Parabolic lemmas on compensated compactness and their applications, Dokl. Math., 81 (2010), 227-232. doi: 10.1134/S1064562410020171.

[14]

V. V. Zhikov and S. E. Pastukhova, Lemmas on compensated compactness in elliptic and parabolic equations, Proceedings of the Steklov Institute of Mathematics, 270 (2010), 104-131. doi: 10.1134/S0081543810030089.

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