Boundedness of global solutions of one dimensional quasilinear degenerate parabolic equations

Ryuichi SUZUKI

doi:10.2969/jmsj/05010119

Ryuichi SUZUKI

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JOURNAL FREE ACCESS

1998 Volume 50 Issue 1 Pages 119-138

DOI https://doi.org/10.2969/jmsj/05010119

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Published: 1998 Received: August 31, 1994 Available on J-STAGE: October 20, 2006 Accepted: - Advance online publication: - Revised: -
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Date of correction: October 20, 2006 Reason for correction: - Correction: CITATION Details: Wrong : 1) N. D. Alikakos and R. Rostamian, Large time behavior of solutions of Neumann boundary value problem for the porous medium equation, Indiana Univ. Math. J. 30 (1981), 749-785.
2) S. Angenent, The zero set of a solution of a parabolic equation, J. Reine Angew. Math. 390 (1988), 79-96.
3) D. G. Aronson, M. G. Crandall and L. A. Peletier, Stabilization of solutions of a degenerate nonlinear diffusion problem, Nonlinear Anal. 6 (1982), 1001-1022.
4) M. Bertsch, R. Kersner and L. A. Peletier, Positivity versus localization in degenerate diffusion equations, Nonlinear Anal. 9 (1985), 987-1008.
5) T. Cazenave and P. L. Lions, Solutions globales d'équations de la chaleur semi linéaires, Comm. Partial Differential Equations 9 (1984), 955-978.
6) X. Y. Chen and H. Matano, Convergence, asymptotic periodicity and finite-point blow-up in one-dimensional semilinear heat equations, J. Differential Equations 78 (1989), 160-190.
7) M. Fila, Boundedness of global solutions of nonlinear diffusion equations, J. Differential Equations 98 (1992), 226-240.
8) Y. Giga, A bound for global solutions of semilinear heat equations, Comm. Math. Phys. 103 (1986), 415-421.
9) T. Imai and K. Mochizuki, On blow-up of solutions for quasilinear degenerate parabolic equations, Publ. RIMS, Kyoto Univ. 27 (1991), 695-709.
10) D. Kröner and J. F. Rodrigues, Global behaviour for bounded solution of a porous media equation of elliptic-parabolic type, J. Math. Pures Appl. 64 (1985), 105-120.
11) O. A. Ladyzenskaja, V. A. Solonikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, Transl. Math. Monographs, 23, AMS Providence, R. I., 1968.
12) K. Mochizuki and R. Suzuki, Blow-up sets and asymptotic behavior of interfaces for quasilinear degenerate parabolic equations in R^N, J. Math. Soc. Japan 44 (1992), 485-504.
13) O. A. Oleinik, A. S. Kalashnikov and Chzou Yui-Lin, The Cauchy problem and boundary problems for equations of the type of nonlinear filtration, Izv. Akad. Nauk. SSSR Ser. Math. 22 (1958), 667-704, (Russian).
14) W. M. Ni, P. E. Sacks and J. Tavantzis, On the asymptotic behavior of solutions of certain quasilinear parabolic equations, J. Differential Equations 54 (1984), 97-120.
15) R. Suzuki, On blow-up sets and asymptotic behavior of interfaces of one dimensional quasilinear degenerate parabolic equations, Publ. RIMS Kyoto Univ. 27 (1991), 375-398.

Right : [1] N. D. Alikakos and R. Rostamian, Large time behavior of solutions of Neumann boundary value problem for the porous medium equation, Indiana Univ. Math. J. 30 (1981), 749-785.
[2] S. Angenent, The zero set of a solution of a parabolic equation, J. Reine Angew. Math. 390 (1988), 79-96.
[3] D. G. Aronson, M. G. Crandall and L. A. Peletier, Stabilization of solutions of a degenerate nonlinear diffusion problem, Nonlinear Anal. 6 (1982), 1001-1022.
[4] M. Bertsch, R. Kersner and L. A. Peletier, Positivity versus localization in degenerate diffusion equations, Nonlinear Anal. 9 (1985), 987-1008.
[5] T. Cazenave and P. L. Lions, Solutions globales d'équations de la chaleur semi linéaires, Comm. Partial Differential Equations 9 (1984), 955-978.
[6] X. Y. Chen and H. Matano, Convergence, asymptotic periodicity and finite-point blow-up in onedimensional semilinear heat equations, J. Differential Equations 78 (1989), 160-190.
[7] M. Fila, Boundedness of global solutions of nonlinear diffusion equations, J. Differential Equations 98 (1992), 226-240.
[8] Y. Giga, A bound for global solutions of semilinear heat equations, Comm. Math. Phys. 103 (1986), 415-421.
[9] T. Imai and K. Mochizuki, On blow-up of solutions for quasilinear degenerate parabolic equations, Publ. RIMS, Kyoto Univ. 27 (1991), 695-709.
[10] D. Kröner and J. F. Rodrigues, Global behaviour for bounded solution of a porous media equation of elliptic-parabolic type, J. Math. Pures Appl. 64 (1985), 105-120.
[11] O. A. Ladyzenskaja, V. A. Solonikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, Transl. Math. Monographs, 23, AMS Providence, R. I., 1968.
[12] K. Mochizuki and R. Suzuki, Blow-up sets and asymptotic behavior of interfaces for quasilinear degenerate parabolic equations in R^N, J. Math. Soc. Japan 44 (1992), 485-504.
[13] O. A. Oleinik, A. S. Kalashnikov and Chzou Yui-Lin, The Cauchy problem and boundary problems for equations of the type of nonlinear filtration, Izv. Akad. Nauk. SSSR Ser. Math. 22 (1958), 667-704, (Russian).
[14] W. M. Ni, P. E. Sacks and J. Tavantzis, On the asymptotic behavior of solutions of certain quasilinear parabolic equations, J. Differential Equations 54 (1984), 97-120.
[15] R. Suzuki, On blow-up sets and asymptotic behavior of interfaces of one dimensional quasilinear degenerate parabolic equations, Publ. RIMS Kyoto Univ. 27 (1991), 375-398.

Date of correction: October 20, 2006 Reason for correction: - Correction: PDF FILE Details: -

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