Blow-up sets and asymptotic behavior of interfaces for quasilinear degenerate parabolic equations in RN

Kiyoshi MOCHIZUKI; Ryuichi SUZUKI

doi:10.2969/jmsj/04430485

Blow-up sets and asymptotic behavior of interfaces for quasilinear degenerate parabolic equations in R^N

Kiyoshi MOCHIZUKI, Ryuichi SUZUKI

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JOURNAL FREE ACCESS

1992 Volume 44 Issue 3 Pages 485-504

DOI https://doi.org/10.2969/jmsj/04430485

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Published: 1992 Received: November 06, 1990 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: SUBTITLE Details: Wrong : Dedicated to Professor Takeshi Kotake on his 60th birthday

Date of correction: October 20, 2006 Reason for correction: - Correction: CITATION Details: Wrong : 1) M. Bertsch, R. Kersner and L. A. Peletie, Positivity versus localization in degenerate diffusion equations, Nonlinear Anal., 9 (1985), 987-1008.
2) L. A. Caffareli, J. L. Vaquez and N. I. Wolanski, Lipschitz continuity of solutions and interfaces of the N-dimensional porous medium equation, Indiana Univ. Math. J., 36 (1987), 373-400.
3) 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.
4) Y.-G. Chen, Blow-up solutions of a semilinear parabolic equation with Neumann and Robin boundary conditions, Hokkaido Univ. Preprint series in Math., 65 (1989).
5) A. Friedman and B. McLeod, Blow-up of positive solutions of semilinear heat equations, Indiana Univ. Math. J., 34 (1985), 425-447.
6) H. Fujita, On the blowing up of solutions of the Cauchy problem for u_t=Δu+u^1+α, J. Fac. Sci. Univ. Tokyo, 13 (1966), 109-124.
7) H. Fujita and Y.-G. Chen, On the set of blow-up points and asymptotic behaviours of blow-up solutions to a semilinear parabolic equation, Preprint 87-13, Univ.Tokyo.
8) V. A. Galaktionov, On localization conditions for unbounded solutions of quasilinear parabolic equations, Soviet Math. Dokl., 23 (1982), 775-780.
9) V. A. Galaktionov, Asymptotic behavior of unbounded solutions of the nonlinear parabolic equation u_t=(u^σu_x)x+u^σ+1, Differents. Urab., 21 (1985), 1126-1134.
10) V. A. Galaktionov, S. P. Kurdyumov, A. P. Mikhailov and A. A. Samarskii, Unbounded solutions of the Cauchy problem for the parabolic equation u_t=Δ(u^σΔu)+u^β, Soviet Phys. Dokl., 25 (1980), 458-459.
11) T. Imai and K. Mochizuki, On blow-up of solutions for quasilinear degenerate parabolic equations, Publ. RIMS, Kyoto Univ., to appear.
12) I. Imai, K. Mochizuki and R. Suzuki, On blow-up sets for the parabolic equation ∂_tβ(u)=Δu+f(u) in a ball, J. Fac. Sci. Shinshu Univ., 25 (1990), 51-58.
13) N. Itaya, A note on the blowup-nonblowup problems in nonlinear parabolic equations, Proc. Japan Acad., 55 (1979), 241-244.
14) N. Itaya, On some subjects related to the blowing-up problem in nonlinear parabolic equations, Lecture Notes Numer. Appl. Anal., 2 (1980), 27-38.
15) 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.
16) H. A. Levine, G. M. Lieberman and P. Meier, On critical exponents for some quasilinear parabolic equations, Math. Mech. Anal. Sci., to appear.
17) 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.
18) R. Suzuki, On blow-up sets and asymptotic behavior of interfaces of one dimensional quasilinear degenerate parabolic equations, Publ. RIMS, Kyoto Univ., to appear.

Right : [1] M. Bertsch, R. Kersner and L. A. Peletie, Positivity versus localization in degenerate diffusion equations, Nonlinear Anal., 9 (1985), 987-1008.
[2] L. A. Caffareli, J. L. Vaquez and N. I. Wolanski, Lipschitz continuity of solutions and interfaces of the N-dimensional porous medium equation, Indiana Univ. Math. J., 36 (1987), 373-400.
[3] 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.
[4] Y. -G. Chen, Blow-up solutions of a semilinear parabolic equation with Neumann and Robin boundary conditions, Hokkaido Univ. Preprint series in Math., 65 (1989).
[5] A. Friedman and B. McLeod, Blow-up of positive solutions of semilinear heat equations, Indiana Univ. Math. J., 34 (1985), 425-447.
[7] H. Fujita and Y.-G. Chen, On the set of blow-up points and asymptotic behaviours of blow-up solutions to a semilinear parabolic equation, Preprint 87-13, Univ. Tokyo.
[8] V. A. Galaktionov, On localization conditions for unbounded solutions of quasilinear parabolic equations, Soviet Math. Dokl., 23 (1982), 775-780.
[9] V. A. Galaktionov, Asymptotic behavior of unbounded solutions of the nonlinear parabolic equation u_t=(u^σu_x)x+u^σ+1, Differents. Urab., 21 (1985), 1126-1134.
[10] V. A. Galaktionov, S. P. Kurdyumov, A. P. Mikhailov and A. A. Samarskii, Unbounded solutions of the Cauchy problem for the parabolic equation u_t=Δ(u^σΔu)+u^β, Soviet Phys. Dokl., 25 (1980), 458-459.
[11] T. Imai and K. Mochizuki, On blow-up of solutions for quasilinear degenerate parabolic equations, Publ. RIMS, Kyoto Univ., to appear.
[12] I. Imai, K. Mochizuki and R. Suzuki, On blow-up sets for the parabolic equation ∂_tβ(u)=Δu+f(u) in a ball, J. Fac. Sci. Shinshu Univ., 25 (1990), 51-58.
[13] N. Itaya, A note on the blowup-nonblowup problems in nonlinear parabolic equations, Proc. Japan Acad., 55 (1979), 241-244.
[14] N. Itaya, On some subjects related to the blowing-up problem in nonlinear parabolic equations, Lecture Notes Numer. Appl. Anal., 2 (1980), 27-38.
[15] 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.
[16] H. A. Levine, G. M. Lieberman and P. Meier, On critical exponents for some quasilinear parabolic equations, Math. Mech. Anal. Sci., to appear.
[17] 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.
[18] R. Suzuki, On blow-up sets and asymptotic behavior of interfaces of one dimensional quasilinear degenerate parabolic equations, Publ. RIMS, Kyoto Univ., to appear.

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