On solutions to the initial-boundary value problem for ∂/∂tu-Δβ(u)=ƒ

Mitsuhiro NAKAO

doi:10.2969/jmsj/03510071

Mitsuhiro NAKAO

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JOURNAL FREE ACCESS

1983 Volume 35 Issue 1 Pages 71-83

DOI https://doi.org/10.2969/jmsj/03510071

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Published: 1983 Received: December 15, 1980 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: TITLE Details: Wrong : On solutions to the initial-boundary value problem for ∂/∂tu-Δβ(u)=f
Right : On solutions to the initial-boundary value problem for ∂/∂tu-Δβ(u)=ƒ

Date of correction: October 20, 2006 Reason for correction: - Correction: CITATION Details: Wrong : 1) D. G. Aronson, Regularity properties of flows through porous media, SIAM J. Appl. Math., 17 (1969), 461-467.
2) F. V. Atkinson and L. A. Peletier, Similarity solutions of the nonlinear diffusion equations, Arch. Rational Mech. Anal., 54 (1974) , 373-392.
3) J. E. Boillet, D. A. D. Saravia and L. T. Villa, Similarity solutions of the equation of one-dimensional heat conduction, J. Differential Equations, 35 (1980), 55-65.
4) H. Brezis, On some degenerate nonlinear parabolic equations; Nonlinear functional analysis, Proc. Symp. Pure Math. A. M. S., 18 (1970) , 28-38.
5) H. Brezis, Monotonicity method in Hilbert spaces and some applications to nonlinear partial differential equations, Contributions to nonlinear functional analysis, Academic Press, New York-London, 1971, 101-156.
6) M. G. Crandall, Semigroups of nonlinear transformations in Banach spaces, Contributions to nonlinear functional analysis, Academic Press, New York-London, 1971, 157-179.
7) J. A. Dubinskii, Weak convergence in nonlinear elliptic and parabolic equations, Amer. Math. Soc. Transl., (2) 67 (1968), 226-258.
8) S. L. Kamenomostskaya, On stefan's problem, Mat. Sb., 53 (1961) , 489-514.
9) Y. Konishi, Une méthode de résolution d'une équation d'évolution nonlinéaire dégénérée, J. Fac. Sci. Univ. Tokyo Sect. IA, 19 (1972) , 243-255.
10) Y. Konishi, On the nonlinear semi-groups associated with u_t=Δβ(u) and φ(u_t)=Δu, J. Math. Soc. Japan, 25 (1973), 622-628.
11) O. A. Ladyzhenskaya, V. A. Solonnikov and N. N. Ural'tseva, Linear and quasilinear equations of parabolic type, Amer. Math. Soc., Providence, R. I., 1968.
12) J. L. Lions, Quelques méthodes de résolution des problèmes aux limites non-inéaires, Dunod, Paris, 1969.
13) M. Nakao, Convergence of solutions of the wave equation with a nonlinear dissipative term to the steady state, Mem. Fac. Sci. Kyushu Univ. Ser. A, 30 (1976), 257-265.
14) M. Nakao, Decay of solutions of some nonlinear evolution equations, J. Math. Anal. Appl., 60 (1977), 542-549.
15) M. Nakao, A difference inequality and its application to nonlinear evolution equations, J. Math. Soc. Japan, 30 (1978), 747-762.
16) O. A. Oleinik, A. S. Kalashnikov and Chzhou Yui-Lin, The Cauchy problem and boundary problems for equations of the type of nonstationary filtration, Izv. Akad. Nauk SSSR, Ser. Mat., 22 (1958), 667-704.
17) L. A. Peletier, Asymptotic behaviour of temperature profiles of a class of non-linear heat conduction problems, Quart. J. Mech. Appl. Math., 23 (1970), 441-447.
18) P. A. Raviart, Sur la résolution et l'approximation de certaines équations paraboliques non linéaires dégénérées, Arch. Rat. Mech. Anal., 25 (1967), 64-80.

Right : [1] D. G. Aronson, Regularity properties of flows through porous media, SIAM J. Appl. Math., 17 (1969), 461-467.
[2] F. V. Atkinson and L. A. Peletier, Similarity solutions of the nonlinear diffusion equations, Arch. Rational Mech. Anal., 54 (1974) , 373-392.
[3] J. E. Boillet, D. A. D. Saravia and L. T. Villa, Similarity solutions of the equation of one-dimensional heat conduction, J. Differential Equations, 35 (1980), 55-65.
[4] H. Brezis, On some degenerate nonlinear parabolic equations; Nonlinear functional analysis, Proc. Symp. Pure Math. A. M. S., 18 (1970), 28-38.
[5] H. Brezis, Monotonicity method in Hilbert spaces and some applications to nonlinear partial differential equations, Contributions to nonlinear functional analysis, Academic Press, New York-London, 1971, 101-156.
[6] M. G. Crandall, Semigroups of nonlinear transformations in Banach spaces, Contributions to nonlinear functional analysis, Academic Press, New York-London, 1971, 157-179.
[7] J. A. Dubinskii, Weak convergence in nonlinear elliptic and parabolic equations, Amer. Math. Soc. Transl., (2) 67 (1968), 226-258.
[8] S. L. Kamenomostskaya, On stefan's problem, Mat. Sb., 53 (1961) , 489-514.
[9] Y. Konishi, Une méthode de résolution d'une équation d'évolution nonlinéaire dégénérée, J. Fac. Sci. Univ. Tokyo Sect. IA, 19 (1972), 243-255.
[10] Y. Konishi, On the nonlinear semi-groups associated with u_t=Δβ(u) and φ(u_t)=Δu, J. Math. Soc. Japan, 25 (1973), 622-628.
[11] O. A. Ladyzhenskaya, V. A. Solonnikov and N. N. Ural'tseva, Linear and quasilinear equations of parabolic type, Amer. Math. Soc., Providence, R. I., 1968.
[12] J. L. Lions, Quelques méthodes de résolution des problèmes aux limites non-inéaires, Dunod, Paris, 1969.
[13] M. Nakao, Convergence of solutions of the wave equation with a nonlinear dissipative term to the steady state, Mem. Fac. Sci. Kyushu Univ. Ser. A, 30 (1976), 257-265.
[14] M. Nakao, Decay of solutions of some nonlinear evolution equations, J. Math. Anal. Appl., 60 (1977), 542-549.
[15] M. Nakao, A difference inequality and its application to nonlinear evolution equations, J. Math. Soc. Japan, 30 (1978), 747-762.
[16] O. A. Oleinik, A. S. Kalashnikov and Chzhou Yui-Lin, The Cauchy problem and boundary problems for equations of the type of nonstationary filtration, Izv. Akad. Nauk SSSR, Ser. Mat., 22 (1958), 667-704.
[17] L. A. Peletier, Asymptotic behaviour of temperature profiles of a class of non-linear heat conduction problems, Quart. J. Mech. Appl. Math., 23 (1970), 441-447.
[18] P. A. Raviart, Sur la résolution et l'approximation de certaines équations paraboliques non linéaires dégénérées, Arch. Rat. Mech. Anal., 25 (1967), 64-80.

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