Bounded, periodic and almost periodic classical solutions of some nonlinear wave equations with a dissipative term

Mitsuhiro NAKAO

doi:10.2969/jmsj/03030375

Mitsuhiro NAKAO

Author information

JOURNAL FREE ACCESS

1978 Volume 30 Issue 3 Pages 375-394

DOI https://doi.org/10.2969/jmsj/03030375

Details

Published: 1978 Received: February 03, 1976 Available on J-STAGE: October 20, 2006 Accepted: - Advance online publication: - Revised: -
Correction information
Date of correction: October 20, 2006 Reason for correction: - Correction: CITATION Details: Wrong : 1) S. Agmon, Lectures on elliptic boundary value problems, Van Nostrand C., 1965.
2) S. Agmon, A. Douglis and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary value conditions, I, Comm. Pure Appl. Math., 12 (1959), 623-727.
3) L. Amerio and G. Prouse, Almost periodic functions and functional equations, Van Nostrand C., 1971.
4) M. Biroli, Sulla esistenza ed unicita della soluzione limitata e della soluzione quasi periodica per una equazione parabolica con termine dissipativo non linear discontinuo, Ricerche Mat., 19 (1970), 93-110.
5) M. Biroli, Sur l'equation des ondes avec un terme non lineaire monotone dans la fonction inconnue, Nota 1, Atti. Accad. Naz. Lincei, 53 (1972), 359-361.
6) J. C. Clements, Existence theorems for some non-linear equations of evolution, Can. J. Math., 22-4 (1970), 726-745.
7) Y. Ebihara, M. Nakao and T. Nanbu, On the existence of global classical solution for _??_u-u³=f, Pacific J. Math., 61 (1975), 63-70.
8) W. S. Hall, On the existence of periodic solutions for the equations D_tt+(-1)^pD_x^2pu =εf (•,•,u), J. Differential Equations, 7 (1970), 509-526.
9) J. L. Lions, Quelques methodes de resolution des problemes aux limites nonlineaire, Dunod Paris, 1969.
10) S. Mizohata, Theory of partial differential equations, (in Japanese), Iwanami, Tokyo, 1965, Cambridge UP., 1973.
11) M. Nakao, On some regularity of periodic solutions of nonlinear wave equations, J. Differential Equations, 17 (1975), 187-197.
12) M. Nakao, Periodic solutions of linear and nonlinear wave equations, Arch. Rational Mech. Anal., 62 (1976), 87-98.
13) M. Nakao, On boundedness, periodicity and almost periodicity of solutions of some nonlinear parabolic equations, J. Differential Equations, 19 (1975), 371-385.
14) M. Nakao, Bounded, periodic or almost periodic solutions of nonlinear hyperbolic partial differential equations, J. Differential Equations, 23 (1977), 368-386.
15) M. Nakao, Asymptotic stability of the bounded or almost periodic solution of the wave equation with a nonlinear dissipative term, J. Math. Anal. Appl., 58 (1977), 336-343.
16) M. Nakao, Convergence of solutions of the wave equation with a nonlinear dissipative term to the steady state, Mem, Fac. Sci. Kyushu Univ., 30 (1976), 257-265.
17) M. Nakao and T. Nanbu, Bounded or almost periodic classical solutions for some non-linear parabolic equations, Mem. Fac. Sci. Kyushu Univ., 30 (1976), 191-211.
18) P. H. Rabinowitz, Periodic solutions of nonlinear hyperbolic partial differential equations, Comm. Pure Appl. Math., 20 (1967), 145-205.
19) P. H. Rabinowitz, Periodic solutions of nonlinear hyperbolic partial differential equations, II, Comm. Pure Appl. Math., 22 (1969), 15-39.
20) J. Sather, The existence of a global classical solution of the initial boundary value problem for _??_u+u³=f, Arch. Rational Mech. Anal., 22 (1966), 292-307.
21) D. H. Sattinger, Stability of nonlinear hyperbolic equations, Arch. Rational Mech. Anal., 28 (1968), 226-244.
22) G. Torelli, Soluzioni periodiche dell'equazione nonlineare u_tt-u_xx+_εF(x,t,u)=0, Rend. Ist. Mat. Univ. Trieste, 1969, 123-137.
23) O. Vejvoda, Periodic solutions of a linear and weakly nonlinear wave equation in one dimension, Czech. Math. J., 14 (1964), 341-382.
24) W. V. Wahl, Periodic solutions of nonlinear wave, equations with a dissipative term, Math. Ann., 190 (1971), 313-322.
25) M. Yamaguchi, On the existence of quasiperiodic solutions of nonlinear partial differential equations, Proc. Fac. Sci. Tokai Univ., 10 (1974), 1-12.
26) A. Matsumura, Global existence and asymptotics of the solutions of the second-order quasilinear hyperbolic equations with the first-order dissipation, Publ. Res. Inst. Math. Sci. A, 13 (1977), 349-379.

Right : [1] S. Agmon, Lectures on elliptic boundary value problems, Van Nostrand C., 1965.
[2] S. Agmon, A. Douglis and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary value conditions, I, Comm. Pure Appl. Math., 12 (1959), 623-727.
[3] L. Amerio and G. Prouse, Almost periodic functions and functional equations, Van Nostrand C., 1971.
[4] M. Biroli, Sulla esistenza ed unicita della soluzione limitata e della soluzione quasi periodica per una equazione parabolica con termine dissipativo non linear discontinuo, Ricerche Mat., 19 (1970), 93-110.
[5] M. Biroli, Sur l'equation des ondes avec un terme non lineaire monotone dans la fonction inconnue, Nota 1, Atti. Accad. Naz. Lincei, 53 (1972), 359-361.
[6] J. C. Clements, Existence theorems for some non-linear equations of evolution, Can. J. Math., 22-4 (1970), 726-745.
[7] Y. Ebihara, M. Nakao and T. Nanbu, On the existence of global classical solution for _??_u-u³=f, Pacific J. Math., 61 (1975), 63-70.
[8] W. S. Hall, On the existence of periodic solutions for the equations D_tt+(-1)^pD_x^2pu=εf (·,·,u), J. Differential Equations, 7 (1970), 509-526.
[9] J. L. Lions, Quelques methodes de resolution des problemes aux limites nonlineaire, Dunod Paris, 1969.
[10] S. Mizohata, Theory of partial differential equations, (in Japanese), Iwanami, Tokyo, 1965, Cambridge UP., 1973.
[11] M. Nakao, On some regularity of periodic solutions of nonlinear wave equations, J. Differential Equations, 17 (1975), 187-197.
[12] M. Nakao, Periodic solutions of linear and nonlinear wave equations, Arch. Rational Mech. Anal., 62 (1976), 87-98.
[13] M. Nakao, On boundedness, periodicity and almost periodicity of solutions of some nonlinear parabolic equations, J. Differential Equations, 19 (1975), 371-385.
[14] M. Nakao, Bounded, periodic or almost periodic solutions of nonlinear hyperbolic partial differential equations, J. Differential Equations, 23 (1977), 368-386.
[15] M. Nakao, Asymptotic stability of the bounded or almost periodic solution of the wave equation with a nonlinear dissipative term, J. Math. Anal. Appl., 58 (1977), 336-343.
[16] M. Nakao, Convergence of solutions of the wave equation with a nonlinear dissipative term to the steady state, Mem, Fac. Sci. Kyushu Univ., 30 (1976), 257-265.
[17] M. Nakao and T. Nanbu, Bounded or almost periodic classical solutions for some non-linear parabolic equations, Mem. Fac. Sci. Kyushu Univ., 30 (1976), 191-211.
[18] P. H. Rabinowitz, Periodic solutions of nonlinear hyperbolic partial differential equations, Comm. Pure Appl. Math., 20 (1967), 145-205.
[19] P. H. Rabinowitz, Periodic solutions of nonlinear hyperbolic partial differential equations, II, Comm. Pure Appl. Math., 22 (1969), 15-39.
[20] J. Sather, The existence of a global classical solution of the initial boundary value problem for _??_u+u³=f, Arch. Rational Mech. Anal., 22 (1966), 292-307.
[21] D. H. Sattinger, Stability of nonlinear hyperbolic equations, Arch. Rational Mech. Anal., 28 (1968), 226-244.
[22] G. Torelli, Soluzioni periodiche dell'equazione nonlineare u_tt-u_xx+_εF(x,t,u)=0, Rend. Ist. Mat. Univ. Trieste, 1969, 123-137.
[23] O. Vejvoda, Periodic solutions of a linear and weakly nonlinear wave equation in one dimension, Czech. Math. J., 14 (1964), 341-382.
[24] W. V. Wahl, Periodic solutions of nonlinear wave equations with a dissipative term, Math. Ann., 190 (1971), 313-322.
[25] M. Yamaguchi, On the existence of quasiperiodic solutions of nonlinear partial differential equations, Proc. Fac. Sci. Tokai Univ., 10 (1974), 1-12.
[26] A. Matsumura, Global existence and asymptotics of the solutions of the second-order quasilinear hyperbolic equations with the first-order dissipation, Publ. Res. Inst. Math. Sci. A, 13 (1977), 349-379.

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

Corresponding author

Correction information

Register with J-STAGE for free!