Finite energy travelling waves for nonlinear damped wave equations
Author:
Eduard Feireisl
Journal:
Quart. Appl. Math. 56 (1998), 55-70
MSC:
Primary 35L70; Secondary 35A15, 35B40
DOI:
https://doi.org/10.1090/qam/1604876
MathSciNet review:
MR1604876
Full-text PDF Free Access
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- Antonio Ambrosetti and Paul H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Functional Analysis 14 (1973), 349–381. MR 0370183, DOI https://doi.org/10.1016/0022-1236%2873%2990051-7
- Vieri Benci and Giovanna Cerami, Positive solutions of some nonlinear elliptic problems in exterior domains, Arch. Rational Mech. Anal. 99 (1987), no. 4, 283–300. MR 898712, DOI https://doi.org/10.1007/BF00282048
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- H. Berestycki and P.-L. Lions, Nonlinear scalar field equations. I. Existence of a ground state, Arch. Rational Mech. Anal. 82 (1983), no. 4, 313–345. MR 695535, DOI https://doi.org/10.1007/BF00250555
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B. Gidas, W. Ni, and L. Nirenberg, Symmetry and related properties by the maximum principle, Comm. Math. Phys. 68, 209–243 (1979)
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A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal. 14, 349–381 (1973)
V. Benci and G. Cerami, Positive solutions of some nonlinear elliptic problems in exterior domains. Arch. Rational Mech. Anal. 99 (4), 283–300 (1987)
H. Berestycki and P. L. Lions, Nonlinear scalar field equations, I. Existence of a ground state, Arch. Rational Mech. Anal. 82, 313–346 (1983)
H. Berestycki and P. L. Lions, Nonlinear scalar field equations, II. Existence of infinitely many solutions, Arch. Rational Mech. Anal. 82, 347–376 (1983)
S. Coleman, V. Glazer, and A. Martin, Action minima among solutions to a class of Euclidean scalar field equations, Commun. Math. Phys. 58 (2), 211–221 (1978)
M. J. Esteban and P. L. Lions, Existence and non-existence results for semilinear elliptic problems in unbounded domains, Proc. Royal Soc. Edinburgh 93 A, 1–14 (1982)
E. Feireisl, Convergence to an equilibrium for semilinear wave equations on unbounded intervals, Dynamic Syst. Appl. 3 (3), 423–434 (1994)
J. M. Ghidaglia and R. Temam, Attractors for damped nonlinear hyperbolic equations, J. Math. Pures Appl. 66, 273–319 (1987)
J. M. Ghidaglia and R. Temam, Regularity of the solutions of second order evolution equations and their attractors, Annali Scuola Normale Sup. Pisa Cl. Sci. 14, 485–511 (1987)
B. Gidas, W. Ni, and L. Nirenberg, Symmetry and related properties by the maximum principle, Comm. Math. Phys. 68, 209–243 (1979)
C. Keller, Stable and unstable manifolds for the nonlinear wave equation with dissipation, J. Differential Equations 50 (3), 330–347 (1983)
M. K. Kwong, Uniqueness of positive solutions of $\Delta u - u + {u^p} = 0$ in ${R^N}$, Arch. Rational Mech. Anal. 105, 243–266 (1989)
H. A. Levine, Instability and nonexistence of global solutions of nonlinear wave equations of the form $P{u_{tt}} = - Au + F\left ( u \right )$, Trans. Amer. Math. Soc. 192, 1–21 (1974)
P. L. Lions, On positive solutions of semilinear elliptic equations on unbounded domains, in Nonlinear Diffusion Equations and Their Equilibrium States, II, W. M. Ni, L. A. Peletier, J. Serrin, Editors, Math. Sci. Res. Inst. Publ. 13, Springer-Verlag, New York, Heidelberg, Berlin, 1988
J. L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications, I., Dunod, Paris, 1968
L. E. Payne and D. H. Sattinger, Saddle points and instability of nonlinear hyperbolic equations, Israel J. Math. 22, 273–303 (1975)
A. Pazy, Semigroups of linear operators and applications to partial differential equations, Appl. Math. Sci., vol. 44, Springer-Verlag, New York, Berlin, Heidelberg, Tokyo, 1983
M. Remoissenet, Waves called solitons. Concepts and Experiments, Springer-Verlag, Berlin, Heidelberg, 1994
J. Shatah, Unstable ground state of nonlinear Klein-Gordon equations, Trans. Amer. Math. Soc. 290 (2), 701–711 (1985)
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