The existence of small global (in time) solutions to an abstract evolution equation containing a damping term is proved. The result is then applied to fully nonlinear telegraph equations and to nonlinear equations involving operators with time delay.
@article{104482, author = {Eduard Feireisl}, title = {Global in time solutions to quasilinear telegraph equations involving operators with time delay}, journal = {Applications of Mathematics}, volume = {36}, year = {1991}, pages = {456-468}, zbl = {0752.45012}, mrnumber = {1134922}, language = {en}, url = {http://dml.mathdoc.fr/item/104482} }
Feireisl, Eduard. Global in time solutions to quasilinear telegraph equations involving operators with time delay. Applications of Mathematics, Tome 36 (1991) pp. 456-468. http://gdmltest.u-ga.fr/item/104482/
The global solvability of unilateral problems for quasilinear operators of the hyperbolic type (Russian), Dokl. Akad. Nauk SSSR 298 (5) (1988), 1033-1036. (1988) | MR 0939671
Global (in time) solution of the approximate non-linear string equation of G. F. Carrier and R. Narasimha, Comment. Math. Univ. Carolinae 26 (1) (1985), 169-172. (1985) | MR 0797299
Linear second order differential equations in Hilbert spaces - the Cauchy problem and asymptotic behaviour for large time:, Arch. Rational Mech. Anal. 86 (2) (1984), 147-180. (1984) | Article | MR 0751306 | Zbl 0563.35041
Remark on the decay for damped string and beam equations, Nonlinear Anal. 10 (9) (1984), 839-842. (1984) | Article | MR 0856867
Small time-periodic solutions to a nonlinear equation of a vibrating string, Apl. mat. 32 (6) (1987), 480-490. (1987) | MR 0916063 | Zbl 0653.35063
On the Cauchy problem for quasilinear hyperbolic systems with a retarded argument, Ann. Mat. Рurа Appl. 143 (4) (1986), 235 - 246. (1986) | Article | MR 0859605
Quasilinear equations of evolution with applications to partial differential equations, Lectures Notes in Mathematics 448, 25 - 70. Springer, Berlin 1975. (1975) | MR 0407477
Hard implicit function theorem and small periodic solutions to partial differential equations, Comment. Math. Univ. Carolinae 25 (1984), 519-536. (1984) | MR 0775567
Problèmes aux limites non homogènes et applications I, Dunod, Paris 1968. (1968)
Global existence and asymptotics of the solutions of the second-order quasilinear hyperbolic equations with the first-order dissipation, Publ. RIMS Kyoto Univ. 13 (1977), 349-379. (1977) | MR 0470507 | Zbl 0371.35030
On a new class of nonlinear wave equations, J. Math. Anal. Appl. 69 (1) (1979), 252-262. (1979) | MR 0535295 | Zbl 0407.35051
Time periodic solutions of telegraph equations in n spatial variables, Časopis Pěst. Mat. 109 (1984), 60-73. (1984) | MR 0741209
Bounded solutions of nonlinear hyperbolic equations with delay, Lecture Notes in Pure and Appl. Math. 109 (Dekker), 1987. (1987) | MR 0912327
Periodic solutions of nonlinear hyperbolic partial differential equations II, Comm. Pure Appl. Math. 22 (1969), 15-39. (1969) | Article
Local existence of solution for the initial boundary value problem of fully nonlinear wave equation, Nonlinear Anal. 11 (3) (1987), 335-365. (1987) | Article | MR 0881723
Small time-periodic solutions to fully nonlinear telegraph equations in more spatial dimensions, preprint. Ann. Inst. Henri Poincaré 6 (3) (1989), 209-232. (1989) | MR 0995505
Partial differential equations: Time periodic solutions, Martinus Nijhoff Publ., 1982. (1982) | Zbl 0501.35001