Bounded, almost-periodic, and periodic solutions to fully nonlinear telegraph equations. (English) Zbl 0762.35066

Feireisl, Eduard

Czech. Math. J. 40(115), No. 3, 514-527 (1990).

The author considers the nonlinear partial differential equations of hyperbolic type: \[ {\mathcal L}u+F(u_{xx},u_ x,u,u_{tt},u_{xt},u_ t)=f(x,t),\quad x\in(0,L),\quad t\in\mathbb{R}^ 1, \] where \({\mathcal L}u=u_{tt}+du_ t-au_{xx}\), \(a,d>0\), together with the boundary conditions \(u(0,t)=u(L,t)=0\).
The problem of existence of global in time solutions is studied. The existence theorem for the solutions belonging to the space of continuous functions is proved under some conditions on \(F\) when \(f(x,t)\) is sufficiently small. The existence theorems for periodic and almost- periodic solutions are proved when \(f(x,t)\) is a periodic or almost- periodic function of \(t\), respectively.

Reviewer: V.S.Rabinovich (Rostov-na-Donu)

MSC:

35L70

Second-order nonlinear hyperbolic equations

Keywords:

global in time solutions; existence theorem

References:

[1]	Amerio L., Prouse G.: Almost-periodic functions and functional equations. Van Nostrand New York 1971. · Zbl 0215.15701
[2]	Arosio A.: Linear second order differential equations in Hilbert spaces - the Cauchy problem and asymptotic behaviour for large time. Arch. Rational Mech. AnaI. 86 (2) (1984), pp. 147-180. · Zbl 0563.35041
[3]	Kato T.: Locally coercive nonlinear equations, with applications to some periodic solutions. Duke Math. J. 51 (4) (1984), pp. 923-936. · Zbl 0571.47051
[4]	Kato T.: Quasilinear equations of evolution with applications to partial differential equations. Lecture Notes in Math., Springer Berlin 1975, pp. 25 - 70.
[5]	Krejčí P.: Hard implicit function theorem and small periodic solutions to partial differential equations. Comment. Math. Univ. Carolinae 25 (1984), pp. 519-536. · Zbl 0567.35007
[6]	Lions J. L., Magenes E.: Problèmes aux limites non homogènes et applications I. Dunod Paris 1968. · Zbl 0165.10801
[7]	Matsumura A.: Global existence and asymptotics of the second-order quasilinear hyperbolic equations with the first-order dissipation. Publ. RIMS Kyoto Univ. 13 (1977), pp. 349-379. · Zbl 0371.35030
[8]	Milani A.: Time periodic smooth solutions of hyperbolic quasilinear equations with dissipation term and their approximation by parabolic equations. Ann. Mat. Pura Appl. 140 (4) (1985), pp. 331-344. · Zbl 0578.35060
[9]	Petzeltová H., Štědrý M.: Time periodic solutions of telegraph equations in n spatial variables. Časopis Pěst. Mat. 109 (1984), pp. 60 - 73. · Zbl 0544.35011
[10]	Rabinowitz P. H.: Periodic solutions of nonlinear hyperbolic partial differential equations II. Comm. Pure Appl. Math. 22 (1969), pp. Ĩ5-39. · Zbl 0157.17301
[11]	Shibata Y.: On the global existence of classical solutions of mixed problem for some second order non-linear hyperbolic operators with dissipative term in the interior domain. Funkcialaj Ekvacioj 25 (1982), pp. 303-345. · Zbl 0524.35070
[12]	Shibata Y., Tsutsumi Y.: Local existence of solution for the initial boundary value problem of fully nonlinear wave equation. Nonlinear Anal. 11 (3) 1987, pp. 335-365. · Zbl 0651.35053
[13]	Štědrý M.: Small time-periodic solutions to fully nonlinear telegraph equations in more spatial dimensions. Ann. Inst. Henri Poincaré 6 (3) (1989), pp. 209-232. · Zbl 0679.34038
[14]	Vejvoda O., al.: Partial differential equations: Time periodic solutions. Martinus Nijhoff PubI. 1982. · Zbl 0501.35001