Two remarks on hyperbolic dissipative problems. (English) Zbl 0579.35057

Haraux, A.

Nonlinear partial differential equations and their applications, Coll. de France Semin., Vol. VII, Paris 1983-84, Res. Notes Math. 122, 161-179 (1985).

[For the entire collection see Zbl 0559.00005.]
The author considers the semi-linear dissipative hyperbolic equation \[ u_{tt}-\Delta u+f(u)+g(u_ t)=h(t,x)\quad on\quad R_+\times \Omega \quad and\quad u=0\quad on\quad R_+\times \partial \Omega. \] He gives a simpler proof of the result of Americo-Prouse on boundedness of the energy of the solution u. In the second part one proves the uniform ultimate boundedness of the trajectories (improving a result of Babin- Vishik).

Reviewer: N.H.Pavel

MSC:

35L70	Second-order nonlinear hyperbolic equations
35B35	Stability in context of PDEs

Keywords:

semi-linear dissipative hyperbolic equation; boundedness of the energy; ultimate boundedness of the trajectories

Citations:

Zbl 0559.00005