Asymptotic for a Semilinear Hyperbolic Equation with Asymptotically Vanishing Damping Term, Convex Potential and Integrable Source

Mounir Balti and Ramzi May – GJM, Volume 7, Issue 1 (2022), 39-45.

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• septembre 10, 2022
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We investigate the long time behavior of solutions to semilinear hyperbolic equations of the form: \displaystyle u^{\prime\prime}(t)+\gamma(t)u^{\prime}(t)+Au(t)+f(u(t))=g(t),~t\geq 0     (E_\alpha) where A is a self-adjoint nonnegative operator, f a function which is the gradient of a regular convex function, and \gamma a nonnegative function which behaves, for t large enough, as  {K\over t^{\alpha}}, with K>0 and \alpha \in\lbrack0,1[ . We obtain sufficient conditions on the source term g(t), that ensure the weak and strong convergence of any solution u(t) of (E_{\alpha}), as t\rightarrow+\infty, to a solution of the stationary equation Av+f(v)=0, if one exists.

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Received: July 28, 2022
Accepted: September 10, 2022
Published online: October 10, 2022

Authors:

Mounir Balti
Institut Préparatoire aux Etude Scientifiques et Techniques,
Université de Carthage, Bp 51 La Marsa, Tunisia.
Ramzi May
Mathematics Department, College of Science,
King Faisal University, P.O. 380,
Ahsaa 31982, , Kingdom of Saudi Arabia.

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