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Zeitschrift für Analysis und ihre Anwendungen


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Volume 37, Issue 1, 2018, pp. 83–99
DOI: 10.4171/ZAA/1604

Published online: 2018-01-08

Stability of Global Bounded Solutions to a Nonautonomous Nonlinear Second Order Integro-Differential Equation

Hassan Yassine[1]

(1) Lebanese University, Zahleh, Lebanon

We study the long-time behavior as time goes to infinity of global bounded weak solutions to the following integro-differential equation $$\ddot u+k*\dot u+\nabla E(u)=g, $$ in finite dimensions, where the nonlinear potential $E$ satisfies the Łojasiewicz inequality near some equilibrium point. Based on an appropriate new Lyapunov function and Łojasiewicz inequality we prove that any global bounded weak solution converges to a steady state. We also obtain the rate of convergence according to the Łojasiewicz exponent and the time-dependent right-hand side $g$.

Keywords: Evolutionary integral equation, semilinear, stabilization, Lojasiewicz-Simon inequality.

Yassine Hassan: Stability of Global Bounded Solutions to a Nonautonomous Nonlinear Second Order Integro-Differential Equation. Z. Anal. Anwend. 37 (2018), 83-99. doi: 10.4171/ZAA/1604