Asymptotic for a semilinear hyperbolic equation with asymptotically\n vanishing damping term, convex potential, and integrable source

2016/08/31 by Mounir Balti, Balti, Mounir, Ramzi May +1
Computer Science · Engineering · Mathematics · #34D05 #34G20 #35B40 #35L71 #Advanced Mathematical Modeling in Engineering #Advanced Mathematical Physics Problems #FOS: Mathematics #Nonlinear Partial Differential Equations #Optimization and Control (math.OC) #Stability and Controllability of Differential Equations

paper · pdf · doi:10.48550/arxiv.1608.08760

openalex publication_date 2016/08/31 · openalex created_date 2022/10/03 · openalex updated_date 2026/07/28

Abstract

We investigate the long time behavior of solutions to semilinear hyperbolic\nequation (E_\α): ν^\′\′(t)+\γ(t)u^\′(t)+Au(t)+f(u(t))=g(t),~t\≥0, where\nA is a self-adjoint nonnegative operator, f a function which derives from a\nconvex function, and \γ a nonnegative function which behaviors, for t\nlarge enough, as fracKt^\α with K>0 and \α \∈ lbrack0,1[.\nWe obtain sufficient conditions on the source term g(t), ensuring the weak or\nthe strong convergence of any solution u(t) of (E_\α) as\nt\→+\∞ to a solution of the stationary equation Av+f(v)=0 if\none exists.\n

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