# Long-Time Stabilization of Solutions to a Nonautonomous Semilinear Viscoelastic Equation

Long-Time Stabilization of Solutions to a Nonautonomous Semilinear Viscoelastic Equation We study the long-time behavior as time goes to infinity of global bounded solutions to the following nonautonomous semilinear viscoelastic equation: \begin{aligned} |u_t |^\rho u_{tt} -\Delta u_{tt}-\Delta u_{t}-\Delta u +\int ^\tau _0 k(s) \Delta u(t-s)ds+ f(x,u)=g, \ \tau \in \{t, \infty \}, \end{aligned} | u t | ρ u t t - Δ u t t - Δ u t - Δ u + ∫ 0 τ k ( s ) Δ u ( t - s ) d s + f ( x , u ) = g , τ ∈ { t , ∞ } , in $${\mathbb {R}}^+\times \Omega$$ R + × Ω , with Dirichlet boundary conditions, where $$\Omega$$ Ω is a bounded domain in $${\mathbb {R}}^n$$ R n and the nonlinearity f is analytic. Based on an appropriate (perturbed) new Lyapunov function and the Łojasiewicz–Simon inequality we prove that any global bounded solution converges to a steady state. We discuss also the rate of convergence which is polynomial or exponential, depending on the Łojasiewicz exponent and the decay of the term g . http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Applied Mathematics and Optimization Springer Journals

# Long-Time Stabilization of Solutions to a Nonautonomous Semilinear Viscoelastic Equation

, Volume 73 (2) – Apr 1, 2016
19 pages

/lp/springer_journal/long-time-stabilization-of-solutions-to-a-nonautonomous-semilinear-0kQMNmtakH
Publisher
Springer US
Subject
Mathematics; Calculus of Variations and Optimal Control; Optimization; Systems Theory, Control; Theoretical, Mathematical and Computational Physics; Mathematical Methods in Physics; Numerical and Computational Physics
ISSN
0095-4616
eISSN
1432-0606
D.O.I.
10.1007/s00245-015-9301-9
Publisher site
See Article on Publisher Site

### Abstract

We study the long-time behavior as time goes to infinity of global bounded solutions to the following nonautonomous semilinear viscoelastic equation: \begin{aligned} |u_t |^\rho u_{tt} -\Delta u_{tt}-\Delta u_{t}-\Delta u +\int ^\tau _0 k(s) \Delta u(t-s)ds+ f(x,u)=g, \ \tau \in \{t, \infty \}, \end{aligned} | u t | ρ u t t - Δ u t t - Δ u t - Δ u + ∫ 0 τ k ( s ) Δ u ( t - s ) d s + f ( x , u ) = g , τ ∈ { t , ∞ } , in $${\mathbb {R}}^+\times \Omega$$ R + × Ω , with Dirichlet boundary conditions, where $$\Omega$$ Ω is a bounded domain in $${\mathbb {R}}^n$$ R n and the nonlinearity f is analytic. Based on an appropriate (perturbed) new Lyapunov function and the Łojasiewicz–Simon inequality we prove that any global bounded solution converges to a steady state. We discuss also the rate of convergence which is polynomial or exponential, depending on the Łojasiewicz exponent and the decay of the term g .

### Journal

Applied Mathematics and OptimizationSpringer Journals

Published: Apr 1, 2016

### References

• Decay estimates to equilibrium for some asymptotically autonomous semilinear evolution equations
Ben Hassen, I

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