Optimal decay rates for semi-linear non-autonomous evolution equations with vanishing damping
We consider a second order non-autonomous evolution equation in a Hilbert space Hu′′(t)+Au(t)+γ(t)u′(t)+∇F(u(t))=0,where A is a self-adjoint and nonnegative operator on H with a nontrivial kernel, the nonlinear term ∇F is the gradient of a differentiable function F:D(A1/2)→R, and γ(t) is a damping c...
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Veröffentlicht in: | Nonlinear analysis 2023-05, Vol.230, p.113247, Article 113247 |
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Format: | Artikel |
Sprache: | eng |
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Zusammenfassung: | We consider a second order non-autonomous evolution equation in a Hilbert space Hu′′(t)+Au(t)+γ(t)u′(t)+∇F(u(t))=0,where A is a self-adjoint and nonnegative operator on H with a nontrivial kernel, the nonlinear term ∇F is the gradient of a differentiable function F:D(A1/2)→R, and γ(t) is a damping coefficient, vanishing as t goes to infinity. We focus on a limit case γ(t)=α/t (with some α>0), which turns out to be quite tricky to deal with. When α exceeds a critical value, we obtain the optimal decay rate of the solution of the equation, in terms of the exponent associated with F. Moreover, we present two examples to illustrate our theoretical results. |
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ISSN: | 0362-546X 1873-5215 |
DOI: | 10.1016/j.na.2023.113247 |