Volume 27, 2021
|Number of page(s)||24|
|Published online||07 June 2021|
Optimal energy decay rates for abstract second order evolution equations with non-autonomous damping*
College of Science, University of Shanghai for Science and Technology,
200093, PR China.
2 Shanghai Key Laboratory for Contemporary Applied Mathematics, School of Mathematical Sciences, Fudan University, Shanghai 200433, PR China.
** Corresponding author: email@example.com
Accepted: 22 April 2021
We consider an abstract second order non-autonomous evolution equation in a Hilbert space H : u″ + Au + γ(t)u′ + f(u) = 0, where A is a self-adjoint and nonnegative operator on H, f is a conservative H-valued function with polynomial growth (not necessarily to be monotone), and γ(t)u′ is a time-dependent damping term. How exactly the decay of the energy is affected by the damping coefficient γ(t) and the exponent associated with the nonlinear term f? There seems to be little development on the study of such problems, with regard to non-autonomous equations, even for strongly positive operator A. By an idea of asymptotic rate-sharpening (among others), we obtain the optimal decay rate of the energy of the non-autonomous evolution equation in terms of γ(t) and f. As a byproduct, we show the optimality of the energy decay rates obtained previously in the literature when f is a monotone operator.
Mathematics Subject Classification: 35B35 / 93D20 / 34G20 / 35L70 / 35L90
Key words: Non-autonomous / abstract second order evolution equation / time dependent damping / energy estimates / slow solutions / nonlinear source / Hilbert space
© The authors. Published by EDP Sciences, SMAI 2021
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