Ultimate bound of solutions to second-order evolution equation with nonlinear damping term

Faouzia Aloui ()
Additional contact information
Faouzia Aloui: Université de Monastir

Partial Differential Equations and Applications, 2021, vol. 2, issue 4, 1-16

Abstract: Abstract We estimate the ultimate bound of the energy of the solutions to the equation $$\begin{aligned}u''(t)+ Au(t)+g(u'(t))=f(t),\;\; t\ge 0, \end{aligned}$$ u ′ ′ ( t ) + A u ( t ) + g ( u ′ ( t ) ) = f ( t ) , t ≥ 0 , where A is a positive selfadjoint operator on a Hilbert space H, g is a nonlinear damping operator and f is a bounded forcing term with values in H. The paper concerns the asymptotic behavior of the solution. We prove that the bound of the solution is of the form $$\begin{aligned} C(1+\Vert f\Vert ^{2(a+2)/(a+1)}_{\infty }) \end{aligned}$$ C ( 1 + ‖ f ‖ ∞ 2 ( a + 2 ) / ( a + 1 ) ) where the constant $$a\ge 0$$ a ≥ 0 appears in the coercivity condition on the nonlinear damping operator g. and $$\Vert f\Vert $$ ‖ f ‖ stands for the $$L^{\infty }$$ L ∞ norm of f with values in H. Moreover, we investigate other cases, for instance, in the case of a special class of damping operator and the case of antiperiodic solutions. The last part of this paper is devoted to study the rate of decay for the solution of the equation as $$t\rightarrow +\infty $$ t → + ∞ .

Keywords: Second-order equation; Nonlinear damping; Bounded solution; 35B40; 35L20; 35B45 (search for similar items in EconPapers)
Date: 2021
References: View complete reference list from CitEc
Citations:

Downloads: (external link)
http://link.springer.com/10.1007/s42985-021-00110-x Abstract (text/html)
Access to the full text of the articles in this series is restricted.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:pardea:v:2:y:2021:i:4:d:10.1007_s42985-021-00110-x

Ordering information: This journal article can be ordered from
https://www.springer.com/journal/42985/

DOI: 10.1007/s42985-021-00110-x

Access Statistics for this article

Partial Differential Equations and Applications is currently edited by Zhitao Zhang

More articles in Partial Differential Equations and Applications from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().