A Study on Damped Elastic Systems in Banach Spaces

Abstract

In this article, we concern with the existence of mild solutions for damped elastic systems. The discussion is based on the operator semigroups theory and fixed point theorem. New forms of the corresponding first order evolution equation are introduced, and their well-posed property is proved by means of the operator semigroup theory. Further, we consider analyticity and exponential of the semigroup associated with the elastic system (1.1(1.1) $$\{\begin{array}{c}u″(t)+\rho \mathit{Bu}\prime (t)+\mathit{Au}(t)=f(t,u(t)), 0<t<a,\hfill \\ u(0)=u_0\in D(A), u\prime (0)=u_1\in E,\hfill \end{array}$$(1.1) ). Finally, two examples are given to illustrate availability of our results.

Keywords:

• Damped elastic systems
• mild solution
• analytic and exponentially stable
• existence of solutions

Mathematics Subject Classification:

• 26A33
• 34K37
• 34A08
• 34A37

Acknowledgments

The authors would like to thank the referees for their useful suggestions which have significantly improved the paper.

Additional information

Funding

This study was supported by the National Natural Science Foundation of China (Grant No.11661071).