Global existence and nonexistence of solutions for a viscoelastic wave equation with nonlinear boundary source term. Issue 11 (29th February 2016)

• Record Type: Journal Article
• Authors: Di, Huafei; Shang, Yadong; Peng, Xiaoming
• Is Part Of: Mathematische Nachrichten. Volume 289:Issue 11/12(2016)
• Page Start: 1408
• Page End: 1432
• Publication Date: 2016-02-29
• Subjects: Viscoelastic equation -- nonlinear boundary source -- potential wells -- invariant sets -- 35B44 -- 35D30 -- 35L30 -- 35L82; Mathematics -- Periodicals; 510.5
• DOI: 10.1002/mana.201500169
• Languages: English
• ISSNs: 0025-584X

Abstract

Abstract : In this paper, we consider the initial boundary value problem for a viscoelastic wave equation with nonlinear boundary source term. First of all, we introduce a family of potential wells and prove the invariance of some sets. Then we establish the existence and nonexistence of global weak solution with small initial energy under suitable assumptions on the relaxation function g ( · ), nonlinear function f ( · ), the initial data and the parameters in the equation. Furthermore, we obtain the global existence of weak solution for the problem with critical initial conditions I ( u 0 ) ≥ 0 and E ( 0 ) = d .