Global well-posedness for strongly damped viscoelastic wave equation

Abstract

We study the initial boundary value problem for the nonlinear viscoelastic wave equation with strong damping term and dispersive term. By introducing a family of potential wells we not only obtain the invariant sets, but also prove the existence and nonexistence of global weak solution under some conditions with low initial energy. Furthermore, we establish a blow-up result for certain solutions with arbitrary positive initial energy (high energy case)

Keywords:

• viscoelastic wave equation
• global existence
• blow-up
• strong damping
• arbitrary positive initial energy

AMS Subject Classification:

• 35K35

Acknowledgements

This work was supported by National Natural Science Foundation of China (10871055, 10926149), Ph.D. Programs Foundation of Ministry of Education of China (20102304120022); Natural Science Foundation of Heilongjiang Province (A201014), Foundational Science Foundation of Harbin Engineering University and Fundamental Research Funds for the Central Universities (HEUCF20111101).