Methods and Applications of Analysis

Volume 22 (2015)

Number 1

Global existence and decay property for solutions in nonlinear elastic solids with voids

Pages: 101 – 130

DOI: http://dx.doi.org/10.4310/MAA.2015.v22.n1.a4

Author

Belkacem Said-Houari (Mathematics and Natural Sciences Department, ALHOSN University, Abu Dhabi, United Arab Emirates)

Abstract

In this paper, we consider a nonlinear Cauchy problem for a system of elastic solids with voids. First, we prove that the damping in the porous equation alone is weak and the solutions of the corresponding system are of regularity-loss type. In addition, we show a global existence result for solutions in $H^s (\mathbb{R})$ for large $s$. Second, we prove that by considering an additional viscoelastic damping, then the solutions can gain some regularity and all solutions in $H^s$ with $s \geq 4$ are global in time.

Keywords

decay rate, stability, regularity-loss, regularity gain, energy method

2010 Mathematics Subject Classification

35B35, 35L55, 74D05, 93D15, 93D20

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