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Arbitrary decays for a viscoelastic equation

Shun-Tang Wu

Author Affiliations

General Education Center National Taipei University of Technology Taipei 106, Taiwan

Boundary Value Problems 2011, 2011:28  doi:10.1186/1687-2770-2011-28

Published: 6 October 2011


In this paper, we consider the nonlinear viscoelastic equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/28/mathml/M1">View MathML</a>, in a bounded domain with initial conditions and Dirichlet boundary conditions. We prove an arbitrary decay result for a class of kernel function g without setting the function g itself to be of exponential (polynomial) type, which is a necessary condition for the exponential (polynomial) decay of the solution energy for the viscoelastic problem. The key ingredient in the proof is based on the idea of Pata (Q Appl Math 64:499-513, 2006) and the work of Tatar (J Math Phys 52:013502, 2010), with necessary modification imposed by our problem.

Mathematical Subject Classification (2010): 35B35, 35B40, 35B60

Viscoelastic equation; Kernel function; Exponential decay; Polynomial decay