Arbitrary decays for a viscoelastic equation
<p>Abstract</p> <p>In this paper, we consider the nonlinear viscoelastic equation <inline-formula><m:math name="1687-2770-2011-28-i1" xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mo class="MathClass-rel">∣</m:mo> &l...
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| Format: | Article |
| Language: | English |
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SpringerOpen
2011-01-01
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| Series: | Boundary Value Problems |
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| Online Access: | http://www.boundaryvalueproblems.com/content/2011/1/28 |
| Summary: | <p>Abstract</p> <p>In this paper, we consider the nonlinear viscoelastic equation <inline-formula><m:math name="1687-2770-2011-28-i1" xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mo class="MathClass-rel">∣</m:mo> <m:msub> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:mi>t</m:mi> </m:mrow> </m:msub> <m:msup> <m:mrow> <m:mo class="MathClass-rel">∣</m:mo> </m:mrow> <m:mrow> <m:mi>ρ</m:mi> </m:mrow> </m:msup> <m:msub> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:mi>t</m:mi> <m:mi>t</m:mi> </m:mrow> </m:msub> <m:mo class="MathClass-bin">-</m:mo> <m:mi>Δ</m:mi> <m:mi>u</m:mi> <m:mo class="MathClass-bin">-</m:mo> <m:mi>Δ</m:mi> <m:msub> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:mi>t</m:mi> <m:mi>t</m:mi> </m:mrow> </m:msub> <m:mo class="MathClass-bin">+</m:mo> <m:msubsup> <m:mrow> <m:mo class="MathClass-op"> ∫ </m:mo> </m:mrow> <m:mrow> <m:mn>0</m:mn> </m:mrow> <m:mrow> <m:mi>t</m:mi> </m:mrow> </m:msubsup> <m:mi>g</m:mi> <m:mrow> <m:mo class="MathClass-open">(</m:mo> <m:mrow> <m:mi>t</m:mi> <m:mo class="MathClass-bin">-</m:mo> <m:mi>s</m:mi> </m:mrow> <m:mo class="MathClass-close">)</m:mo> </m:mrow> <m:mi>Δ</m:mi> <m:mi>u</m:mi> <m:mrow> <m:mo class="MathClass-open">(</m:mo> <m:mrow> <m:mi>s</m:mi> </m:mrow> <m:mo class="MathClass-close">)</m:mo> </m:mrow> <m:mstyle class="text"> <m:mtext class="textsf" mathvariant="sans-serif">d</m:mtext> </m:mstyle> <m:mi>s</m:mi> <m:mspace width="0.3em" class="thinspace"/> <m:mo class="MathClass-bin">+</m:mo> <m:mo class="MathClass-rel">∣</m:mo> <m:mi>u</m:mi> <m:msup> <m:mrow> <m:mo class="MathClass-rel">∣</m:mo> </m:mrow> <m:mrow> <m:mi>p</m:mi> </m:mrow> </m:msup> <m:mi>u</m:mi> <m:mo class="MathClass-rel">=</m:mo> <m:mn>0</m:mn> </m:math> </inline-formula>, in a bounded domain with initial conditions and Dirichlet boundary conditions. We prove an arbitrary decay result for a class of kernel function <it>g </it>without setting the function <it>g </it>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.</p> <p><b>Mathematical Subject Classification (2010): </b>35B35, 35B40, 35B60</p> |
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| ISSN: | 1687-2762 1687-2770 |