Semilinear structural damped waves. (11th July 2013)

• Record Type: Journal Article
• Authors: D'Abbicco, M.; Reissig, M.
• Is Part Of: Mathematical methods in the applied sciences. Volume 37:Number 11(2014:Jul. 30)
• Page Start: 1570
• Page End: 1592
• Publication Date: 2013-07-11
• Subjects: Mathematics -- Periodicals; Technology -- Mathematics -- Periodicals; 519
• DOI: 10.1002/mma.2913
• Languages: English
• ISSNs: 0170-4214

Abstract

<abstract abstract-type="main" id="mma2913-abs-0001"> <title> <x xml:space="preserve">Abstract</x> </title> <p id="mma2913-para-0001">We study the Cauchy problem for the semilinear structural damped wave equation with source term <disp-formula content-type="mathematics" id="mma2913-disp-0001"><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pghmhfpzsz" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="block" altimg="urn:x-wiley:01704214:media:mma2913:mma2913-math-0001" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mrow><mml:mi>u</mml:mi></mml:mrow><mml:mrow><mml:mi>tt</mml:mi></mml:mrow></mml:msub><mml:mo class="MathClass-bin">−</mml:mo><mml:mo class="MathClass-rel">△</mml:mo><mml:mi>u</mml:mi><mml:mo class="MathClass-bin">+</mml:mo><mml:mi>μ</mml:mi><mml:msup><mml:mrow><mml:mrow><mml:mo class="MathClass-open">(</mml:mo><mml:mrow><mml:mo class="MathClass-bin">−</mml:mo><mml:mi>Δ</mml:mi></mml:mrow><mml:mo class="MathClass-close">)</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:mi>σ</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mrow><mml:mi>u</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo class="MathClass-rel">=</mml:mo><mml:mi>f</mml:mi><mml:mrow><mml:mo class="MathClass-open">(</mml:mo><mml:mrow><mml:mi>u</mml:mi></mml:mrow><mml:mo class="MathClass-close">)</mml:mo></mml:mrow><mml:mo class="MathClass-punc">, </mml:mo><mml:mspace width="1em" class="quad" /><mml:mi>u</mml:mi><mml:mrow><mml:mo class="MathClass-open">(</mml:mo><mml:mrow><mml:mn>0</mml:mn><mml:mo class="MathClass-punc">, </mml:mo><mml:mi>x</mml:mi></mml:mrow><mml:mo class="MathClass-close">)</mml:mo></mml:mrow><mml:mo class="MathClass-rel">=</mml:mo><mml:msub><mml:mrow><mml:mi>u</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mrow><mml:mo class="MathClass-open">(</mml:mo><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mo class="MathClass-close">)</mml:mo></mml:mrow><mml:mo class="MathClass-punc">, </mml:mo><mml:mspace width="1em" class="quad" /><mml:msub><mml:mrow><mml:mi>u</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo class="MathClass-open">(</mml:mo><mml:mrow><mml:mn>0</mml:mn><mml:mo class="MathClass-punc">, </mml:mo><mml:mi>x</mml:mi></mml:mrow><mml:mo class="MathClass-close">)</mml:mo></mml:mrow><mml:mo class="MathClass-rel">=</mml:mo><mml:msub><mml:mrow><mml:mi>u</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mrow><mml:mo class="MathClass-open">(</mml:mo><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mo class="MathClass-close">)</mml:mo></mml:mrow><mml:mo class="MathClass-punc">, </mml:mo></mml:math></alternatives></disp-formula></p> <p id="mma2913-para-0002">with <italic>σ</italic> ∈ (0, 1] in space dimension <italic>n</italic> ≥ 2 and with a positive constant <italic>μ</italic>. We are interested in the influence of <italic>σ</italic> on the critical exponent <italic>p</italic><sub>crit</sub> in | <italic>f</italic>(<italic>u</italic>) | ≈ | <italic>u</italic> | <sup><italic>p</italic></sup>. This critical exponent is the threshold between global existence in time of small data solutions and blow‐up behavior for some suitable range of <italic>p</italic>. Our results are optimal for <italic>σ</italic> = 1 ∕ 2. Copyright © 2013 John Wiley & Sons, Ltd.</p> </abstract>