Stability Results for a Weakly Dissipative Viscoelastic Equation with Variable-Exponent Nonlinearity: Theory and Numerics
In this paper, we study the long-time behavior of a weakly dissipative viscoelastic equation with variable exponent nonlinearity of the form <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mstyle scriptlevel="...
| Main Authors: | , , , |
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| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2023-01-01
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| Series: | Mathematical and Computational Applications |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2297-8747/28/1/5 |
| Summary: | In this paper, we study the long-time behavior of a weakly dissipative viscoelastic equation with variable exponent nonlinearity of the form <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mstyle scriptlevel="0" displaystyle="true"><mrow><msub><mi>u</mi><mrow><mi>t</mi><mi>t</mi></mrow></msub><mo>+</mo><msup><mo>Δ</mo><mn>2</mn></msup><mi>u</mi><mo>−</mo><msubsup><mo>∫</mo><mn>0</mn><mi>t</mi></msubsup><mi>g</mi><mrow><mo>(</mo><mi>t</mi><mo>−</mo><mi>s</mi><mo>)</mo></mrow><mo>Δ</mo><mi>u</mi><mrow><mo>(</mo><mi>s</mi><mo>)</mo></mrow><mi>d</mi><mi>s</mi><mo>+</mo><mi>a</mi><msup><mrow><mo stretchy="false">|</mo><msub><mi>u</mi><mi>t</mi></msub><mo stretchy="false">|</mo></mrow><mrow><mi>n</mi><mo>(</mo><mo>·</mo><mo>)</mo><mo>−</mo><mn>2</mn></mrow></msup><msub><mi>u</mi><mi>t</mi></msub><mo>−</mo><mo>Δ</mo><msub><mi>u</mi><mi>t</mi></msub><mo>=</mo><mn>0</mn><mo>,</mo></mrow></mstyle></semantics></math></inline-formula> where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>(</mo><mo>.</mo><mo>)</mo></mrow></semantics></math></inline-formula> is a continuous function satisfying some assumptions and <i>g</i> is a general relaxation function such that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>g</mi><mo>′</mo></msup><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>≤</mo><mo>−</mo><mi>ξ</mi><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mi mathvariant="double-struck">G</mi><mrow><mo>(</mo><mi>g</mi><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>)</mo></mrow><mo>,</mo></mrow></semantics></math></inline-formula> where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ξ</mi></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="double-struck">G</mi></semantics></math></inline-formula> are functions satisfying some specific properties that will be mentioned in the paper. Depending on the nature of the decay rate of <i>g</i> and the variable exponent <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>(</mo><mo>.</mo><mo>)</mo></mrow></semantics></math></inline-formula>, we establish explicit and general decay results of the energy functional. We give some numerical illustrations to support our theoretical results. Our results improve some earlier works in the literature. |
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| ISSN: | 1300-686X 2297-8747 |