General Stability for the Viscoelastic Wave Equation with Nonlinear Time-Varying Delay, Nonlinear Damping and Acoustic Boundary Conditions
This paper is focused on energy decay rates for the viscoelastic wave equation that includes nonlinear time-varying delay, nonlinear damping at the boundary, and acoustic boundary conditions. We derive general decay rate results without requiring the condition <inline-formula><math xmlns=&q...
| Main Authors: | , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2023-11-01
|
| Series: | Mathematics |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2227-7390/11/22/4593 |
| _version_ | 1827639398450069504 |
|---|---|
| author | Mi Jin Lee Jum-Ran Kang |
| author_facet | Mi Jin Lee Jum-Ran Kang |
| author_sort | Mi Jin Lee |
| collection | DOAJ |
| description | This paper is focused on energy decay rates for the viscoelastic wave equation that includes nonlinear time-varying delay, nonlinear damping at the boundary, and acoustic boundary conditions. We derive general decay rate results without requiring the condition <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>a</mi><mn>2</mn></msub><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula> and without imposing any restrictive growth assumption on the damping term <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>f</mi><mn>1</mn></msub><mo>,</mo></mrow></semantics></math></inline-formula> using the multiplier method and some properties of the convex functions. Here we investigate the relaxation function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ψ</mi></semantics></math></inline-formula>, namely <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>ψ</mi><mo>′</mo></msup><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>≤</mo><mo>−</mo><mi>μ</mi><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mi>G</mi><mrow><mo>(</mo><mi>ψ</mi><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, where <i>G</i> is a convex and increasing function near the origin, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>μ</mi></semantics></math></inline-formula> is a positive nonincreasing function. Moreover, the energy decay rates depend on the functions <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>μ</mi></semantics></math></inline-formula> and <i>G</i>, as well as the function <i>F</i> defined by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>f</mi><mn>0</mn></msub></semantics></math></inline-formula>, which characterizes the growth behavior of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>f</mi><mn>1</mn></msub></semantics></math></inline-formula> at the origin. |
| first_indexed | 2024-03-09T16:37:55Z |
| format | Article |
| id | doaj.art-2995dcfa261548c882edce8d4ad9f5df |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-09T16:37:55Z |
| publishDate | 2023-11-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-2995dcfa261548c882edce8d4ad9f5df2023-11-24T14:54:08ZengMDPI AGMathematics2227-73902023-11-011122459310.3390/math11224593General Stability for the Viscoelastic Wave Equation with Nonlinear Time-Varying Delay, Nonlinear Damping and Acoustic Boundary ConditionsMi Jin Lee0Jum-Ran Kang1Department of Mathematics, Pusan National University, Busan 46241, Republic of KoreaDepartment of Applied Mathematics, Pukyong National University, Busan 48513, Republic of KoreaThis paper is focused on energy decay rates for the viscoelastic wave equation that includes nonlinear time-varying delay, nonlinear damping at the boundary, and acoustic boundary conditions. We derive general decay rate results without requiring the condition <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>a</mi><mn>2</mn></msub><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula> and without imposing any restrictive growth assumption on the damping term <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>f</mi><mn>1</mn></msub><mo>,</mo></mrow></semantics></math></inline-formula> using the multiplier method and some properties of the convex functions. Here we investigate the relaxation function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ψ</mi></semantics></math></inline-formula>, namely <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>ψ</mi><mo>′</mo></msup><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>≤</mo><mo>−</mo><mi>μ</mi><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mi>G</mi><mrow><mo>(</mo><mi>ψ</mi><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, where <i>G</i> is a convex and increasing function near the origin, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>μ</mi></semantics></math></inline-formula> is a positive nonincreasing function. Moreover, the energy decay rates depend on the functions <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>μ</mi></semantics></math></inline-formula> and <i>G</i>, as well as the function <i>F</i> defined by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>f</mi><mn>0</mn></msub></semantics></math></inline-formula>, which characterizes the growth behavior of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>f</mi><mn>1</mn></msub></semantics></math></inline-formula> at the origin.https://www.mdpi.com/2227-7390/11/22/4593optimal decayviscoelastic wave equationnonlinear time-varying delaynonlinear dampingacoustic boundary conditions |
| spellingShingle | Mi Jin Lee Jum-Ran Kang General Stability for the Viscoelastic Wave Equation with Nonlinear Time-Varying Delay, Nonlinear Damping and Acoustic Boundary Conditions Mathematics optimal decay viscoelastic wave equation nonlinear time-varying delay nonlinear damping acoustic boundary conditions |
| title | General Stability for the Viscoelastic Wave Equation with Nonlinear Time-Varying Delay, Nonlinear Damping and Acoustic Boundary Conditions |
| title_full | General Stability for the Viscoelastic Wave Equation with Nonlinear Time-Varying Delay, Nonlinear Damping and Acoustic Boundary Conditions |
| title_fullStr | General Stability for the Viscoelastic Wave Equation with Nonlinear Time-Varying Delay, Nonlinear Damping and Acoustic Boundary Conditions |
| title_full_unstemmed | General Stability for the Viscoelastic Wave Equation with Nonlinear Time-Varying Delay, Nonlinear Damping and Acoustic Boundary Conditions |
| title_short | General Stability for the Viscoelastic Wave Equation with Nonlinear Time-Varying Delay, Nonlinear Damping and Acoustic Boundary Conditions |
| title_sort | general stability for the viscoelastic wave equation with nonlinear time varying delay nonlinear damping and acoustic boundary conditions |
| topic | optimal decay viscoelastic wave equation nonlinear time-varying delay nonlinear damping acoustic boundary conditions |
| url | https://www.mdpi.com/2227-7390/11/22/4593 |
| work_keys_str_mv | AT mijinlee generalstabilityfortheviscoelasticwaveequationwithnonlineartimevaryingdelaynonlineardampingandacousticboundaryconditions AT jumrankang generalstabilityfortheviscoelasticwaveequationwithnonlineartimevaryingdelaynonlineardampingandacousticboundaryconditions |