Global existence and new decay results of a viscoelastic wave equation with variable exponent and logarithmic nonlinearities
In this paper, we consider the following viscoelastic problem with variable exponent and logarithmic nonlinearities: $ u_{tt}-\Delta u+u+ \int_0^tb(t-s)\Delta u(s)ds+|u_t|^{{\gamma}(\cdot)-2}u_t = u\ln{\vert u\vert^{\alpha}}, $ where $ {\gamma}(.) $ is a function satisfying some conditions. W...
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| Format: | Article |
| Language: | English |
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AIMS Press
2021-07-01
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| Series: | AIMS Mathematics |
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| Online Access: | https://aimspress.com/article/doi/10.3934/math.2021587?viewType=HTML |
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| author | Mohammad M. Al-Gharabli Adel M. Al-Mahdi Mohammad Kafini |
| author_facet | Mohammad M. Al-Gharabli Adel M. Al-Mahdi Mohammad Kafini |
| author_sort | Mohammad M. Al-Gharabli |
| collection | DOAJ |
| description | In this paper, we consider the following viscoelastic problem with variable exponent and logarithmic nonlinearities:
$ u_{tt}-\Delta u+u+ \int_0^tb(t-s)\Delta u(s)ds+|u_t|^{{\gamma}(\cdot)-2}u_t = u\ln{\vert u\vert^{\alpha}}, $
where $ {\gamma}(.) $ is a function satisfying some conditions. We first prove a global existence result using the well-depth method and then establish explicit and general decay results under a wide class of relaxation functions and some specific conditions on the variable exponent function. Our results extend and generalize many earlier results in the literature. |
| first_indexed | 2024-12-22T14:36:29Z |
| format | Article |
| id | doaj.art-872b0ce504d7484196ae25dbc9f375f5 |
| institution | Directory Open Access Journal |
| issn | 2473-6988 |
| language | English |
| last_indexed | 2024-12-22T14:36:29Z |
| publishDate | 2021-07-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | AIMS Mathematics |
| spelling | doaj.art-872b0ce504d7484196ae25dbc9f375f52022-12-21T18:22:38ZengAIMS PressAIMS Mathematics2473-69882021-07-0169101051012910.3934/math.2021587Global existence and new decay results of a viscoelastic wave equation with variable exponent and logarithmic nonlinearitiesMohammad M. Al-Gharabli0Adel M. Al-Mahdi 1Mohammad Kafini 21. The Preparatory Year Program, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia3. The Interdisciplinary Research Center in Construction and Building Materials, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia1. The Preparatory Year Program, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia3. The Interdisciplinary Research Center in Construction and Building Materials, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia2. Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia 3. The Interdisciplinary Research Center in Construction and Building Materials, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi ArabiaIn this paper, we consider the following viscoelastic problem with variable exponent and logarithmic nonlinearities: $ u_{tt}-\Delta u+u+ \int_0^tb(t-s)\Delta u(s)ds+|u_t|^{{\gamma}(\cdot)-2}u_t = u\ln{\vert u\vert^{\alpha}}, $ where $ {\gamma}(.) $ is a function satisfying some conditions. We first prove a global existence result using the well-depth method and then establish explicit and general decay results under a wide class of relaxation functions and some specific conditions on the variable exponent function. Our results extend and generalize many earlier results in the literature.https://aimspress.com/article/doi/10.3934/math.2021587?viewType=HTMLviscoelasticityrelaxation functiongeneral decaylogarithmic nonlinearityvariable exponent |
| spellingShingle | Mohammad M. Al-Gharabli Adel M. Al-Mahdi Mohammad Kafini Global existence and new decay results of a viscoelastic wave equation with variable exponent and logarithmic nonlinearities AIMS Mathematics viscoelasticity relaxation function general decay logarithmic nonlinearity variable exponent |
| title | Global existence and new decay results of a viscoelastic wave equation with variable exponent and logarithmic nonlinearities |
| title_full | Global existence and new decay results of a viscoelastic wave equation with variable exponent and logarithmic nonlinearities |
| title_fullStr | Global existence and new decay results of a viscoelastic wave equation with variable exponent and logarithmic nonlinearities |
| title_full_unstemmed | Global existence and new decay results of a viscoelastic wave equation with variable exponent and logarithmic nonlinearities |
| title_short | Global existence and new decay results of a viscoelastic wave equation with variable exponent and logarithmic nonlinearities |
| title_sort | global existence and new decay results of a viscoelastic wave equation with variable exponent and logarithmic nonlinearities |
| topic | viscoelasticity relaxation function general decay logarithmic nonlinearity variable exponent |
| url | https://aimspress.com/article/doi/10.3934/math.2021587?viewType=HTML |
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