Existence of global solutions and decay estimates for a viscoelastic Petrovsky equation with a delay term in the non-linear internal feedback
In this article we consider a nonlinear viscoelastic Petrovsky equation in a bounded domain with a delay term in the weakly nonlinear internal feedback: $$\eqalign{ &|u_t|^{l}u_{tt} +\Delta^2 u -\Delta u_{tt} -\int_0^t h(t-s)\Delta^2 u(s)\,ds\cr &+\mu_1g_1(u_t(x,t)) +\mu_2g_2(u_t(x,t-\t...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
| Published: |
Texas State University
2017-02-01
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| Series: | Electronic Journal of Differential Equations |
| Subjects: | |
| Online Access: | http://ejde.math.txstate.edu/Volumes/2017/58/abstr.html |
| _version_ | 1818931519788417024 |
|---|---|
| author | Nadia Mezouar Mama Abdelli Amira Rachah |
| author_facet | Nadia Mezouar Mama Abdelli Amira Rachah |
| author_sort | Nadia Mezouar |
| collection | DOAJ |
| description | In this article we consider a nonlinear viscoelastic Petrovsky equation
in a bounded domain with a delay term in the weakly nonlinear internal feedback:
$$\eqalign{
&|u_t|^{l}u_{tt} +\Delta^2 u -\Delta u_{tt}
-\int_0^t h(t-s)\Delta^2 u(s)\,ds\cr
&+\mu_1g_1(u_t(x,t)) +\mu_2g_2(u_t(x,t-\tau))=0.
}$$
We prove the existence of global solutions in suitable Sobolev spaces by
using the energy method combined with Faedo-Galarkin method under condition
on the weight of the delay term in the feedback and the weight
of the term without delay. Furthermore, we study general stability
estimates by using some properties of convex functions. |
| first_indexed | 2024-12-20T04:17:53Z |
| format | Article |
| id | doaj.art-956e89683fb74f5e935b3ccbca6f6f34 |
| institution | Directory Open Access Journal |
| issn | 1072-6691 |
| language | English |
| last_indexed | 2024-12-20T04:17:53Z |
| publishDate | 2017-02-01 |
| publisher | Texas State University |
| record_format | Article |
| series | Electronic Journal of Differential Equations |
| spelling | doaj.art-956e89683fb74f5e935b3ccbca6f6f342022-12-21T19:53:44ZengTexas State UniversityElectronic Journal of Differential Equations1072-66912017-02-01201758,125Existence of global solutions and decay estimates for a viscoelastic Petrovsky equation with a delay term in the non-linear internal feedbackNadia Mezouar0Mama Abdelli1Amira Rachah2 Djillali Liabes Univ., Sidi Bel Abbes, Algeria Djillali Liabes Univ., Sidi Bel Abbes, Algeria Univ. Paul Sabatier, Cedex 9, France In this article we consider a nonlinear viscoelastic Petrovsky equation in a bounded domain with a delay term in the weakly nonlinear internal feedback: $$\eqalign{ &|u_t|^{l}u_{tt} +\Delta^2 u -\Delta u_{tt} -\int_0^t h(t-s)\Delta^2 u(s)\,ds\cr &+\mu_1g_1(u_t(x,t)) +\mu_2g_2(u_t(x,t-\tau))=0. }$$ We prove the existence of global solutions in suitable Sobolev spaces by using the energy method combined with Faedo-Galarkin method under condition on the weight of the delay term in the feedback and the weight of the term without delay. Furthermore, we study general stability estimates by using some properties of convex functions.http://ejde.math.txstate.edu/Volumes/2017/58/abstr.htmlGlobal solutiondelay termgeneral decaymultiplier methodweak frictional dampingconvexityviscoelastic Petrovsky equation |
| spellingShingle | Nadia Mezouar Mama Abdelli Amira Rachah Existence of global solutions and decay estimates for a viscoelastic Petrovsky equation with a delay term in the non-linear internal feedback Electronic Journal of Differential Equations Global solution delay term general decay multiplier method weak frictional damping convexity viscoelastic Petrovsky equation |
| title | Existence of global solutions and decay estimates for a viscoelastic Petrovsky equation with a delay term in the non-linear internal feedback |
| title_full | Existence of global solutions and decay estimates for a viscoelastic Petrovsky equation with a delay term in the non-linear internal feedback |
| title_fullStr | Existence of global solutions and decay estimates for a viscoelastic Petrovsky equation with a delay term in the non-linear internal feedback |
| title_full_unstemmed | Existence of global solutions and decay estimates for a viscoelastic Petrovsky equation with a delay term in the non-linear internal feedback |
| title_short | Existence of global solutions and decay estimates for a viscoelastic Petrovsky equation with a delay term in the non-linear internal feedback |
| title_sort | existence of global solutions and decay estimates for a viscoelastic petrovsky equation with a delay term in the non linear internal feedback |
| topic | Global solution delay term general decay multiplier method weak frictional damping convexity viscoelastic Petrovsky equation |
| url | http://ejde.math.txstate.edu/Volumes/2017/58/abstr.html |
| work_keys_str_mv | AT nadiamezouar existenceofglobalsolutionsanddecayestimatesforaviscoelasticpetrovskyequationwithadelayterminthenonlinearinternalfeedback AT mamaabdelli existenceofglobalsolutionsanddecayestimatesforaviscoelasticpetrovskyequationwithadelayterminthenonlinearinternalfeedback AT amirarachah existenceofglobalsolutionsanddecayestimatesforaviscoelasticpetrovskyequationwithadelayterminthenonlinearinternalfeedback |