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: $$\eqalign{ &|u_t|^{l}u_{tt} +\Delta^2 u -\Delta u_{tt} -\int_0^t h(t-s)\Delta^2 u(s)\,ds\cr &+\mu_1g_1(u_t(x,t)) +\mu_2g_2(u_t(x,t-\t...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
| Published: |
Texas State University
2017-02-01
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| Series: | Electronic Journal of Differential Equations |
| Subjects: | |
| Online Access: | http://ejde.math.txstate.edu/Volumes/2017/58/abstr.html |
| Summary: | In this article we consider a nonlinear viscoelastic Petrovsky equation
in a bounded domain with a delay term in the weakly nonlinear internal feedback:
$$\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. |
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| ISSN: | 1072-6691 |