Abstract
The current paper deals with the proof of a global solution of a viscoelasticity singular one-dimensional system with localized frictional damping and general source terms, taking into consideration nonlocal boundary condition. Moreover, similar to that in Boulaaras’ recent studies by constructing a Lyapunov functional and use it together with the perturbed energy method in order to prove a general decay result.
Highlights
The evolution problem with integral conditions is related with many branches of sciences ([1,2,3,4,5,6])
By the motivation of this work, some authors gave necessary and sufficient conditions for the hyperbolic equation with source term. This manuscript is devoted to the study of the global existence and decay for a system of two singular one-dimensional nonlinear viscoelastic equations with general source terms Journal of Function Spaces
We used the techniques in [11]; we have proven in [8] the existence of a global solution using the potential well theory for the following viscoelastic system with nonlocal boundary condition and localized frictional damping
Summary
The evolution problem with integral conditions is related with many branches of sciences ([1,2,3,4,5,6]). We used the techniques in [11]; we have proven in [8] the existence of a global solution using the potential well theory for the following viscoelastic system with nonlocal boundary condition and localized frictional damping aðxÞut jvjq+1jujp−1u, in 1 x ðxvxðx, aðxÞvt jvjq+1jujp−1v, in TÞ, uðx, vðx, uðα, u0ðxÞ, utðx, 0Þ = u1ðxÞ, x ∈ ð0, αÞ, v0ðxÞ, vtðx, 0Þ = v1ðxÞ, x ∈ ð0, αÞ, ðα ðα vðα, tÞ = 0, xuðx, tÞdx = xvðx, t. In view of the articles mentioned above in ([8, 9, 11]), much less effort has been devoted to the existence of a global solution to the system of two singular nonlinear equations which arise in generalized viscoelasticity with localized frictional damping terms using the potential-well theory.
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