Abstract
In this work, we consider a quasilinear system of viscoelastic equations with degenerate damping, dispersion, and source terms under Dirichlet boundary condition. Under some restrictions on the initial datum and standard conditions on relaxation functions, we study global existence and general decay of solutions. The results obtained here are generalization of the previous recent work.
Highlights
Let Ω be a bounded domain with a sufficiently smooth boundary in Rnðn ≥ 1Þ: We investigate a quasilinear system of two viscoelastic equations in the presence of degenerate damping, dispersion, and source terms, namely, 8 >>>>>>>>>>>
We studied the general decay of solutions
As far as we know, there have not been any global existences and general decay results in the literature known for quasilinear viscoelastic equations with degenerate damping terms
Summary
Let Ω be a bounded domain with a sufficiently smooth boundary in Rnðn ≥ 1Þ: We investigate a quasilinear system of two viscoelastic equations in the presence of degenerate damping, dispersion, and source terms, namely,. To motivate our problem (1), it can trace back to the initial boundary value problem for the single viscoelastic equation of the form ðt jutjηutt − Δu + hðt − sÞΔuðsÞds − Δutt + gðu, utÞ = f ðuÞ: ð6Þ This type problem appears a variety of mathematical models in applied science. UtÞ = 0 and without dispersion term, problem (5) has been investigated by Song [6], and the blow-up result for positive initial energy has been proved. Pișkin and Ekinci [10] studied a general decay and blow-up of solutions with nonpositive initial energy for problem (1) case (Kirchhoff-type instead of Δu and without dispersion term). We studied the general decay of solutions
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