Abstract
This paper deals with the global existence and energy decay of solutions for some coupled system of higher-order Kirchhoff-type equations with nonlinear dissipa- tive and source terms in a bounded domain. We prove the existence of global solutions for this problem by constructing a stable set in H m1 0 (W) H m2 0 (W) and give the decay estimate of global solutions by applying a lemma of V. Komornik. with initial data u(x, 0) = u0(x), ut(x, 0) = u1(x), x2 W, (1.3) v(x, 0) = v0(x), vt(x, 0) = v1(x), x2 W,
Highlights
In this paper we investigate the following system of nonlinear higher-order Kirchhoff-type equations utt + Φ( Dm1 u 2 + Dm2 v 2)(−∆)m1 u + a|ut|q−2ut = f1(u, v), vtt + Φ( Dm1 u 2 + Dm2 v 2)(−∆)m2 v + a|vt|q−2vt = f2(u, v), x ∈ Ω, t > 0, x ∈ Ω, t > 0, (1.1) (1.2)
Ω is a bounded domain in Rn with smooth boundary ∂Ω so that the divergence theorem can be applied, ν denotes the unit outward normal vector on
Motivated by the above researches, in this paper, we prove the global existence for the problem (1.1)–(1.6) by constructing a stable set in H0m1 (Ω) × H0m2 (Ω) and give the energy decay of global solutions by applying a lemma of V
Summary
Tsai [30] considered the system (1.7)– (1.11) with Φ( ∇u 2 + ∇v 2) = Φ( ∇u 2) in (1.7) and Φ( ∇u 2 + ∇v 2) = Φ( ∇v 2) in (1.8), respectively They obtain the existence of local and global solutions and give the blowup result for small positive initial energy. Wu [31] discusses the existence, asymptotic behavior and blow-up of solutions of the problem (1.7)–(1.11) under some conditions He gives the decay estimates of the energy function and the estimates for the lifespan of solutions. For the initial boundary value problem of a single nonlinear higher-order wave equation of Kirchhoff-type utt + Φ( Dmu 2)(−∆)mu + a|ut|q−2ut = b|u|p−2u, x ∈ Ω, t > 0,.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
