In this paper we consider a mixed problem for the nonlinear wave equations with transmission acoustic conditions, that is, the wave propagation over bodies consisting of two physically different types of materials, one of which is clamped. | Turk J Math (2018) 42: 3211 – 3231 © TÜBİTAK doi: Turkish Journal of Mathematics Research Article Existence and nonexistence of global solutions for nonlinear transmission acoustic problem 1 Akbar ALIEV1 ,, Sevda ISAYEVA2,∗ Department of Differential Equations, Institute of Mathematics and Mechanics of the National Academy of Sciences of Azerbaijan, Baku, Azerbaijan 2 Department of Higher Mathematics, Baku State University, Baku, Azerbaijan Received: • Accepted/Published Online: • Final Version: Abstract: In this paper we consider a mixed problem for the nonlinear wave equations with transmission acoustic conditions, that is, the wave propagation over bodies consisting of two physically different types of materials, one of which is clamped. We prove the existence of a global solution. Under the condition of positive initial energy we show that the solution for this problem blows up in finite time. Key words: Nonlinear wave equation, transmission acoustic conditions, locally reacting boundary, existence of global solutions, blow up result 1. Introduction Let Ω be a bounded domain in Rn (n ≥ 1) with smooth boundary Γ1 , Ω2 ⊂ Ω is a subdomain with smooth boundary Γ2 , and Ω1 = Ω\Ω2 is a subdomain with boundary Γ = Γ1 ∪ Γ2 , Γ1 ∩ Γ2 =Ø. The nonlinear transmission acoustic problem considered here is q1 −1 ut = |u| q2 −1 υt = |υ| utt − ∆u + |ut | υtt − ∆υ + |υt | p−1 u in Ω1 × (0, ∞) , () p−1 υ in Ω2 × (0, ∞) , () M δtt + Dδt + Kδ = −ut on Γ2 × (0, ∞) , () u = 0 on Γ1 × (0, ∞) , () u = υ, δt = ∂u ∂υ − on Γ2 × (0, ∞) , ∂ν ∂ν () ¯ 1, u (x, 0) = u0 (x) , ut (x, 0) = u1 (x) , x ∈ Ω () ¯ 2, υ (x, 0) = υ0 (x) , υt (x, 0) = υ1 (x) , x ∈ Ω () ∗Correspondence: isayevasevda@ 2010 AMS Mathematics Subject Classification: 35L10, 74G25 3211 This work is licensed under a Creative Commons Attribution International .
