Abstract
This paper is concerned with the study of damped wave equation of Kirchhoff type u t t −M( ∥ ∇ u ( t ) ∥ 2 2 )△u+ u t =g(u) in Ω×(0,∞), with initial and Dirichlet boundary condition, where Ω is a bounded domain of R 2 having a smooth boundary ∂ Ω. Under the assumption that g is a function with exponential growth at infinity, we prove global existence and the decay property as well as blow-up of solutions in finite time under suitable conditions.MSC: 35L70, 35B40, 35B44.
Highlights
For all u ∈ W ,n( ), n eα|u| n– ∈ L ( ), for all α > , and there exist positive constants Cn and αn such that sup u
Let be a bounded domain with smooth boundary ∂, we are concerned with the initialboundary value problem ⎧ ⎪⎪⎨utt – M( ∇u(t) u + ut = g(u) in × (, ∞),⎪⎪⎩uu(( t,xx))== un, ∂ut(, x) = u (x), × (, ∞), x∈ ( . )where g is a source term with exponential growth at the infinity to be specified later, M(s) is a positive C class function in s ≥
It is known that Kirchhoff [ ] first investigated the following nonlinear vibration of an elastic string for δ = f = :
Summary
For all u ∈ W ,n( ), n eα|u| n– ∈ L ( ), for all α > , and there exist positive constants Cn and αn such that sup u We have the following global existence and decay result. We have the following energy decay estimate: E(t) ≤ E( )e–κ[t– ]+ on [ , ∞), where κ is a positive constant.
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