Global existence and asymptotic behaviour for a mildly degenerate dissipative hyperbolic equation of Kirchhoff type

Abstract

We investigate the evolution problem

u"+δu'+m(|A^1/2u|²)Au=0,

u(0)=u₀,  u'(0)=u₁,

where H is a Hilbert space, A is a self‐adjoint non‐negative operator on H with domain D(A), δ>0 is a parameter, and m :[0,+∞[→[0,+∞[ is a locally Lipschitz continuous function. We prove that this problem has a unique global solution for positive times, provided that the initial data (u₀,u₁)∈D(A)×D(A^1/2) satisfy a suitable smallness assumption and the non‐degeneracy condition m(|A^1/2u₀|²)>0. Moreover (u(t),u′(t),u″(t))→(u_∞,0,0) in D(A)×D(A^1/2)×H as t→+∞, where |A^1/2u_∞|m(|A^1/2u_∞|²)=0. These results apply to degenerate hyperbolic PDEs with non‐local non‐linearities.