Affiliations: Università degli Studi di Pisa, Dipartimento di Matematica, via M. Buonarroti 2, 56127 Pisa, Italy E‐mail: [email protected] | Università degli Studi di Pisa, Dipartimento di Matematica Applicata “Ulisse Dini”, via Bonanno 25/B, 56126 Pisa, Italy E‐mail: [email protected]
Abstract: We investigate the evolution problem u"+δu'+m(|A1/2u|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(A1/2) satisfy a suitable smallness assumption and the non‐degeneracy condition m(|A1/2u0|2)>0. Moreover (u(t),u′(t),u″(t))→(u∞,0,0) in D(A)×D(A1/2)×H as t→+∞, where |A1/2u∞|m(|A1/2u∞|2)=0. These results apply to degenerate hyperbolic PDEs with non‐local non‐linearities.
