Global Existence and Asymptotics of the Solutions of the Second-Order Quasilinear Hyperbolic Equations with the First-Order Dissipation

Abstract

Here the coefficients a tj are smooth and satisfy Z fl«X^ ^' 3^)C^^ flWLS, fl(0)>0 for all xeR n , teR 1 , yeR n+2 , Recently, we investigated the global existence and decay of the solutions of the semilinear wave equations (2) u tt-Au + au t +b(Du) = Q x<=R n , r>0, a>0

On the existence of weak solutions to the Cauchy problem for a class of quasilinear hyperbolic equations with a source term.

“…To complete the proof of theorem 1.1, we show that the entropy inequality (9) is satisfied by the entropy-entropy flux pair defined by (10).…”

“…We now assume that φ ∈ C 2 ([0, T [; H 2 (R)) (cf. [10]). By multiplying equation (14) by φ xx , integrating in R and integrating by parts, we get…”

Long-time behavior of small solutions to quasilinear dissipative hyperbolic equations

“…When f ≡ 0, global existence results for strong solutions of (1.1) corresponding to small initial values (1.2) were given by Matsumura [9], by means of direct energy estimates which yield an a priori bound on any local solution. In the case of a bounded domain, this method shows that solutions decay exponentially to 0.…”

Systems of Hyperbolic Conservation Laws With Memory

“…E.g., see [12,15,19,20,21,22,23,27,29,17]. Global weak solutions have also been constructed for special equations of the form (1.3) mainly in L ∞ by the method of compensated compactness.…”