Abstract. In this work, the nonexistence of the global solutions to a class of initial boundary value problems with dissipative terms in the boundary conditions is considered for a quasilinear hyperbolic equation. The nonexistence proof is achieved by the use of a lemma due to O. Ladyzhenskaya and V. K. Kalantarov and by the usage of the so called concavity method. In this method one writes down a functional which reflects the properties of dissipative boundary conditions and represents the norm of the solution in some sense, then proves that this functional satisfies the hypotheses of Ladyzhenskaya-Kalantarov lemma. Hence from the conclusion of the lemma one concludes that in finite time t 2 , this functional and hence the norm of the solution blows up.

No abstract

We investigate the blow-up phenomena for the two-component generalizations of Camassa-Holm equation on the real line. We establish some a local-in-space blow-up criterion for system of coupled equations under certain natural initial profiles. Presented result extends and specifies the earlier blow-up criteria for such type systems.

In this 'MJrk, the nonexistence of the global solutions of a quasilinear hyperbolic boundary value problem with dissipative term in the equation is considered. In one space dimension this initial value problem models the behavior of a riser vibrating dl1e to the effects of\Wves and current. The nonexistence proof is achieved by the use of the so called concavity method. In thiS method one writes do""n a functional which represents the norm of the solution in some sense. Then it is proved that this functional satisfies the hypotheses of the concavity lemma. Hence one concludes that one cannot continue the solution for all time by sho•.•.•-ing that this functional and hence the norm of the solution, lMJuld otherwise blow up in finite time.

No abstract