Narrow Results

Following a recent paper of the authors in Communications in Partial Differential Equations, this paper establishes the global existence of weak solutions to a nonlinear variational wave equation under relaxed conditions on the initial data so that the solutions can contain singularities (blow-up). Propagation of local oscillations along one family of characteristics remains under control despite singularity formation in the other family of characteristics.

The authors investigate the global propeties of general autonomous systems on the plane and establish criteria for the nonexistence, existence and uniqueness of limit cycles. As application examples, the limit cycles for some polynomial systems are studied.

The authors study a generalized thin film equation. Under some assumptions on the initial value, the existence of weak solutions is established by the time-discrete method. The uniqueness and asymptotic behavior of solutions are also discussed.