Asymptotic variational wave equations (Q1006794)

The initial boundary value problem for the equation \((u_{t}+f(u)_{x})_{x}=\frac{1}{2}f''(u)u_{x}^2\) on \((0,T)\times \mathbb{R}^{+}\) is investigated. The authors prove that conservative solutions can be globally defined. Under non-degeneracy condition the conservative solution is unique. If flux \(f(u)\) is convex, then there exists a continuous semigroup of dissipative solutions. If \(f(u)\) is not convex, the solutions do not depend continuously on the initial data, in general.