Riemann problem for rate-type materials with non-constant initial conditions

In this paper, using the compatible theory of differential invariants, a class of exact solutions is obtained for nonhomogeneous quasilinear hyperbolic system of partial differential equations (PDEs) describing rate type materials; these solutions exhibit genuine nonlinearity that leads to the formation of discontinuities such as shocks and rarefaction waves. For certain nonconstant and smooth initial data, the solution to the Riemann problem is presented providing a complete characterisation of the solutions.

Generalized hyperbolic mean curvature flow in Minkowski space $R^{1,1}$

This paper concerns the generalized hyperbolic mean curvature flow for spacelike curves in Minkowski $R^{1,1}$. Base on the derived quasilinear hyperbolic system, we investigate the formation of singularities in the motion of these curves. In particular, under the generalized hyperbolic mean curvature flow, we prove that the motion of periodic spacelike curves with small variation on one period and small initial velocity blows up in finite time. Some blowup results have been obtained and the estimates on the life-span of the solutions are given.

Life-Span of Classical Solutions to Hyperbolic Inverse Mean Curvature Flow

In this paper, we investigate the life-span of classical solutions to hyperbolic inverse mean curvature flow. Under the condition that the curve can be expressed in the form of a graph, we derive a hyperbolic Monge–Ampère equation which can be reduced to a quasilinear hyperbolic system in terms of Riemann invariants. By the theory on the local solution for the Cauchy problem of the quasilinear hyperbolic system, we discuss life-span of classical solutions to the Cauchy problem of hyperbolic inverse mean curvature.

Mixed problem for quasilinear hyperbolic system with coefficients functionally dependent on solution

The mixed problem for quasilinear hyperbolic system with coefficients functionally dependent on the solution is studied. We assume that the coefficients are continuous nonlinear operators in the Banach space C1(ℝ) satisfying some additional assumptions. Under these assumptions we prove the uniqueness and existence of local in time C1 solution, provided that the initial data are also of class C1.