NONLINEARLY DEGENERATE WAVE EQUATION Vtt = c(lv1s-1v)xx

Abstract

It is well known that the generalized solutions for the Cauchy problem (1.4)-(1.5) are also the solutions of the nonlinearly degenerate wave equation Vtt = c(lvls-1v)xx with the initial data v0(x). In this paper, we first study the strong and weak entropies of system (1.4), then the H-1 compactness of η( vɛ, uɛ)t + q( vɛ, uɛ)x of these entropy-entropy flux pairs with respect to the viscosity solutions of the Cauchy problem (1.7)-(1.5). Finally, suppose that for fixed point (x, t), the support set of the Young measure Vx,t is concentrated on either the region v ≥ 0 or the region v ≤ 0, then vx,t must be a Dirac measure by using the theory of compensated compactness coupled with the kinetic formulation by Lions, Perthame, Souganidis and Tadmor [LPS, LPT].