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 V_tt = c(lvl^s-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( v^ɛ, u^ɛ)t + q( v^ɛ, u^ɛ)_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 V_x,t is concentrated on either the region v ≥ 0 or the region v ≤ 0, then v_x,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].