Radially symmetric weak solutions for a quasilinear wave equation in two space dimensions

We prove the convergence of the radially symmetric solutions to the Cauchy problem for the viscoelasticity equationsφtt-Δφ-div(13|∇φ|2∇φ)=ɛΔφt,as ɛ→0, with radially symmetric initial data φɛ(x,0)=φ0ɛ(r), φtɛ(x,0)=φ1ɛ(r), r=(x12+x22)1/2, where φ0rɛ⇀φ0r, φ1ɛ⇀φ1, to a weak solution of the Cauchy problem for the corresponding limit equation with ɛ=0, and initial data φ(x,0)=φ0(r), φt(x,0)=φ1(r). Our analysis is based on energy estimates and the method of compensated compactness closely following Serre and Shearer (Convergence with physical viscosity for nonlinear elasticity, 1993, unpublished).