Convergence of a linearly regularized nonlinear wave equation to the p-system

We consider a second-order nonlinear wave equation with linear convolution term. When the operator is taken as identity operator, our reduces to classical elasticity which can be written $p$-system of first-order differential equations. first establish local well-posedness Cauchy problem. then investigate behavior solutions problem in limit kernel function integral approaches Dirac delta function, that is, vanishing dispersion limit. two different types behaviors for depending on form function. In both cases, we show converge strongly corresponding equation.