Existence of a global solution to a scalar conservation law with a source term for large data

We are concerned with a scalar conservation law with a source term. This equation is proposed to describe the qualitative behavior of waves for a general system in resonance with the source term by T.P. Liu. The goal in the present paper is to provide a condition that the Cauchy problem has a global entropy solution for large data. The key point is to obtain the bounded estimate of solutions. To deduce this, we introduce some functions of x as the lower and upper bounds. Therefore, our bounded estimate depends on the space variable. Moreover, we use the vanishing viscosity method to construct approximate solutions and derive the convergence by the compensated compactness.