Traveling waves to one-dimensional Cauchy problems for scalar parabolic-hyperbolic conservation laws

Abstract

In this paper, we introduce traveling waves to one-dimensional Cauchy problems (CP) for scalar parabolic-hyperbolic conservation laws. The equation is regarded as a linear combination of the scalar hyperbolic conservation laws and the porous medium type equations. Thus, this equation has both properties of hyperbolic equations and those of parabolic equations. Accordingly, it is difficult to investigate the behavior of solutions to (CP). Therefore, it is necessary to construct particular solutions and investigate their properties. In pure hyperbolic case, Riemann solutions are well studied because they are self-similar solutions. However, it cannot be expected in this paper. Hence, we focus on the traveling wave structure instead of the self-similar structure.At first, we construct traveling waves to (CP) and investigate their properties. Next, we discuss the asymptotic behavior of entropy solutions to (CP) using the constructed traveling waves. Finally, we estimate the propagation speed of support for entropy solutions to (CP) using the modified traveling waves.