Particular solutions to one-dimensional Cauchy problems for scalar parabolic–hyperbolic conservation laws and their applications

In this paper, we introduce traveling waves with multiple discontinuities to one-dimensional Cauchy problems (CP) for scalar parabolic–hyperbolic conservation laws. Since the equation has nonlinear convective term and degenerate diffusion term, it has both properties of hyperbolic equations and those of parabolic equations. Therefore, it is difficult to investigate the behavior of solutions to (CP). One way to overcome difficulties is 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. In fact, we succeeded in constructing shock wave type traveling waves with at most 1-discontinuity under some restriction of the diffusion term. In the present paper, we remove the restriction of the diffusion term and construct shock wave type traveling waves with multiple discontinuities to (CP) and investigate their properties. Next, we discuss the stability of the constructed traveling waves. Thirdly, we construct rarefaction wave type sub-, super-solutions to (CP) and prove the stability of them. Finally, we estimate the propagation speed of support for entropy solutions to (CP) using the constructed functions.