Slant kink-wave solutions and spreading of the free boundary of the inhomogeneous pressureless Euler equations

In this paper, we consider the inhomogeneous pressureless Euler equations. First, we present a class of self-similar analytical solutions to the 1D Cauchy problem and investigate the large-time behavior of the solutions, and particularly, we obtain slant kink-wave solutions for the inhomogeneous Burgers (InhB) type equation. Next, we prove the integrability of the InhB equation in the sense of Lax pair. Furthermore, we study the spreading rate of the moving domain occupied by mass for the 1D Cauchy problem with compact support initial density. We find that the expanding domain grows exponentially in time, provided that the solutions exist and smooth at all time. Finally, we extend the corresponding results of the inhomogeneous pressureless Euler equations to the radially symmetric multi-dimensional case. Copyright © 2025 Editorial Committee of Applied Mathematics.