Stability of entropy solutions to the Cauchy problem for a class of nonlinear hyperbolic-parabolic equations (Q2782952)

The Cauchy problem for the nonlinear equation of hyperbolic-parabolic type NEWLINE\[NEWLINE \partial_t u + \frac{1}{2}{\mathbf{a}} \cdot \nabla_x u^2 = \Delta u_+ , NEWLINE\]NEWLINE where \(\mathbf{a}\) is a constant vector, \(u_+ = \max\{u,0\}\), in domain \( S_T=\mathbb{R}^N \times (0,T],\) \(T>0\) is considered. This equation features states of ideal fluid; in ``a viscous phase'' \([u < 0]\) it is of hyperbolic type and in ``a non-viscous'' \([u > 0]\) it is of parabolic type. Provided that the solution of the Cauchy problem is unique, the upper bound as \( x \to \infty\) for an ``entropy'' of the solution in a weighted space \(L^1(\mathbb{R}^N)\) is established.