Chen, Gui-Qiang; DiBenedetto, Emmanuele Stability of entropy solutions to the Cauchy problem for a class of nonlinear hyperbolic-parabolic equations. (English) Zbl 1027.35080 SIAM J. Math. Anal. 33, No. 4, 751-762 (2001). The Cauchy problem for the nonlinear equation of hyperbolic-parabolic type \[ \partial_t u + \frac{1}{2}{\mathbf{a}} \cdot \nabla_x u^2 = \Delta u_+ , \] 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. Reviewer: Kamil Sabitov (Sterlitamak) Cited in 24 Documents MSC: 35M10 PDEs of mixed type 35B35 Stability in context of PDEs 35K65 Degenerate parabolic equations Keywords:entropy solutions; uniqueness PDFBibTeX XMLCite \textit{G.-Q. Chen} and \textit{E. DiBenedetto}, SIAM J. Math. Anal. 33, No. 4, 751--762 (2001; Zbl 1027.35080) Full Text: DOI