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Stability of entropy solutions to the Cauchy problem for a class of nonlinear hyperbolic-parabolic equations. (English) Zbl 1027.35080

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.

MSC:

35M10 PDEs of mixed type
35B35 Stability in context of PDEs
35K65 Degenerate parabolic equations
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