We present counter-intuitive examples of a viscous regularizations of a two-dimensional strictly hyperbolic system of conservation laws. The regularizations are obtained using two different viscosity matrices. While for both of the constructed ``viscous'' systems waves propagating in either $x$- or $y$-directions are stable, oblique waves may be linearly unstable. Numerical simulations fully corroborate these analytical results. To the best of our knowledge, this is the first nontrivial result related to the multidimensional Gelfand problem. Our conjectures provide direct answer to Gelfand's problem both in one- and multi-dimensional cases.

翻译：我们给出了二维严格双曲守恒律系统粘性正则化的反直觉例子。该正则化通过两种不同的粘性矩阵实现。尽管对于所构造的两个“粘性”系统，沿$x$或$y$方向传播的波是稳定的，但斜向波可能呈现线性不稳定性。数值模拟完全证实了这些分析结果。据我们所知，这是与多维Gelfand问题相关的首个非平凡结论。我们提出的猜想为一维及多维情形下的Gelfand问题提供了直接解答。