We prove existence of the largest and the smallest entropy solutions to the Cauchy problem for a nonlinear degenerate anisotropic parabolic equation. Applying this result, we establish the comparison principle in the case when at least one of the initial functions is periodic. In the case when initial function vanishes at infinity (in the sense of strong average) we prove the long time decay of an entropy solution under exact nonlinearity-diffusivity condition.

中文翻译：

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关于退化非线性各向异性抛物线方程熵解的一些性质

我们证明了非线性退化各向异性抛物线方程柯西问题的最大和最小熵解的存在性。应用这一结果，我们建立了至少一个初始函数是周期性的情况下的比较原理。在初始函数在无穷远处消失的情况下（在强平均的意义上），我们证明了在精确的非线性扩散条件下熵解的长时间衰减。