论文标题
各向异性快速扩散方程
Anisotropic Fast Diffusion Equations
论文作者
论文摘要
我们证明了在合适的快速扩散范围内的各向异性多孔介质方程的自相似基本解决方案（SSF）的存在。这种SSF溶液的每个溶液都是由其质量唯一决定的。我们还根据自相似基本解决方案的家族来获得所有有限质量解决方案的渐近行为。时间衰减速率以及解决方案的其他特性，例如定量界限，积极性和规律性。自我相似性和各向异性的结合在我们的分析中至关重要，并造成了严重的数学困难，这些难度通过新方法解决。
We prove the existence of self-similar fundamental solutions (SSF) of the anisotropic porous medium equation in the suitable fast diffusion range. Each of such SSF solutions is uniquely determined by its mass. We also obtain the asymptotic behaviour of all finite-mass solutions in terms of the family of self-similar fundamental solutions. Time decay rates are derived as well as other properties of the solutions, like quantitative boundedness, positivity and regularity. The combination of self-similarity and anisotropy is essential in our analysis and creates serious mathematical difficulties that are addressed by means of novel methods.
