论文标题
全球存在解决方案和平滑效应的反应扩散方程类别对流形的效果
Global existence of solutions and smoothing effects for classes of reaction-diffusion equations on manifolds
论文作者
论文摘要
我们考虑带有能量样反应项的多孔培养基方程，该方程构成了riemannian歧管上。根据$ p $和$ m $（1.1）的某些假设，对于足够小的非负初始数据，我们证明了全球时间解决方案的存在，前提是Sobolev不平等存在于歧管上。此外，当Sobolev和Poincaré不平等都保持时，相似的结果在强迫术语中的假设较弱下。通过相同的功能分析方法，我们研究了$ {\ Mathbb r}^n $中源项和可变密度的多孔介质方程解决方案的全局存在。
We consider the porous medium equation with a power-like reaction term, posed on Riemannian manifolds. Under certain assumptions on $p$ and $m$ in (1.1), and for small enough nonnegative initial data, we prove existence of global in time solutions, provided that the Sobolev inequality holds on the manifold. Furthermore, when both the Sobolev and the Poincaré inequality hold, similar results hold under weaker assumptions on the forcing term. By the same functional analytic methods, we investigate global existence for solutions to the porous medium equation with source term and variable density in ${\mathbb R}^n$.
