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 inequalities 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 \({{\mathbb {R}}}^n\).

中文翻译：

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流形上反应扩散方程类的解和平滑效应的整体存在

我们考虑在黎曼流形上具有幂次反应项的多孔介质方程。在（1.1）中关于p和m的某些假设下，对于足够小的非负初始数据，只要Sobolev不等式保持在流形上，我们证明存在全局及时解。此外，当Sobolev不等式和Poincaré不等式都成立时，在强迫项的假设较弱的情况下，也会得到类似的结果。通过相同的函数分析方法，我们研究了\（{{\ mathbb {R}}} ^ n \中具有源项和变量密度的多孔介质方程解的整体存在性。