We consider reaction-diffusion equations either posed on Riemannian manifolds or in the Euclidean weighted setting, with power-type nonlinearity and slow diffusion of porous medium type. We consider the particularly delicate case $p<m$ in problem (1.1), a case presently largely open even when the initial datum is smooth and compactly supported. We prove global existence for L^m data, and that solutions corresponding to such data are bounded at all positive times with a quantitative bound on their L^∞ norm. We also show that on Cartan-Hadamard manifolds with curvature pinched between two strictly negative constants, solutions corresponding to sufficiently large L^m data give rise to solutions that blow up pointwise everywhere in infinite time, a fact that has no Euclidean analogue. The methods of proof are functional analytic in character, as they depend solely on the validity of the Sobolev and of the Poincaré inequalities. As such, they are applicable to different situations, among which we single out the case of (mass) weighted reaction-diffusion equation in the Euclidean setting. In this latter setting we also consider, with stronger results for large times, the case of globally integrable weights.

我们考虑在黎曼流形上或在欧几里得加权设置中提出的反应扩散方程，具有幂型非线性和多孔介质类型的缓慢扩散。我们考虑特别微妙的情况$磷<米$在问题 (1.1) 中，即使初始数据是光滑且紧致支撑的情况下，目前也有很大程度打开的情况。我们证明了 L ^m数据的全局存在性，并且对应于这些数据的解在所有正时间都有界，其 ​​L ^∞范数有定量界限。我们还表明，在曲率夹在两个严格负常数之间的 Cartan-Hadamard 流形上，解对应于足够大的 L ^m数据产生的解在无限时间内随处爆炸，这是没有欧几里得类似物的事实。证明方法在性质上是函数分析的，因为它们完全依赖于 Sobolev 和 Poincaré 不等式的有效性。因此，它们适用于不同的情况，其中我们在欧几里德设置中挑选了（质量）加权反应扩散方程的情况。在后一种设置中，我们还考虑了全局可积分权重的情况，在长时间内获得更强的结果。