Nonlinear Analysis ( IF 1.3 ) Pub Date : 2020-11-14 , DOI: 10.1016/j.na.2020.112187 Giulia Meglioli , Fabio Punzo
We are concerned with nonnegative solutions to the Cauchy problem for the porous medium equation with a variable density and a power-like reaction term with . The density decays fast at infinity, in the sense that as with In the case when , if is bigger than , we show that, for large enough initial data, solutions blow-up in finite time and for small initial datum, solutions globally exist. On the other hand, in the case when , we show that existence of global in time solutions always prevails. The case of slowly decaying density at infinity, i.e. , is examined in Meglioli and Punzo (2020).
中文翻译:
具有反应和快速衰减密度的多孔介质方程解的爆破和整体存在
我们关注密度可变的多孔介质方程的柯西问题的非负解 和类似幂的反应项 与 。密度在无限远处快速衰减, 如 与 在这种情况下 如果 大于 ,我们表明,对于足够大的初始数据,解决方案会在有限的时间内爆炸,而对于小的初始基准面,则存在全局解决方案。另一方面,当,我们证明了全局及时解决方案的存在总是占主导地位。在无限远处缓慢衰减密度的情况,即,在Meglioli和Punzo(2020)中进行了研究。