Abstract We study existence of global solutions and finite time blow-up of solutions to the Cauchy problem for the porous medium equation with a variable density ρ ( x ) and a power-like reaction term ρ ( x ) u p with p > 1 ; this is a mathematical model of a thermal evolution of a heated plasma (see [29] ). The density decays slowly at infinity, in the sense that ρ ( x ) ≲ | x | − q as | x | → + ∞ with q ∈ [ 0 , 2 ) . We show that for large enough initial data, solutions blow-up in finite time for any p > 1 . On the other hand, if the initial datum is small enough and p > p ¯ , for a suitable p ¯ depending on ρ , m , N , then global solutions exist. In addition, if p p _ , for a suitable p _ ≤ p ¯ depending on ρ , m , N , then the solution blows-up in finite time for any nontrivial initial datum; we need the extra hypothesis that q ∈ [ 0 , ϵ ) for ϵ > 0 small enough, when m ≤ p p _ . Observe that p _ = p ‾ , if ρ ( x ) is a multiple of | x | − q for | x | large enough. Such results are in agreement with those established in [48] , where ρ ( x ) ≡ 1 , and are related to some results in [32] , [33] . The case of fast decaying density at infinity, i.e. q ≥ 2 , is examined in [36] .

中文翻译：

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具有反应和缓慢衰减密度的多孔介质方程解的爆破和全局存在性

摘要 我们研究了具有变密度 ρ ( x ) 和类幂反应项 ρ ( x ) 且 p > 1 的多孔介质方程的柯西问题的全局解的存在性和有限时间解的膨胀；这是加热等离子体热演化的数学模型（参见 [29]）。密度在无穷远处缓慢衰减，即 ρ ( x ) ≲ | × | − q 为 | × | → + ∞ 与 q ∈ [ 0 , 2 ) 。我们表明，对于足够大的初始数据，任何 p > 1 的解决方案都会在有限时间内爆炸。另一方面，如果初始数据足够小并且 p > p¯ ，对于取决于 ρ , m , N 的合适 p ¯ ，则存在全局解。此外，如果 pp _ ，对于一个合适的 p _ ≤ p ¯ 取决于 ρ , m , N ，那么对于任何非平凡的初始数据，解在有限时间内爆炸；当 m ≤ pp _ 时，我们需要额外的假设 q ∈ [ 0 , ϵ ) for ϵ > 0 足够小。观察 p _ = p ‾ ，如果 ρ ( x ) 是 | 的倍数 × | − q 为 | × | 足够大。这些结果与 [48] 中建立的结果一致，其中 ρ ( x ) ≡ 1 ，并且与 [32] 、 [33] 中的一些结果有关。在 [36] 中检查了无穷远处快速衰减密度的情况，即 q ≥ 2。