具有空间非齐次强吸收的非线性扩散方程的支撑自相似收缩性和非灭绝性
Self-similar shrinking of supports and non-extinction for a nonlinear diffusion equation with spatially inhomogeneous strong absorption
摘要：
本文研究了对$(t, x) \in (0,\infty) \times \mathbb{R}^N$提出的具有强吸收的多孔介质方程$$\partial_t u=Δu^m-|x|^σu^q,$$的动力学性质，用$m > 1$，$q \in (0, 1)$和$σ> 2(1-q)/(m-1)$ .考虑具有非负初始条件的Cauchy问题，建立了解在任意$u_0 \in L^\infty(\mathbb{R}^N)$处的瞬时收缩和局部化。利用这一性质，证明了具有代数衰减的非负紧支撑径向对称正向自相似解的存在唯一性。最后，它表明，有限时间灭绝不会发生广泛的一类初始条件，这种独特的自相似解决方案是这些一般解决方案的大时间行为的模式。
英文摘要：
We study the dynamics of the following porous medium equation with strong absorption $$\partial_t u=Δu^m-|x|^σu^q,$$ posed for $(t, x) \in (0,\infty) \times \mathbb{R}^N$, with $m > 1$, $q \in (0, 1)$ and $σ> 2(1-q)/(m-1)$. Considering the Cauchy problem with non-negative initial condition $u_0 \in L^\infty(\mathbb{R}^N)$ instantaneous shrinking and localization of supports for the solution $u(t)$ at any $t > 0$ are established. With the help of this property, existence and uniqueness of a nonnegative compactly supported and radially symmetric forward self-similar solution with algebraic decay in time are proven. Finally, it is shown that finite time extinction does not occur for a wide class of initial conditions and this unique self-similar solution is the pattern for large time behavior of these general solutions.
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