AI 中文总结
研究带燃烧反应的多孔介质方程，针对径向、非负且紧支集的初始数据。建立有界解长时间行为分类，给出二维和三维及以上的收敛情况、排除\(\theta\)作为过渡极限及自由边界传播速度估计，揭示退化燃烧动力学的维度二分法。
AI 中文摘要
我们研究带燃烧型反应的多孔介质方程\[ u_t=\Delta u^m+f(u),\qquad x\in\mathbb R^N,\ t>0, \]对于径向、非负且具有紧支集的初始数据。建立了有界解长时间行为的完整分类。在二维中，此类解局部一致收敛到常数0、\(\theta\)或1之一；对于有序初始数据集存在唯一阈值参数区分消失与扩散，临界解收敛到点火温度\(\theta\)。在三维及以上，在基态中心值集的自然完全不连通条件下，有界解局部一致收敛到0、\(\theta\)、1或径向基态\(U\in\mathcal S\)。此外，通过新的归一化环形扰动论证在三维及以上排除点火状态\(\theta\)作为过渡极限。对于过渡解，给出了自由边界传播速度的估计：二维中\[ b(t)\asymp \frac{\sqrt t}{(\log t)^{\frac{m-1}{2m}}}, \]三维及以上，当极限是基态时，\[ b(t)\asymp t^{\frac{m}{N(m-1)+2}}. \]这些结果揭示了退化燃烧动力学中尖锐的维度二分法。
英文摘要
We study the porous medium equation with a combustion-type reaction, \[ u_t=Δu^m+f(u),\qquad x\in\mathbb R^N,\ t>0, \] for radial, nonnegative, compactly supported initial data. A complete classification of the long-time behaviour of bounded solutions is established. In dimension $N=2$, every such solution converges locally uniformly to one of the constants $0$, $θ$, or $1$; for ordered families of initial data there exists a unique threshold parameter separating vanishing from spreading, and the critical solution converges to the ignition temperature $θ$. In dimensions $N\ge3$, under a natural total-disconnectedness condition on the set of central values of ground states, every bounded solution converges locally uniformly to either $0$, $θ$, $1$, or a radial ground state $U\in\mathcal S$. Moreover, the ignition state $θ$ is excluded as a transition limit in $N\ge3$ via a novel normalized annular perturbation argument. For transition solutions, we provide estimates for the propagation speed of the free boundary: in dimension two, \[ b(t)\asymp \frac{\sqrt t}{(\log t)^{\frac{m-1}{2m}}}, \] and in dimensions $N\ge3$, whenever the limit is a ground state, \[ b(t)\asymp t^{\frac{m}{N(m-1)+2}}. \] These results reveal a sharp dimensional dichotomy in the degenerate combustion dynamics.
