Singularity and decay estimates in superlinear problems via liouville-type theorems. Part II: Parabolic equations
Singularity and decay estimates in superlinear problems via liouville-type theorems. Part II: Parabolic equations
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DOI：
10.1512/iumj.2007.56.2911
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发表时间：
2007-06
影响因子：
1.1
通讯作者：
P. Polácik;P. Quittner;P. Souplet
中科院分区：
文献类型：
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作者：
P. Polácik;P. Quittner;P. Souplet
本文研究了抛物型Liouvile型定理与超线性抛物型问题非负经典解的局部和整体性质之间的一些新的联系，无论有无边界条件。也就是说，我们发展了一种由Liouvile型定理导出解的普适的逐点先验估计的一般方法，它统一和改进了许多关于先验界、衰减估计以及初始和最终爆破率的结果。例如，对于区域Ω上的方程ut-≤u=up，我们得到了形式为u(x，t)ΩC(Ω，p)(1+t-1/(p-1)+(T-t)-1/(p-1))的初值和终值爆破率估计。我们的方法是基于重定标量结合一个关键的“加倍”性质，并利用抛物型Liouvile型定理来简化整个空间或半空间。作为普适估计的应用，我们证明了一个初边值问题的非唯一性结果。
In this paper, we study some new connections between parabolic Liouville-type theorems and local and global properties of nonnegative classical solutions to superlinear parabolic problems, with or without boundary conditions. Namely, we develop a general method for derivation of universal, pointwise a priori estimates of solutions from Liouville-type theorems, which unifies and improves many results concerning a priori bounds, decay estimates and initial and final blow-up rates. For example, for the equation u t - Δu = u p on a domain Ω, possibly unbounded and not necessarily convex, we obtain initial and final blow-up rate estimates of the form u(x, t) ≤ C(Ω, p)(1 + t -1/(p-1) + (T - t) -1/(p-1) ). Our method is based on rescaling arguments combined with a key "doubling" property, and it is facilitated by parabolic Liouville-type theorems for the whole space or the half-space. As an application of our universal estimates, we prove a nonuniqueness result for an initial boundary value problem.