Optimal Liouville theorems for superlinear parabolic problems
Optimal Liouville theorems for superlinear parabolic problems
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超线性抛物线问题的最优刘维尔定理
DOI：
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发表时间：
2020
期刊：
影响因子：
--
通讯作者：
P. Quittner
中科院分区：
文献类型：
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作者：
P. Quittner
缩放不变非线性抛物方程和系统的Liouville定理（即方程或系统不具有正全解）保证了相关初值和初边值问题解的最优全称估计。在非线性热方程$$u_t-Delta u=u^pquadhbox{in}quad {R}^n imes{R}, qquad p>1, $$的情况下，在亚临界范围内不存在正经典解$p(n-2)<n+2$已经推测了很长时间，但所有已知的结果都需要在$p$上有更严格的假设或处理一类特殊的解（时间无关或径向对称或满足适当的衰变条件）。我们解决了这个开放问题，并且用同样的论证证明了一类超线性抛物系统的最优Liouville定理。在非线性热方程的情况下，我们的刘维尔定理的直接应用解决了几个相关的长期问题。例如，它们保证了古解的最优Liouville定理，相应Cauchy问题全局解的最优衰减估计，非凸域解的最优爆破率估计，相应初边值问题解的最优全称估计。我们的主要结果的证明是基于对适当重新缩放的解决方案的精细能量估计。
Liouville theorems for scaling invariant nonlinear parabolic equations and systems (saying that the equation or system does not possess positive entire solutions) guarantee optimal universal estimates of solutions of related initial and initial-boundary value problems. In the case of the nonlinear heat equation $$u_t-Delta u=u^pquadhbox{in}quad {R}^n imes{R}, qquad p>1, $$ the nonexistence of positive classical solutions in the subcritical range $p(n-2)<n+2$ has been conjectured for a long time, but all known results require either a more restrictive assumption on $p$ or deal with a special class of solutions (time-independent or radially symmetric or satisfying suitable decay conditions). We solve this open problem and -- by using the same arguments -- we also prove optimal Liouville theorems for a class of superlinear parabolic systems. In the case of the nonlinear heat equation, straightforward applications of our Liouville theorem solve several related long-standing problems. For example, they guarantee an optimal Liouville theorem for ancient solutions, optimal decay estimates for global solutions of the coresponding Cauchy problem, optimal blow-up rate estimate for solutions in non-convex domains, optimal universal estimates for solutions of the corresponding initial-boundary value problems. The proof of our main result is based on refined energy estimates for suitably rescaled solutions.
影响因子：
1.7
作者：
Manuel del Pino;M. Musso;Juncheng Wei
通讯作者：
Manuel del Pino;M. Musso;Juncheng Wei
超临界半线性热方程的完整且古老的解
DOI：
10.3934/dcds.2020136
发表时间：
2019
期刊：
Discrete & Continuous Dynamical Systems - A
影响因子：
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作者：
Poláčik, Peter;Quittner, Pavol
通讯作者：
Quittner, Pavol
影响因子：
2.2
作者：
M. Pino;M. Musso;Juncheng Wei
通讯作者：
M. Pino;M. Musso;Juncheng Wei