Liouville theorems for scaling invariant nonlinear parabolic problems in the whole space and/or the halfspace (saying that the problem does not posses positive bounded solutions defined for all times \(t\in (-\infty ,\infty )\)) guarantee optimal estimates of solutions of related initial-boundary value problems in general domains. We prove an optimal Liouville theorem for the linear equation in the halfspace complemented by the nonlinear boundary condition \(\partial u/\partial \nu =u^q\), \(q>1\).

中文翻译：

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具有非线性边界条件的线性热方程的最优Liouville定理

用于在整个空间和/或半空间中缩放不变非线性抛物线问题的Liouville定理（假设问题不具有在所有时间都定义的正有界解\（t \ in（-\ infty，\ infty）\）），可以保证最优一般领域中相关初边值问题的解的估计。我们证明了半空间中线性方程的最佳Liouville定理，该方程由非线性边界条件\（\ partial u / \ partial \ nu = u ^ q \），\（q> 1 \）补充。