论文标题
通用估计和liouville定理，无需扩展不变性
Universal estimates and Liouville theorems for superlinear problems without scale invariance
论文作者
论文摘要
我们对非线性椭圆和抛物线问题进行重新续订方法，并表明，通过适当的修改，它们可以用于即使是渐近的，即使是渐进的，其行为也可以远离能力。 在这个扩大的框架中，通过调整[37，38]的加倍响应方法，我们表明在通用估计和liouville定理之间发现的等价性仍然有效。在抛物线案例中，我们还证明了一个liouville型定理，用于一系列相当大的非规模不变非线性。这导致了许多新的结果，即有关空间或时空奇异性估计，初始和最终爆炸率，通用和先验范围的非尺度椭圆形和抛物线问题，全球解决方案以及空间和/或时间的衰减率。 我们通过许多示例来说明我们的方法，这反过来又表明了估计值和假设的最佳性。
We revisit rescaling methods for nonlinear elliptic and parabolic problems and show that, by suitable modifications, they may be used for nonlinearities that are not scale invariant even asymptotically and whose behavior can be quite far from power like. In this enlarged framework, by adapting the doubling-rescaling method from [37, 38], we show that the equivalence found there between universal estimates and Liouville theorems remains valid. In the parabolic case we also prove a Liouville type theorem for a rather large class of non scale invariant nonlinearities. This leads to a number of new results for non scale invariant elliptic and parabolic problems, concerning space or space-time singularity estimates, initial and final blow-up rates, universal and a priori bounds for global solutions, and decay rates in space and/or time. We illustrate our approach by a number of examples, which in turn give indication about the optimality of the estimates and of the assumptions.
