论文标题

通过Kato型引理的半连接阻尼波动方程的溶液的热和波状的寿命估计

Heat-like and wave-like lifespan estimates for solutions of semilinear damped wave equations via a Kato's type lemma

论文作者

Lai, Ning-An, Schiavone, Nico Michele, Takamura, Hiroyuki

论文摘要

在本文中,我们研究了几个具有“亚临界”非线性的半线性阻尼波方程,重点是展示能量溶液的寿命估计。我们的主要关注点是与规模不变的阻尼和质量的方程式。在对初始数据施加的不同假设下,清楚地显示了上面的寿命估计值。关键事实是我们找到了“过渡表面”,该表面将寿命估计区域估计在“波浪状”和“热”行为之间。此外,我们猜测,对“过渡表面”的寿命估计可以在对数上得到改善。作为直接后果,我们重组了“过渡表面”退化为“过渡曲线”的无质量情况的爆炸结果和寿命估计。此外,与已知结果相比,我们在一个空间维度中获得了改进的寿命估计值。我们还研究了半连接波方程的散射阻尼和负质量项,并发现,如果质量项的衰减速率等于2,则寿命估计值与一个方程的特殊情况相同,具有尺度不变的阻尼和正质量。证明的主要策略是由Kato的类型引理组成的整体形式,该形式是由迭代论点建立的。

In this paper we study several semilinear damped wave equations with "subcritical" nonlinearities, focusing on demonstrating lifespan estimates for energy solutions. Our main concern is on equations with scale-invariant damping and mass. Under different assumptions imposed on the initial data, lifespan estimates from above are clearly showed. The key fact is that we find "transition surfaces", which distinguish lifespan estimates between "wave-like" and "heat-like" behaviours. Moreover we conjecture that the lifespan estimates on the "transition surfaces" can be logarithmically improved. As direct consequences, we reorganize the blow-up results and lifespan estimates for the massless case in which the "transition surfaces" degenerate to "transition curves". Furthermore, we obtain improved lifespan estimates in one space dimension, comparing to the known results. We also study semilinear wave equations with the scattering damping and negative mass term, and find that if the decay rate of the mass term equals to 2, the lifespan estimate is the same as one special case of the equations with the scale-invariant damping and positive mass. The main strategy of the proof consists of a Kato's type lemma in integral form, which is established by iteration argument.

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