Heat-like and wave-like lifespan estimates for solutions of semilinear damped wave equations via a Kato's type lemma
Heat-like and wave-like lifespan estimates for solutions of semilinear damped wave equations via a Kato's type lemma
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DOI：
10.1016/j.jde.2020.08.020
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发表时间：
2020-03
期刊：
影响因子：
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通讯作者：
Ning-An Lai;N. Schiavone;H. Takamura
中科院分区：
文献类型：
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作者：
Ning-An Lai;N. Schiavone;H. Takamura
在这项工作中，我们考虑几个半线性阻尼波方程的“亚临界”非线性，重点研究寿命估计的能量解决方案。我们主要关注的是具有标度不变阻尼和质量的方程。通过对初始数据施加不同的假设，我们证明了上面的寿命估计，区分了“波浪状”和“热状”行为。此外，我们猜想对数改善的“过渡表面”分离的两种行为的估计。作为一个直接的结果，我们重新整理了无质量情形下的blow-up结果和寿命估计，特别是改进了一维情形下的寿命估计.我们还研究了具有散射阻尼和负质量项的半线性波动方程，发现如果质量项的衰减率等于2，寿命估计与尺度不变阻尼方程的一个特殊情形下的寿命估计一致，证明中使用的主要工具是通过迭代论证建立的Kato型引理.
In this work we consider several semilinear damped wave equations with “subcritical” nonlinearities, focusing on studying lifespan estimates for energy solutions. Our main concern is on equations with scale-invariant damping and mass. By imposing different assumptions on the initial data, we prove lifespan estimates from above, distinguishing between “wave-like” and “heat-like” behaviours. Furthermore, we conjecture logarithmic improvements for the estimates on the “transition surfaces” separating the two behaviours. As a direct consequence, we reorganize the blow-up results and lifespan estimates for the massless case, and we obtain in particular improved lifespan estimates for the one dimensional case, compared to the known results.We also study semilinear wave equations with scattering damping and negative mass term, finding that if the decay rate of the mass term equals to 2, the lifespan estimate coincides with the one in a special case of scale-invariant damped equation.The main tool employed in the proof is a Kato's type lemma, established by iteration argument.