Critical exponent for the wave equation with a time-dependent scale invariant damping and a cubic convolution
Critical exponent for the wave equation with a time-dependent scale invariant damping and a cubic convolution
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DOI：
10.1016/j.jde.2020.08.047
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发表时间：
2020-03
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影响因子：
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通讯作者：
M. Ikeda;Tomoyuki Tanaka;Kyouhei Wakasa
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文献类型：
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作者：
M. Ikeda;Tomoyuki Tanaka;Kyouhei Wakasa
本文研究了具有含时尺度不变阻尼项2 1+ t t v和三次卷积的波动方程的Cauchy问题（|X| − γ <$v2）v，其中γ∈（− 12，3），初始数据（v（x，0），<$tv（x，0））∈ C2（R3）× C1（R3），有紧支集，其中v= v（x，t）是R3 ×[0，T）上的未知函数.本文的第一个目的是证明在超临界情形γ∈（0，3）下柯西问题的相关积分方程解的唯一整体（在时间上）存在性和渐近性，并给出在临界或次临界情形γ∈（− 12，0]下寿命的较低估计.他们证明的基本部分是在比没有阻尼的情况（见[22]）更弱的条件下（命题3.1和定理3.1）导出一个更弱的估计，并通过耗散项的影响恢复弱点。本文的第二个目的是证明在次临界情形γ∈（− 1 2，0）下，紧支集下正数据的小数据爆破和寿命的几乎尖锐的上估计。证明的基本部分是改进[4]中定理6.1的证明的论证，以获得寿命的上估计。我们的两个结果确定了区分小解整体存在和爆破的临界指数γ c为0，即γ c= 0.结果表明，由于尺度不变阻尼项的影响，临界指数从2变为0。
In the present paper, we study the Cauchy problem for the wave equation with a time-dependent scale invariant damping term 2 1+ t∂ t v and a cubic convolution (| x|− γ⁎ v 2) v with γ∈(− 1 2, 3) in three spatial dimension for initial data (v (x, 0),∂ t v (x, 0))∈ C 2 (R 3)× C 1 (R 3) with a compact support, where v= v (x, t) is an unknown function to the problem on R 3×[0, T). Here T denotes the maximal existence time of v. The first aim of the present paper is to prove unique global (in time) existence and asymptotic behavior of the solutions of associated integral equations to the Cauchy problem in the supercritical case γ∈(0, 3), and show a lower estimate of the lifespan in the critical or subcritical case γ∈(− 1 2, 0]. The essential part for their proofs is to derive a weaker estimate under a weaker condition (Proposition 3.1 and Theorem 3.1) than the case without damping (see [22]) and to recover the weakness by the effect of the dissipative term. The second aim of the present paper is to prove a small data blow-up and the almost sharp upper estimate of the lifespan for positive data with a compact support in the subcritical case γ∈(− 1 2, 0). The essential part for the proof is to refine the argument for the proof of Theorem 6.1 in [4] to obtain an upper estimate of the lifespan. Our two results determine that a critical exponent γ c which divides global existence and blow-up for small solutions is 0, namely γ c= 0. As the result, we can see that the critical exponent shifts from 2 to 0 due to the effect of the scale invariant damping term.