论文标题
带有局部初始数据的波方程的爆炸结果：与非线性相结合的标尺不变性阻尼和质量项
A blow-up result for the wave equation with localized initial data: the scale-invariant damping and mass term with combined nonlinearities
论文作者
论文摘要
在\ textit {scale-Invariant案例}中，我们对本文感兴趣，以研究阻尼波方程，并具有质量项和两个合并的非线性。更准确地说，我们考虑以下等式：$$（e）{1cm} u_ {tt}-Δu+\fracμ{1+t} u_t+\ frac {ν^2} {（1+t）{（1+t） \ Mathbb {r}^n \ times [0，\ infty），$$，带有少量初始数据。根据质量和阻尼系数的一些假设，分别为$ν$和$μ> 0 $，我们表明$（e）$解决方案的爆炸区域和寿命与\ cite {our2}获得的解决方案相同。 此外，在部分使用$（e）$完成的计算中，我们在Glassey猜想上以$（e）$的解决方案来增强\ cite {palmieri}的结果，并省略了非线性术语$ | u |^q $。确实，爆炸区域从$ p \ in（1，p_g（n+σ）] $延长，其中$σ$由（1.12）以下（1.12）给出，$ p \ in（1，p_g（n+μ）] $产量，因此，当$（μ-1）^2-2-2-4c con sance con con con中，lifespan的更好估计是一个更好的限制，否则con con con con con con，live con con con，我们均为两个。 {\它对没有非线性项$ | u |^q $）的$（e）$（e）$（e）$的动力学没有影响，以及我们在\ cite {our2}中做出的猜想对在这里获得的全球存在区域之间的阈值之间的阈值。
We are interested in this article in studying the damped wave equation with localized initial data, in the \textit{scale-invariant case} with mass term and two combined nonlinearities. More precisely, we consider the following equation: $$ (E) {1cm} u_{tt}-Δu+\fracμ{1+t}u_t+\frac{ν^2}{(1+t)^2}u=|u_t|^p+|u|^q, \quad \mbox{in}\ \mathbb{R}^N\times[0,\infty), $$ with small initial data. Under some assumptions on the mass and damping coefficients, $ν$ and $μ>0$, respectively, we show that blow-up region and the lifespan bound of the solution of $(E)$ remain the same as the ones obtained in \cite{Our2} in the case of a mass-free wave equation, it i.e. $(E)$ with $ν=0$. Furthermore, using in part the computations done for $(E)$, we enhance the result in \cite{Palmieri} on the Glassey conjecture for the solution of $(E)$ with omitting the nonlinear term $|u|^q$. Indeed, the blow-up region is extended from $p \in (1, p_G(N+σ)]$, where $σ$ is given by (1.12) below, to $p \in (1, p_G(N+μ)]$ yielding, hence, a better estimate of the lifespan when $(μ-1)^2-4ν^2<1$. Otherwise, the two results coincide. Finally, we may conclude that the mass term {\it has no influence} on the dynamics of $(E)$ (resp. $(E)$ without the nonlinear term $|u|^q$), and the conjecture we made in \cite{Our2} on the threshold between the blow-up and the global existence regions obtained holds true here.
