论文标题
用比例不变的阻尼和联合非线性对波方程的爆破进行改进
Improvement on the blow-up of the wave equation with the scale-invariant damping and combined nonlinearities
论文作者
论文摘要
我们在本文中考虑了抑制波方程，在\ textIt {scale-casiniant案例}中具有两个合并的非线性，如下所示：\ begin {displaymath} \ d（e）\ hspace \ hspace {1cm} {1cm} u__ {tt} u { \ quad \ mbox {in} \ \ \ r^n \ times [0，\ infty），\ end {display-math}带有少量数据。被删除，第二个可以从$（0，μ_*/2）$延长至$（0，μ_*）$，其中$μ_*> 0 $是$（q-1）\ left的解决方案\ left（（（n+μ_* - 1）p-2 \ 1）p-2 \ right）= 4 $。的确，由于对阻尼术语在解决方案的全球动态中的影响有更好的理解，我们认为$μ$的新间隔更好地描述了爆破和全球存在区域之间的阈值。此外，利用问题$（e）$中所用的技术，我们还与Glassey猜想的$（e）$相关的\ cite {lt2，palmieri}的结果改进了结果，但没有非线性项$ | u | u |^q $。更确切地说，我们将爆炸区域从$ p \ in（1，p_g（n+σ）] $扩展，其中$σ$由下面的\ eqref {sigma}给出，到$ p \ in（1，p_g（n+μ）]，因此在这种情况下可以更好地估计生命的估计。
We consider in this article the damped wave equation, in the \textit{scale-invariant case} with combined two nonlinearities, which reads as follows: \begin{displaymath} \d (E) \hspace{1cm} u_{tt}-Δu+\fracμ{1+t}u_t=|u_t|^p+|u|^q, \quad \mbox{in}\ \R^N\times[0,\infty), \end{displaymath} with small initial data.\\ Compared to our previous work \cite{Our}, we show in this article that the first hypothesis on the damping coefficient $μ$, namely $μ< \frac{N(q-1)}{2}$, can be removed, and the second one can be extended from $(0, μ_*/2)$ to $(0, μ_*)$ where $μ_*>0$ is solution of $(q-1)\left((N+μ_*-1)p-2\right) = 4$. Indeed, owing to a better understanding of the influence of the damping term in the global dynamics of the solution, we think that this new interval for $μ$ describe better the threshold between the blow-up and the global existence regions. Moreover, taking advantage of the techniques employed in the problem $(E)$, we also improve the result in \cite{LT2,Palmieri} in relationship with the Glassey conjecture for the solution of $(E)$ but without the nonlinear term $|u|^q$. More precisely, we extend the blow-up region from $p \in (1, p_G(N+σ)]$, where $σ$ is given by \eqref{sigma} below, to $p \in (1, p_G(N+μ)]$ giving thus a better estimate of the lifespan in this case.
