论文标题
具有比例不变的阻尼和联合非线性的波动方程的爆炸
Blow-up for wave equation with the scale-invariant damping and combined nonlinearities
论文作者
论文摘要
在本文中，我们研究了\ textIt {scale-casiniant案例}中阻尼波方程的爆炸，并在存在两个非线性的情况下进行了爆炸。更准确地说，我们考虑以下等式：$$ u_ {tt}-ΔU+\ \ \ \ \ \ \ \ \fracμ{1+t} u_t = | u_t | u_t |^p+| u | u |^q，\ quad \ quad \ quad \ quad \ mbox {in} \ \ \ \ \ \ \ r^n \ times [0，\ iffty），$初始数据。 \ frac {n（q-1）} {2} $和$μ\ in（0，μ_*）$，其中$μ_*> 0 $是取决于非lineArties的功率和空间维度（$ _*$ _*$ _*$ _*$（Q-1）\（q-1）行为就像没有耗散的那样（$μ= 0 $）。我们的结果完成了以前的研究，在$ \fracμ{（1+t）^β} u_t给出耗散的情况下； \β> 1 $（\ cite {lt3}），与我们在当前工作中获得的相反，阻尼的效果在动力学中并不重要。有趣的是，在我们的情况下，阻尼项$ \fracμ{1+t} u_t $的影响很重要。
In this article, we study the blow-up of the damped wave equation in the \textit{scale-invariant case} and in the presence of two nonlinearities. More precisely, we consider the following equation: $$u_{tt}-Δu+\fracμ{1+t}u_t=|u_t|^p+|u|^q, \quad \mbox{in}\ \R^N\times[0,\infty), $$ with small initial data.\\ For $μ< \frac{N(q-1)}{2}$ and $μ\in (0, μ_*)$, where $μ_*>0$ is depending on the nonlinearties' powers and the space dimension ($μ_*$ satisfies $(q-1)\left((N+2μ_*-1)p-2\right) = 4$), we prove that the wave equation, in this case, behaves like the one without dissipation ($μ=0$). Our result completes the previous studies in the case where the dissipation is given by $\fracμ{(1+t)^β}u_t; \ β>1$ (\cite{LT3}), where, contrary to what we obtain in the present work, the effect of the damping is not significant in the dynamics. Interestingly, in our case, the influence of the damping term $\fracμ{1+t}u_t$ is important.
