论文标题
非线性抑制$σ$进化方程的关键指数
The critical exponent for nonlinear damped $σ$-evolution equations
论文作者
论文摘要
In this paper, we derive suitable optimal $L^p-L^q$ decay estimates, $1\leq p\leq q\leq \infty$, for the solutions to the $σ$-evolution equation, $σ>1$, with structural damping and power nonlinearity $|u|^{1+α}$ or $|u_t|^{1+α}$, \[ u_ {tt}+（ - δ）^σu+（ - δ）^θu_t= \ begin {case} | u |^{1+α}，\\ | u_t |^{1+α}，\ end end end {cases} \]，其中$ t \ geq0 $和$ x $ and $ x \ n $和$ x \ n $ x \ in $ x \ in \ in \ in \ in $使用这些估计值，我们可以解决在所谓的非有效情况下以上两个非线性问题的关键指数的问题，$θ\ in（σ/2，σ] $。后者比有效的情况$θ\在[0，σ/2）$中的有效情况要困难得多，因为该解决方案的差异尤其有差异，并且是一个差异的组成部分。本文中的新颖思想包括分别处理两个组成部分，以忽略两个组成部分相互作用所产生的衰减率的损失。我们通过将振荡出现的低频定位在扩展的相空间中进行处理。这种策略使我们能够恢复一个准缩放属性，以取代方程式缺乏同质性。
In this paper, we derive suitable optimal $L^p-L^q$ decay estimates, $1\leq p\leq q\leq \infty$, for the solutions to the $σ$-evolution equation, $σ>1$, with structural damping and power nonlinearity $|u|^{1+α}$ or $|u_t|^{1+α}$, \[ u_{tt}+(-Δ)^σu +(-Δ)^θu_t=\begin{cases} |u|^{1+α}, \\ |u_t|^{1+α}, \end{cases}\] where $t\geq0$ and $x\in\mathbb{R}^n$. Using these estimates, we can solve the problem of finding the critical exponents for the two nonlinear problems above in the so-called non-effective case, $θ\in(σ/2,σ]$. This latter is more difficult than the effective case $θ\in[0,σ/2)$, since the asymptotic profile of the solution involves a diffusive component and an oscillating one. The novel idea in this paper consists in treating separately the two components to neglect the loss of decay rate created by the interplay of the two components. We deal with the oscillating component, by localizing the low frequencies, where oscillations appear, in the extended phase space. This strategy allows us to recover a quasi-scaling property which replaces the lack of homogeneity of the equation.
