论文标题
从富士馆到一类半线性演化方程的Kato Type指数，并与时间相关的阻尼
The move from Fujita to Kato type exponent for a class of semilinear evolution equations with time-dependent damping
论文作者
论文摘要
在本文中，我们得出合适的最佳$ l^p-l^q $衰减估计，$ 1 \ leq p \ leq 2 \ leq q \ leq q \ leq \ leq \ infty $，用于$σ$ - 循环方程的解决方案，$σ> 1 $，带有比例 - invariant时间相关的damping and domented damping and lineleareity和lineleareity unlinearity〜 $ $ | u | u_ {tt}+（ - δ）^σu+\fracμ{1+t} u_t = | u |^{p}，\]，其中$μ> 0 $，$ p> 1 $。全局（及时）存在针对库奇问题的小数据解决方案的关键指数$ p = p_c $与解决方案的长时间行为有关，解决方案的长期行为相应地改变了（0，1）$> 1 $。在$ l^1 \ cap l^2 $中的较小初始数据的假设下，我们找到了关键指数\ [p_c = 1+\ max \ max \ left \ {\ frac {\ frac {2σ} {[n-σ+σμ] _+} \ frac {2σ} {[[n-σ+σμ] _+}，\quadμ\ in（0，1）\\ 1+\ frac {2σ} {n}，\quadμ> 1。 \ end {case} \] 对于$μ> 1 $，它众所周知是富士式类型的指数，而对于$μ\ in（0，1）$，一个人可以将其读为KATO指数的转换。
In this paper, we derive suitable optimal $L^p-L^q$ decay estimates, $1\leq p\leq 2\leq q\leq \infty$, for the solutions to the $σ$-evolution equation, $σ>1$, with scale-invariant time-dependent damping and power nonlinearity~$|u|^p$, \[ u_{tt}+(-Δ)^σu + \fracμ{1+t} u_t= |u|^{p}, \] where $μ>0$, $p>1$. The critical exponent $p=p_c$ for the global (in time) existence of small data solutions to the Cauchy problem is related to the long time behavior of solutions, which changes accordingly $μ\in (0, 1)$ or $μ>1$. Under the assumption of small initial data in $L^1\cap L^2$, we find the critical exponent \[ p_c=1+ \max \left\{\frac{2σ}{[n-σ+σμ]_+}, \frac{2σ}{n} \right\} =\begin{cases} 1+ \frac{2σ}{[n-σ+σμ]_+}, \quad μ\in (0, 1)\\ 1+ \frac{2σ}{n}, \quad μ>1. \end{cases} \] For $μ>1$ it is well known as Fujita type exponent, whereas for $μ\in (0, 1)$ one can read it as a shift of Kato exponent.
