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The move from Fujita to Kato type exponent for a class of semilinear evolution equations with time-dependent damping

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.