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

TL;DR

This paper establishes optimal decay estimates and identifies the critical exponent for global existence of solutions to a class of semilinear evolution equations with time-dependent damping, revealing a transition from Fujita to Kato type exponents.

Contribution

It derives the optimal $L^p-L^q$ decay estimates and determines the critical exponent for global solutions, highlighting the shift from Fujita to Kato type exponents based on damping parameter.

Findings

Derived optimal decay estimates for solutions.

Identified the critical exponent $p_c$ depending on damping parameter.

Revealed the transition from Fujita to Kato type exponents.

Abstract

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 $\sigma$-evolution equation, $\sigma>1$, with scale-invariant time-dependent damping and power nonlinearity~$|u|^p$, \[ u_{tt}+(-\Delta)^\sigma u + \frac{\mu}{1+t} u_t= |u|^{p}, \] where $\mu>0$, $p>1$. 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 $\mu \in (0, 1)$ or $\mu>1$. Under the assumption of small initial data in $L^1\cap L^2$, we find the critical exponent \[ p_c=1+ \max \left\{\frac{2\sigma}{[n-\sigma+\sigma\mu]_+}, \frac{2\sigma}{n} \right\} =\begin{cases} 1+ \frac{2\sigma}{[n-\sigma+\sigma\mu]_+}, \quad \mu \in (0, 1)\\ 1+ \frac{2\sigma}{n}, \quad \mu>1. \end{cases} \] For $\mu>1$ it is well known as Fujita type exponent, whereas for $\mu \in (0, 1)$ one can read it as a shift of Kato exponent.