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

Abstract

In this paper, we derive suitable optimal $L^p-L^q$ decay estimates, $1\le p\le 2\le q\le \infty$, for the solutions to the $\sigma$-evolution equation, $\sigma >1$, with scale-invariant time-dependent damping and power nonlinearity $|u|^p$,

$$\begin{aligned} u_{tt}+(-\Delta )^\sigma u + \frac{\mu }{1+t} u_t= |u|^{p}, \quad t\ge 0, \quad x\in {{\mathbb {R}}}^n, \end{aligned}$$

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 $L^m({{\mathbb {R}}}^n)\cap L^2({{\mathbb {R}}}^n), m=1,2$, we find the critical exponent at low space dimension n with respect to $\sigma$, namely,

$$\begin{aligned} p_c= \max \left\{ {{\bar{p}}}(\gamma _{m}), {{\bar{p}}} (\gamma _{m}+\mu -1) \right\} , \quad \gamma _{m}{\mathrm {\,:=\,}}\frac{n}{m\sigma }, \quad \mu >1-\gamma _m, \end{aligned}$$

where ${{\bar{p}}}(\gamma ){\mathrm {\,:=\,}}1+ \frac{2}{\gamma }$ is the well known Fujita exponent. Hence, $p_c={{\bar{p}}}(\gamma _{m})$ if $\mu >1$, whereas $p_c={{\bar{p}}} (\gamma _{m}+\mu -1)$ is a shift of Fujita type exponent if $\mu \in (0, 1)$.

Appendix

In this section we include notations and properties of special functions used throughout the paper.

Notation 1

By $[x]_+$ we denote the non-negative part of $x\in {{\mathbb {R}}}$, i.e. $[x]_+=\max \{x,0\}$.

Notation 2

We write $f\lesssim g$ if there exists a constant $C>0$ such that $f\le C g$, and $f\approx g$ if $g\lesssim f \lesssim g$.

Notation 3

We denote by ${{\hat{f}}}={\mathfrak {F}} f$ or ${{\hat{f}}}(t,\cdot )={\mathfrak {F}} f(t,\cdot )$ the partial Fourier transform, with respect to the space variable x, of a tempered distribution $\mathcal {S^{\prime }}({\mathbb {R}}^n)$ or of a function, in the appropriate distributional or functional sense and its inverse transform by ${\mathfrak {F}}^{-1}$.

Notation 4

By $L^p=L^p({{\mathbb {R}}}^n)$, $p\in [1,\infty ]$, we denote the space of measurable functions f such that $|f|^p$ has finite integral over ${{\mathbb {R}}}^n$, if $p\in [1,\infty )$, or has finite essential supremum over ${{\mathbb {R}}}^n$ if $p=\infty$. We denote by $W^{m,p}$, $m\in {{\mathbb {N}}}$, the space of $L^p$ functions with weak derivatives up to the m-th order in $L^p$. We denote by ${H}^s({{\mathbb {R}}}^n)$ and ${\dot{H}}^s({{\mathbb {R}}}^n)$, $s\ge 0$, the spaces of tempered distributions $\mathcal {S^{\prime }}({\mathbb {R}}^n)$ with $(1+{|\xi |}^2)^{\frac{s}{2}}\,{{\hat{u}}} \in L^2$ and ${|\xi |}^{s}\,{{\hat{u}}} \in L^2$, respectively.

In [1] one can find the following properties for Bessel and Hankel functions:

Lemma 4.1

The function

$$\begin{aligned} \Gamma _{\gamma }(\tau )=\tau ^{-\gamma }J_{\gamma }(\tau ),\end{aligned}$$

where $J_{\gamma }(\tau )$ is the Bessel function, is entire in $\gamma$ and $\tau$, in particular,

$$\begin{aligned} |J_{\gamma }(\tau )| \lesssim \tau ^{\gamma }, \quad 0<\tau <1. \end{aligned}$$

(4.6)

The Weber’s function $Y_{\gamma }(\tau )$ satisfies for every integer n

$$\begin{aligned} Y_{n}(\tau )= \frac{2}{\pi }J_{n}(\tau )\ln \tau + A_n(\tau ),\end{aligned}$$

where $\tau ^{n}A_{n}(\tau )$ is entire, non-null for $\tau =0$ and

$$\begin{aligned} |A_n(\tau )| \lesssim \tau ^{-n}, \quad 0<\tau <1. \end{aligned}$$

(4.7)

The Hankel functions $H^{\pm }_{\gamma }=J_{\gamma }\pm iY_{\gamma }$ satisfy

$$\begin{aligned}2(H^{\pm }_{\gamma })'(\tau )=H^{\pm }_{\gamma -1}(\tau )- H^{\pm }_{\gamma +1}(\tau ), \quad and \quad \tau (H^{\pm }_{\gamma })'(\tau )=\tau H^{\pm }_{\gamma -1}(\tau )-\gamma H^{\pm }_{\gamma }(\tau ).\end{aligned}$$

Moreover, $H^{\pm }_{\gamma }(\tau ), \tau \ge K$ can be written as

$$\begin{aligned} H^{\pm }_{\gamma }(\tau )=e^{\pm i\tau } a_{\gamma }^{\pm }(\tau ), \end{aligned}$$

(4.8)

where $a_{\gamma }^{\pm }(\tau ) \in S^{-\frac{1}{2}}(K, \infty )$ is a classical symbol of order $-\frac{1}{2}$.

For small arguments $0<\tau \le K <1$ we have

$$\begin{aligned} |H^{\pm }_{\gamma }(\tau )|\lesssim {\left\{ \begin{array}{ll} \tau ^{-|\gamma |}, \qquad if \ {} &{} \ \gamma \ne 0\\ -\ln (\tau ), \qquad if &{} \gamma = 0. \end{array}\right. } \end{aligned}$$

(4.9)