Blow-Up Solutions for the Space-Time Fractional Evolution Equation

Abstract

This paper focuses on the blow-up solutions of the space-time fractional equations with Riemann–Liouville type nonlinearity in arbitrary-dimensional space. Using the Banach mapping principle and the test function method, we establish the local well-posedness and overcome the difficulties caused by the fractional operators to obtain the blow-up results. Furthermore, we get the precise lifespan of blow-up solutions under special initial conditions.

1 Introduction

In this paper, we consider the following problem for the space-time fractional evolution equation:

$$\begin{aligned} {\left\{ \begin{array}{ll} \textbf{D}_{0 \mid t}^\alpha \psi (x,t) +(-\Delta )^{\beta / 2} \psi (x,t)=I_{0 \mid t}^{1-\alpha }\left( |\psi |^p\right) , &{} x \in \mathbb {R}^N, t>0, \\ \psi (x, 0)=\psi _0(x), &{} x \in \mathbb {R}^N,\end{array}\right. } \end{aligned}$$

(1)

where $\alpha \in (0,1)$, $\beta \in (0,2]$, $p>1$, $\psi _0(x)\in C_0\left( \mathbb {R}^N\right)$. The nonlocal fractional Laplacian operator $(-\Delta )^{\beta /2}$ is realized as a Fourier multiplier with symbol $|\xi |^\beta$: $(-\Delta )^{\beta /2}=\mathfrak {F}^{-1}|\xi |^\beta \mathfrak {F}$ under Fourier transform, $\textbf{D}_{0 \mid t}^\alpha$ is the Caputo fractional derivative operator, and $I_{0 \mid t}^{1-\alpha }$ denotes the Riemann–Liouville (R–L) fractional integral operator; they are defined respectively as

$$\begin{aligned} \textbf{D}_{0 \mid t}^{\alpha }\psi (t){} & {} :=\frac{1}{\Gamma (1-\alpha )}\int _{0}^{t}(t-r)^{-\alpha }\psi ^{'}(r)d r, \quad t>0,\\ I_{0 \mid t}^{1-\alpha }\psi (t){} & {} :=\frac{1}{\Gamma (1-\alpha )}\int _{0}^{t}(t-r)^{-\alpha }\psi (r)d r, \quad t>0, \end{aligned}$$

where $\Gamma$ is the Euler gamma function. $C_0\left( \mathbb {R}^N\right)$ is the normal space in which all continuous functions decay to zero as they approach infinity.

In the past decades, this kind of system has attracted a lot of attentions [1, 2], which is widely used in the fields such as fluid mechanics [3], control theory [4], engineering applications [5], and life sciences [6]. The most recent application encompasses biomedical therapy [7, 8], anomalous diffusion [9], and signal processing [10].

For this, Nagasawa [11], Kobayashi [12], Guedda and Kirane [13] considered the evolution equation involving fractional diffusion

$$\begin{aligned} {\left\{ \begin{array}{ll}\psi _t+(-\Delta )^{\beta /2}\psi =h(t,x)\psi ^{p}, \\ \psi (x, 0)=\psi _0(x).\end{array}\right. } \end{aligned}$$

(2)

Additionally, Cazenave, Dickstein and Weissler [14] proved the sharp blow-up, global existence results for the heat equation with nonlinear memory,

$$\begin{aligned} \psi _t-\Delta \psi =\int _{0}^{t}(t-r)^{-\gamma }|\psi |^{p-1}\psi (r)d r. \end{aligned}$$

(3)

Fino and Kirane [15] generalized and solved the associated problems of the following equations based on article [14]:

$$\begin{aligned} {\left\{ \begin{array}{ll} \psi _t+(-\Delta )^{\beta / 2} \psi =I_{0 \mid t}^{1-\gamma }\left( |\psi |^{p-1}\psi \right) , \\ \psi (x, 0)=\psi _0(x). \end{array}\right. } \end{aligned}$$

(4)

Using an existence-uniqueness test, they confirmed the validity of the equation and proved the existence of a blow-up solution. When

$$\begin{aligned} p \le 1+\frac{\beta (2-\gamma )}{(N-\beta +\beta \gamma )_{+}} \quad \text{ or } \quad p<\frac{1}{\gamma }, \end{aligned}$$

the solution to problem (4) is blow-up in finite time. In addition, the conditions for local or global solutions have also been established.

Different from previous work, in this paper, we focus on the evolution equations with two fractional forms of the Caputo time and the fractional diffusion. Our interest in this problem is motivated by the studies above to develop a general blow-up theory for (1)with R–L type nonlinear term. For this, we perform a fixed point argument to establish the local well-posedness. Our proof relies on the properties of the space-time fractional operators derived from Mittag–Leffler functions [19]. In addition, by contradiction argument, we obtain sufficient conditions for blow-up. Usually we need to define the mild solution as follows:

Definition 1.1

Let $\psi _0\in C_0(\mathbb {R}^N)$, $\alpha \in (0,1)$, $\beta \in (0,2]$, $p>1$, $T>0$. Assume that $\psi (x,t)\in C([0,T];C_0(\mathbb {R}^N))$ satisfies the integral equation given below,

$$\begin{aligned} \psi (t)=\mathcal {V}_{\alpha ,\beta }(t)\psi _0+\int _{0}^{t}(t-s)^{\alpha -1}\mathcal {K}_{\alpha ,\beta }(t-s)I_{0 \mid s}^{1-\alpha }\left( |\psi |^p\right) ds,\quad t\in [0,T], \end{aligned}$$

(5)

then $\psi$ is a mild solution to the problem (1). For the specific definition of $\mathcal {V}_{\alpha ,\beta }, \mathcal {K}_{\alpha ,\beta }$, see the preliminaries.

However, due to the lack of space-time estimates for $\mathcal {V}_{\alpha ,\beta }, \mathcal {K}_{\alpha ,\beta }$, we cannot directly derive the blow-up of the solutions. In order to overcome the technical difficulty, we introduce an integral test function which allows us to deal with the nonlinearity properly, and then use the relation of the R–L type operators and the Mittag–Leffler operators to show the equivalence between mild solutions and weak solutions. It turns out that weak solutions work well for achieving our goal and obtaining the threshold of p. It is worth noticing that this threshold will tend to the one obtained in [15] as $\alpha \rightarrow 1$. Moreover, we shall present the upper bound of the lifespan of solutions for some special initial data, the proof of this point is standard.

For simplicity, we use $f\lesssim g$ to denote $f\le Cg$, where C may have different values in different lines. Our main results are summarized below.

Theorem 1.2

Let $\psi _0\in C_0(\mathbb {R}^N)$, $p>1$. Then there exists a positive maximal time $T_{max}$ such that problem (1) has a unique mild solution $\psi \in C([0,T_{max});C_0(\mathbb {R}^N))$. Furthermore, either $T_{max}=+\infty$ or $T_{max}<+\infty$ and $\Vert \psi \Vert _{L^{\infty }((0,t)\times \mathbb {R}^N)}\rightarrow \infty$ as $t\rightarrow T_{max}$. In particular, if $\psi _0\ge 0$, $\psi _0\not \equiv 0$, $0<t<T_{max}$, then $\psi (t)$ is positive.

Definition 1.3

Let $\psi _0 \in L_{loc}^{\infty }(\mathbb {R}^N)$, $p>1$, and $T>0$. If $\psi \in L^p((0,T);L_{loc}^{\infty }(\mathbb {R}^N))$ and for any function $H (x,t)\in C^1([0,T];\mathcal {H}^{\bar{\beta }}(\mathbb {R}^N))$, $\psi$ satisfies the following equation

$$\begin{aligned}&\int _{0}^{T}\int _{\mathbb {R}^N} \psi (-\Delta )^{\beta / 2}H d x d t+\int _{0}^{T}\int _{\mathbb {R}^N} \psi D_{t \mid T}^{\alpha } H d x d t\nonumber \\&\quad =\int _{0}^{T}\int _{\mathbb {R}^N}I_{0 \mid t}^{1-\alpha }\left( |\psi |^p\right) H d x d t+\int _{0}^{T}\int _{\mathbb {R}^N}\psi _0(x)D_{t \mid T}^{\alpha }H d x d t, \end{aligned}$$

(6)

then $\psi$ is a weak solution to the problem (1), where $\mathcal {H}^\beta \left( \mathbb {R}^N\right) =\left\{ f\in \mathcal {S}^{\prime };(-\Delta )^{\beta /2}f\in L^2\left( \mathbb {R}^N\right) \right\}$, $\mathcal {S}^{\prime }$ is Schwartz space, $H(x,T)=0$ for all $x\in \mathbb {R}^N$, $\bar{\beta }>max\{\beta ,s\}$, $s= N(\frac{1}{2}-\frac{1}{q}+\beta )$, q is the conjugate of p.

Theorem 1.4

Let $\psi _0 \in C_0(\mathbb {R}^N)$. Suppose $\psi \in C([0,T];C_0(\mathbb {R}^N))$ is a mild solution of the problem (1), then $\psi$ is also a weak solution.

Theorem 1.5

Let $\psi _0\in C_0(\mathbb {R}^N)$, $\psi _0\ge 0$, and $\psi _0\not \equiv 0$. If $1<p<1+\frac{\beta }{\alpha N}$, or $p<\frac{1}{\alpha }$, then the solution of (1) blows up at a finite time.

Theorem 1.6

Let $0<\alpha <1$, $0<\beta \le 2$, $1<p<1+\frac{\beta }{\alpha N}$ or $p<\frac{1}{\alpha }$. Given $\psi _0=\epsilon f(x)$ with $\epsilon >0$, when

$$\begin{aligned} |x|\ge \epsilon _0,\quad \delta <\frac{\beta }{(p-1)\alpha }, \quad f(x)\gtrsim \vert x\vert ^{-\delta } \end{aligned}$$

holds, the lifespan $T_\epsilon$ of the solution $\psi (x,t)$ of the problem (1) has the following upper bound:

$$\begin{aligned} T_\epsilon \lesssim \epsilon ^{\frac{1}{\kappa }},\kappa = \frac{ min\{\alpha N,\alpha \delta \}}{\beta }-\frac{1}{p-1}. \end{aligned}$$

Corollary 1.7

Let $0<\alpha <1$, $0<\beta \le 2$, $1<p<1+\frac{\beta }{\alpha N}$ or $p<\frac{1}{\alpha }$. Suppose $\psi _0=\chi f(x)$ where $\chi >0$, if

$$\begin{aligned} |x|\le \chi _0, \quad \delta < N, \quad f(x)\gtrsim \vert x\vert ^{-\delta } \end{aligned}$$

is holding for some positive constant $\chi _0$. Then, for any $\chi \in [\chi _0,+\infty )$, the lifespan $T_{\chi }$ of the solution $\psi (x,t)$ to the problem (1) satisfies the following bound:

$$\begin{aligned} T_\chi \lesssim \chi ^{\frac{1}{\kappa }},\kappa =\frac{\alpha \delta }{\beta }-\frac{1}{p-1}. \end{aligned}$$

The remainder of this paper is divided into two parts. In part 2, we collect the necessary definitions and Lemmas. The part 3 is devoted to the proof of our main conclusions.

2 Preliminaries

In this section, we outline and review the relevant properties about the evolution operators, which are essential to prove our main conclusions.

Given $Z(t):=e^{-t(-\Delta )^{\beta / 2}}$, where $(-\Delta )^{\beta / 2}$ is a self-adjoint operator on $L^2(\mathbb {R}^N)$, it follows that Z(t) is a strongly continuous semigroup on $L^2(\mathbb {R}^N)$ generated by $(-(-\Delta )^{\beta / 2})$ (see [16]). $Z(t)\mu =J_\beta (t)*\mu$, where $*$ stands for convolution, and

$$\begin{aligned} J_\beta (t)(x):=\frac{1}{(2 \pi )^{N / 2}} \int _{\mathbb {R}^N} e^{i x. \xi -t|\xi |^\beta } d \xi , \quad t>0. \end{aligned}$$

Using the self-similar form of $J_\beta (x,t)$ and convolution by Young’s inequality, we can conclude

$$\begin{aligned} \left\| Z(t)\mu \right\| _q \le C t^{-\frac{N}{\beta }\left( \frac{1}{s}-\frac{1}{q}\right) }\Vert \mu \Vert _{s}, \end{aligned}$$

(7)

for all $\mu \in L^s(\mathbb {R}^N)$, $1 \le s \le q \le \infty$. Besides, since $(-\Delta )^{\beta / 2}$ is a self-adjoint operator,

$$\begin{aligned} \int _{\mathbb {R}^N} \psi (x)(-\Delta )^{\beta / 2} \varphi (x) d x=\int _{\mathbb {R}^N} \varphi (x)(-\Delta )^{\beta / 2} \psi (x) dx, \end{aligned}$$

(8)

holds for all $\psi , \varphi \in \mathcal {H}^\beta (\mathbb {R}^N)$.

Regarding the representation (5) of the mild solution, it contains the relevant content of the Mainardi’s and Mittag–Leffler functions. In the following, we will present their definitions.

A function of the Mittag–Leffler form with two parameters is defined as

$$\begin{aligned} E_{\alpha ,\beta }(z)=\sum _{n=0}^{\infty }\frac{z^n}{\Gamma (\alpha n+\beta )},\quad (z\in \mathbb {C},\alpha ,\beta >0). \end{aligned}$$

The semigroup Z(t) whose Mittag–Leffer operators forms are defined as follows:

$$\begin{aligned} \mathcal {V}_{\alpha , \beta }(t)=\int _0^{\infty } \mathcal {M}(\theta ;\,\alpha ) e^{-\theta t^\alpha (-\Delta )^{\beta / 2}} d \theta , \end{aligned}$$

and

$$\begin{aligned} \mathcal {K}_{\alpha , \beta }(t)=\int _0^{\infty } \alpha \theta \mathcal {M}(\theta ;\,\alpha ) e^{-\theta t^\alpha (-\Delta )^{\beta / 2}} d \theta , \end{aligned}$$

where $\mathcal {M}(\theta ;\,\alpha )$ is a Mainardi’s function defined by

$$\begin{aligned} \mathcal {M}(z;\,\alpha )=\sum _{n=0}^{\infty } \frac{(-1)^{n}z^n}{n ! \Gamma (-\alpha n+1-\alpha )}, 0<\alpha <1, \quad z \in \mathbb {C}. \end{aligned}$$

$\mathcal {M}(\theta ;\alpha )\ge 0$ for all $\theta \ge 0$ and satisfies the following properties:

$$\begin{aligned}{} & {} \int _{0}^{\infty }\mathcal {M}(\theta ;\,\alpha )d \theta =1,\quad 0<\alpha <1,\end{aligned}$$

(9)

$$\begin{aligned}{} & {} \int _{0}^{\infty }\theta ^{\sigma }\mathcal {M}(\theta ;\,\alpha )d\theta =\frac{\Gamma (\sigma +1)}{\Gamma (\alpha \sigma +1)},\quad \sigma >-1, 0<\alpha <1. \end{aligned}$$

(10)

From this we can derive the following results.

Lemma 2.1

$$\begin{aligned} \Vert \mathcal {V}_{\alpha , \beta }(t) \psi _0\Vert _{L^\infty (\mathbb {R}^N)}{} & {} \le \Vert \psi _0\Vert _{L^\infty (\mathbb {R}^N)} , \end{aligned}$$

(11)

$$\begin{aligned} \Vert \mathcal {K}_{\alpha , \beta }(t) \psi _0\Vert _{L^\infty (\mathbb {R}^N)}{} & {} \le C \Vert \psi _0\Vert _{L^\infty (\mathbb {R}^N)} . \end{aligned}$$

(12)

Proof

Using (7), (9), and (10), which yields

$$\begin{aligned} \Vert \mathcal {V}_{\alpha , \beta }(t) \psi _0\Vert _{L^\infty (\mathbb {R}^N)}&= \bigg \Vert \int _0^{\infty } \mathcal {M}(\theta ;\,\alpha ) Z(\theta t^\alpha )\psi _0 d \theta \bigg \Vert _{L^\infty (\mathbb {R}^N)} \\&\le \int _0^{\infty } \mathcal {M}(\theta ;\,\alpha ) (\theta t^{\alpha })^{-\frac{N}{\beta s}}\Vert \psi _0\Vert _{L^\infty (\mathbb {R}^N)} d \theta \\&\le \Vert \psi _0\Vert _{L^\infty (\mathbb {R}^N)}, \end{aligned}$$

and

$$\begin{aligned} \Vert \mathcal {K}_{\alpha , \beta }(t) \psi _0\Vert _{L^\infty (\mathbb {R}^N)}&= \bigg \Vert \int _0^{\infty } \alpha \theta \mathcal {M}(\theta ;\,\alpha ) Z(\theta t^\alpha )\psi _0 d \theta \bigg \Vert _{L^\infty (\mathbb {R}^N)} \\&\le \int _0^{\infty }\alpha \theta \mathcal {M}(\theta ;\,\alpha ) (\theta t^{\alpha })^{-\frac{N}{\beta s}}\Vert \psi _0\Vert _{L^\infty (\mathbb {R}^N)} d \theta \\&\le \frac{\alpha \Gamma (2)}{\Gamma (\alpha +1)} \Vert \psi _0\Vert _{L^\infty (\mathbb {R}^N)}\\ {}&\le C\Vert \psi _0\Vert _{L^\infty (\mathbb {R}^N)}. \end{aligned}$$

$\square$

In addition, we review the previous conclusions concerning time fractional operators.

Lemma 2.2

([17]) Assuming $\psi \in L^p((0, T), C_0(\mathbb {R}^N))$, $p>1$. Let

$$\begin{aligned} \xi (t)=\int _0^t(t-r)^{\alpha -1} \mathcal {K}_{\alpha , \beta }(t-r) \psi (r) d r, \end{aligned}$$

then

$$\begin{aligned} I_{0 \mid t}^{1-\alpha } \xi =\int _0^t \mathcal {V}_{\alpha , \beta }(t-r) \psi (r) d r. \end{aligned}$$

Lemma 2.3

([17]) For $\psi _0 \in C_0(\mathbb {R}^N)$, $t>0$, then we have $\mathcal {V}_{\alpha , \beta }(t) \psi _0 \in D(A)$,

$$\begin{aligned} \begin{aligned} \textbf{D}_{0 \mid t}^\alpha \mathcal {V}_{\alpha , \beta }(t) \psi _0 =A \mathcal {V}_{\alpha , \beta }(t) \psi _0, \end{aligned} \end{aligned}$$

where $D(A)=\left\{ \psi \in C_0\left( \mathbb {R}^N\right) \mid -(-\triangle )^{\frac{\beta }{2}} \psi \in C_0\left( \mathbb {R}^N\right) \right\}$.

Next, let us define R–L fractional derivatives and recall several results that will be used in proving the equivalence of the mild solution and the weak one.

Definition 2.4

Let $T>0$, $\psi \in AC[0,T]$, AC stands for the space of absolutely continuous functions. Fractional derivatives of order $\alpha \in (0,1)$ on the left- and right-sides of the R–L are defined as follows:

$$\begin{aligned} D_{0 \mid t}^{\alpha }\psi (t):=\frac{d}{dt}I_{0 \mid t}^{1-\alpha }\psi (t)=\frac{1}{\Gamma (1-\alpha )}\frac{d}{dt}\int _{0}^{t}(t-r)^{-\alpha }\psi (r)d r, \quad t>0, \end{aligned}$$

and

$$\begin{aligned} D_{t \mid T}^{\alpha }\psi (t):=-\frac{d}{dt}I_{t \mid T}^{1-\alpha }\psi (t)=-\frac{1}{\Gamma (1-\alpha )}\frac{d}{dt}\int _{t}^{T}(r-t)^{-\alpha }\psi (r)d r, \quad t<T. \end{aligned}$$

Based on this definition, it is not difficult to derive the following relation between Caputo and R–L derivatives

$$\begin{aligned} \textbf{D}_{0 \mid t}^{\alpha }\psi (\cdot ,t)=D_{0 \mid t}^{\alpha }[\psi (\cdot ,t)-\psi (\cdot ,0)]=\frac{d}{dt}I_{0 \mid t}^{1-\alpha }[\psi (\cdot ,t)-\psi (\cdot ,0)]. \end{aligned}$$

Proposition 2.5

([18]) Let $0<\alpha <1$, $T>0$. Fractional integral formula by parts

$$\begin{aligned} \int _{0}^{T}\psi (t)D_{0 \mid t}^{\alpha }\varphi (t)dt = \int _{0}^{T}\varphi (t)D_{t \mid T}^{\alpha }\psi (t)dt \end{aligned}$$

is valid for every $\psi \in I_{t \mid T}^{\alpha }(L^p(0,T))$ and $\varphi \in I_{0 \mid t}^{\alpha }(L^q(0,T))$ such that $\frac{1}{p}+\frac{1}{q}\le 1+\alpha$ with $p,q>1$, where

$$\begin{aligned} I_{0 \mid t}^{\alpha }(L^p(0,T)) :=\left\{ \psi =I_{0 \mid t}^{\alpha }f,~ f\in L^p(0,T)\right\} . \end{aligned}$$

Proposition 2.6

([19]) Let $0<\alpha <1$, $T>0$. Then we arrive at the following identities:

$$\begin{aligned} D_{0 \mid t}^{\alpha }I_{0 \mid t}^{\alpha }\psi (t)=\psi (t), \text{ a.e. } t\in (0,T), \end{aligned}$$

$\psi \in L^r(0,T)$, $1\leqslant r \leqslant \infty$.

Finally, to prove the blow-up, we need the following lemma.

Lemma 2.7

Let $\omega _1(m)=(1-m/M)^{\gamma }$, $0\le m \le M, \gamma \gg 1$, we have

$$\begin{aligned}&{D}_{m \mid M}^\alpha \omega _1(m)=CM^{-\alpha }\left( 1-\frac{m}{M}\right) ^{\gamma -\alpha } , \nonumber \\&{D}_{m \mid M}^{1-\alpha } \omega _1(m)=CM^{\alpha -1}\left( 1-\frac{m}{M}\right) ^{\gamma +\alpha -1}. \end{aligned}$$

(13)

Proof

Note that

$$\begin{aligned} \int _{m}^{M}(J-m)^{-\alpha }\left( 1-\frac{J}{M}\right) ^{\gamma }d J&=M^{-\alpha }\int _{m}^{M}\left( \frac{J-m}{M}\right) ^{-\alpha }\left( \frac{M-J}{M}\right) ^{\gamma }d J \\&=M^{-\alpha }\left( 1-\frac{m}{M}\right) ^{\gamma -\alpha }\int _{m}^{M}\left( \frac{J-m}{M-m}\right) ^{-\alpha }\left( \frac{M-J}{M-m}\right) ^{\gamma }d J \\&=M^{-\alpha }\left( 1-\frac{m}{M}\right) ^{\gamma -\alpha }(M-m)\int _{0}^{1}x^{-\alpha }(1-x)^{\gamma }dx \\&=M^{-\alpha }\left( 1-\frac{m}{M}\right) ^{\gamma -\alpha }(M-m)B(1-\alpha ,1+\gamma ), \end{aligned}$$

we have

$$\begin{aligned} \frac{d}{d m}\int _{m}^{M}(J-m)^{-\alpha }\left( 1-\frac{J}{M}\right) ^{\gamma }d J=-(1-\alpha +\gamma )B(1-\alpha ,1+\gamma )M^{-\alpha }\left( 1-\frac{m}{M}\right) ^{\gamma -\alpha }. \end{aligned}$$

As a result,

$$\begin{aligned} {D}_{m \mid M}^\alpha \omega _1(m)=CM^{-\alpha }\left( 1-\frac{m}{M}\right) ^{\gamma -\alpha }. \end{aligned}$$

Similarly, we can complete the proof of another equation. $\square$

3 Proof of Main Results

Proof of Theorem 1.2

We prove this conclusion based on the contraction mapping principle. A Banach space $\Pi _T$ is constructed for every $T>0$,

$$\begin{aligned} \Pi _T:=\left\{ \psi \in C([0,T];C_0(\mathbb {R}^N));\Vert \psi \Vert _1\le 2\Vert \psi _0\Vert _\infty \right\} , \end{aligned}$$

where $\Vert \psi \Vert _1:=\Vert \psi \Vert _{L^{\infty }((0,T),L^{\infty }(\mathbb {R}^N))}$. For any given $\psi \in \Pi _T$, we define

$$\begin{aligned} G(\psi ):=\mathcal {V}_{\alpha ,\beta }(t)\psi _0+\int _{0}^{t}(t-s)^{\alpha -1}\mathcal {K}_{\alpha ,\beta }(t-s)I_{0 \mid s}^{1-\alpha }\left( |\psi |^p\right) ds. \end{aligned}$$

$\mathbf {Step ~~1}:$ Let $\psi \in \Pi _T$, we claim $\Vert \cdot \Vert _{\infty }=\Vert \cdot \Vert _{L^{\infty (\mathbb {R}^N)}}$ by using (11) and (12), then

$$\begin{aligned} \Vert G(\psi )\Vert _1&\le \Vert \psi _0\Vert _\infty +C \bigg \Vert \int _{0}^{t}(t-s)^{\alpha -1}\int _{0}^{s}(s-\tau )^{-\alpha }\Vert \psi \Vert _{\infty }^p d\tau ds \bigg \Vert _{L^{\infty }(0,T)}\nonumber \\&\le \Vert \psi _0\Vert _\infty +CT^{\alpha }T^{-\alpha +1}\Vert \psi \Vert _{1}^{p} \nonumber \\&\le \Vert \psi _0\Vert _\infty +CT2^p\Vert \psi _0\Vert _{\infty }^{p-1}\Vert \psi _0\Vert _{\infty }. \end{aligned}$$

(14)

By choosing T small enough, we have

$$\begin{aligned} CT2^p\Vert \psi _0\Vert _{\infty }^{p-1}\le 1, \end{aligned}$$

this implies that $\Vert G(\psi )\Vert _1\le 2 \Vert \psi _0\Vert _{\infty }$. Consequently, we get $G(\psi )\in \Pi _T$.

$\mathbf {Step ~2}:$ For $\psi , \varphi \in \Pi _T$, by using (12), we have the following estimate,

$$\begin{aligned} \Vert G(\psi )-G(\varphi )\Vert _1&\le \bigg \Vert \int _{0}^{t}(t-s)^{\alpha -1}\mathcal {K}_{\alpha ,\beta }(t-s)I_{0\mid s}^{1-\alpha }(|\psi |^p-|\varphi |^p)ds\bigg \Vert _1 \nonumber \\&\le C\bigg \Vert \int _{0}^{t}(t-s)^{\alpha -1}\int _{0}^{s}(s-\tau )^{-\alpha }\Vert |\psi |^p-|\varphi |^p\Vert _{\infty }d \tau ds \bigg \Vert _{L^{\infty }(0,T)} \nonumber \\&\le CT\Vert |\psi |^p-|\varphi |^p\Vert _{1} \nonumber \\&\le CT2^p\Vert \psi _0\Vert _{\infty }^{p-1}\Vert \psi -\varphi \Vert _1, \end{aligned}$$

(15)

due to the inequality

$$\begin{aligned} |x|^p-|y|^p \le C \left|x^{p-1}+y^{p-1}\right||x-y|, \end{aligned}$$

(16)

a choice of small T such that

$$\begin{aligned} CT2^p\Vert \psi _0\Vert _{\infty }^{p-1}\le \frac{1}{2}, \end{aligned}$$

implies that $G(\psi )$ is a contraction mapping on $\Pi _T$. To sum up, we conclude from Banach’s fixed point theorem that there is a mild solution $\psi$ to the problem (1).

$\mathbf {Step ~~3}:$ Concerning the uniqueness issue, we use Gronwall’s inequality to deal with it.

Let $\psi _1, \psi _2$ be two mild solutions in $\Pi _T$. Using (12) and (16), we obtain

$$\begin{aligned} \Vert \psi _1(t)-\psi _2(t)\Vert _{\infty }&\le C\int _{0}^{t}(t-s)^{\alpha -1}\int _{0}^{s}(s-\tau )^{-\alpha }\Vert |\psi _1|^p-|\psi _2|^p\Vert _{\infty }d \tau ds \nonumber \\&\le C \Vert \psi _0\Vert _{\infty }^{p-1}\int _{0}^{t}\int _{0}^{s}(t-s)^{\alpha -1}(s-\tau )^{-\alpha }\Vert \psi _1- \psi _2\Vert _{\infty }d \tau ds \nonumber \\&\le C\Vert \psi _0\Vert _{\infty }^{p-1}\int _{0}^{t}\Vert \psi _1-\psi _2\Vert _{\infty }ds. \end{aligned}$$

(17)

From Gronwall’s inequality, we infer that $\psi _1=\psi _2$. In addition, due to the uniqueness, there must be a solution in the maximal interval $[0,T_{max})$ (see also Fino, Kirane [15]).

If $\psi _0\ge 0$ and $\psi _0\not \equiv 0$, by (9), (10), we can get directly from (5) that $\psi \ge \mathcal {V}_{\alpha ,\beta }(t)\psi _0>0$. This closes the proof. $\square$

Proof of Theorem 1.4

Equation (5) implies that

$$\begin{aligned} \psi -\psi _0=\mathcal {V}_{\alpha ,\beta }(t)\psi _0-\psi _0+\int _0^t(t-s)^{\alpha -1} \mathcal {K}_{\alpha ,\beta }(t-s)I_{0 \mid s}^{1-\alpha }\left( |\psi |^p\right) d s. \end{aligned}$$

By Lemma 2.2, we get

$$\begin{aligned} I_{0 \mid t}^{1-\alpha }(\psi -\psi _0)=I_{0 \mid t}^{1-\alpha }(\mathcal {V}_{\alpha ,\beta }(t) \psi _0-\psi _0)+\int _0^t \mathcal {V}_{\alpha ,\beta }(t-s)I_{0 \mid s}^{1-\alpha }(|\psi |^p) d s. \end{aligned}$$

Integrating the above equation with respect to the variable x yields

$$\begin{aligned}&\int _{\mathbb {R}^N}I_{0 \mid t}^{1-\alpha }(\psi -\psi _0) H d x \\&\quad =\int _{\mathbb {R}^N}I_{0 \mid t}^{1-\alpha }(\mathcal {V}_{\alpha ,\beta }(t) \psi _0-\psi _0) H d x+\int _{\mathbb {R}^N} \int _0^t \mathcal {V}_{\alpha ,\beta }(t-s)I_{0 \mid s}^{1-\alpha }(|\psi |^p) d s H d x \\&\quad = M_1+M_2. \end{aligned}$$

Now, using Lemma 2.3, one has

$$\begin{aligned} \frac{d M_1}{d t}=\int _{\mathbb {R}^N} A(\mathcal {V}_{\alpha ,\beta }(t) \psi _0) H d x+\int _{\mathbb {R}^N}I_{0 \mid t}^{1-\alpha }(\mathcal {V}_{\alpha ,\beta }(t) \psi _0-\psi _0) H_t d x. \end{aligned}$$

Next, we construct the time derivative of $M_2$, let $h>0, t+h \le T$, and obtain

$$\begin{aligned} \begin{aligned}&\frac{1}{h}(M_2(t+h)-M_2(t))\\&\quad = \frac{1}{h} \int _0^{t+h} \int _{\mathbb {R}^N} \mathcal {V}_{\alpha ,\beta }(t+h-s)I_{0 \mid s}^{1-\alpha }(|\psi |^p) d s H(t+h, x) d x \\&\qquad -\frac{1}{h} \int _0^t \int _{\mathbb {R}^N} \mathcal {V}_{\alpha ,\beta }(t-s)I_{0 \mid s}^{1-\alpha }(|\psi |^p) d s H(t, x) d x \\&\quad =\frac{1}{h} \int _{\mathbb {R}^N} \int _t^{t+h} \int _0^{\infty }\mathcal {M}(\theta ;\alpha ) Z((t+h-s)^\alpha \theta )I_{0 \mid s}^{1-\alpha }(|\psi |^p) d \theta d s H(t+h, x) d x, \\&\qquad + \frac{1}{h} \int _{\mathbb {R}^N} \int _0^t \int _0^{\infty } \mathcal {M}(\theta ;\alpha )(Z((t+h-s)^\alpha \theta )-Z((t-s)^\alpha \theta ))I_{0 \mid s}^{1-\alpha }(|\psi |^p) d \theta d s H(t, x) d x, \\&\qquad +\frac{1}{h} \int _{\mathbb {R}^N} \int _0^t \int _0^{\infty } \mathcal {M}(\theta ;\alpha ) Z((t+h-s)^\alpha \theta )I_{0 \mid s}^{1-\alpha }(|\psi |^p) d \theta d s(H(t+h, x)-H(t, x)) d x \\&\quad =J_1+J_2+J_3. \end{aligned} \end{aligned}$$

By dominated convergence theorem, we deduce that when h tends to zero, $J_1$ and $J_3$ respectively converge to

$$\begin{aligned}{} & {} \int _{\mathbb {R}^N}I_{0 \mid s}^{1-\alpha }(|\psi |^p) H d x,\\{} & {} \int _{\mathbb {R}^N} \int _0^t \mathcal {V}_{\alpha ,\beta }(t-s)I_{0 \mid s}^{1-\alpha }(|\psi |^p) d s H_t d x. \end{aligned}$$

Afterwards, we consider the estimation of $J_2$, which can be rewritten as follows:

$$\begin{aligned} \begin{aligned} J_2&= \int _{\mathbb {R}^N} \int _0^t \int _0^{\infty } \int _0^1 \alpha \theta \mathcal {M}(\theta ;\alpha )(t+mh-s)^{\alpha -1} A\left( Z\left( (t+m h-s)^\alpha \theta \right) \right) I_{0 \mid s}^{1-\alpha }(|\psi |^p) d m d \theta d s H d x \\&= \int _{\mathbb {R}^N} \int _0^t \int _0^{\infty } \int _0^1 \alpha \theta \mathcal {M}(\theta ;\alpha )(t+mh-s)^{\alpha -1} Z\left( (t+mh-s)^\alpha \theta \right) I_{0 \mid s}^{1-\alpha }(|\psi |^p) d m d \theta d s A H d x. \end{aligned} \end{aligned}$$

By dominated convergence theorem, we deduce that when h tends to zero, $J_2$ converge to

$$\begin{aligned} \int _{\mathbb {R}^N} \int _0^t(t-s)^{\alpha -1}\mathcal {K}_{\alpha ,\beta }(t-s)I_{0 \mid s}^{1-\alpha }(|\psi |^p) d s A H d x. \end{aligned}$$

Then, we get

$$\begin{aligned} \begin{aligned} \frac{d M_2}{d t}&= \int _{\mathbb {R}^N}I_{0 \mid s}^{1-\alpha }(|\psi |^p) H d x+\int _{\mathbb {R}^N} \int _0^t(t-s)^{\alpha -1} \mathcal {K}_{\alpha ,\beta }(t-s)I_{0 \mid s}^{1-\alpha }(|\psi |^p) d s A H d x \\&\quad +\int _{\mathbb {R}^N} \int _0^t \mathcal {V}_{\alpha ,\beta }(t-s)I_{0 \mid s}^{1-\alpha }(|\psi |^p) d s H_t d x \\&= \int _{\mathbb {R}^N}I_{0 \mid s}^{1-\alpha }(|\psi |^p) H d x+\int _{\mathbb {R}^N} \int _0^t(t-s)^{\alpha -1} \mathcal {K}_{\alpha ,\beta }(t-s)I_{0 \mid s}^{1-\alpha }(|\psi |^p) d s A H d x \\&\quad +\int _{\mathbb {R}^N}I_{0 \mid t}^{1-\alpha }\left( \int _0^t(t-s)^{\alpha -1} \mathcal {K}_{\alpha ,\beta }(t-s)I_{0 \mid s}^{1-\alpha }(|\psi |^p) d s\right) H_t d x. \end{aligned} \end{aligned}$$

Using the integration by parts formula, we derive

$$\begin{aligned} \begin{aligned} \int _{\mathbb {R}^N}I_{0 \mid t}^{1-\alpha }\left( \psi -\psi _0\right) H_t dx&=-\int _{\mathbb {R}^N}\frac{d}{dt}I_{0 \mid t}^{1-\alpha }\left( \psi -\psi _0\right) H dx \\&=-\int _{\mathbb {R}^N}\left( \psi -\psi _0\right) D_{t \mid T}^{\alpha }H(x,t)dx. \end{aligned} \end{aligned}$$

It follows

$$\begin{aligned} \begin{aligned} 0{} & {} = \int _0^T \frac{d}{d t} \int _{\mathbb {R}^N}I_{0 \mid t}^{1-\alpha }\left( \psi -\psi _0\right) H d x d t=\int _0^T \frac{d M_1}{d t}+\frac{d M_2}{d t} d t \\{} & {} = -\int _0^T \int _{\mathbb {R}^N} \psi (-\Delta )^{\frac{\beta }{2}} H d x d t-\int _0^T \int _{\mathbb {R}^N}(\psi -\psi _0)D_{t \mid T}^{\alpha }H d x d t\\{} & {} \quad +\int _0^T \int _{\mathbb {R}^N}I_{0 \mid t}^{1-\alpha }(|\psi |^p) H d x d t. \end{aligned} \end{aligned}$$

The proof is completed. $\square$

Proof of Theorem 1.5

Here, we use the contradiction analysis based on the test functions to verify our conclusion. In what follows, we prove this conclusion in two different cases associated with $p>1$.

$\mathbf {Case 1:}$ Assume $\psi$ is a weak solution to the Eq. (1), then the Eq. (6) holds, $\psi \in C([0,T]; C_0(\mathbb {R}^N))$, $T\gg 1$, and $\psi (t)>0$.

Let $H(x,t):=D_{t \mid T}^{1-\alpha }\omega (x,t)$ with $\omega \in C^1([0,T];H^{\bar{\beta }}(\mathbb {R}^N))$, $\omega =\omega _1(t)\varphi ^{l}(x)$, $\omega _1(t)=(1-\frac{t}{T})^{\gamma }$, and $\varphi (x)=\Psi (\frac{\vert x \vert }{T^{\alpha /\beta }})$, where $l\gg 1,\gamma \gg 1$. The function $\Psi$ is smooth, non-increasing, and satisfying

$$\begin{aligned} \Psi (x)=\left\{ \begin{array}{lll} 1 &{} \text{ if } &{} x \le 1, \\ \searrow &{} \text{ if } &{} 1 \le x \le 2,\\ 0 &{} \text{ if } &{} x \ge 2. \end{array}\right. \end{aligned}$$

After that, we take $Q_T=[0,T]\times Q$, $Q=\{x\in \mathbb {R}^N; \vert x\vert \le 2T^{1/\beta }\}$ and substitute the test function into (6),

$$\begin{aligned}&\int _{Q_T} \psi (-\Delta )^{\beta / 2}D_{t \mid T}^{1-\alpha }\omega dxdt+\int _{Q_T} \psi D_{t \mid T}^{\alpha }H d x d t \nonumber \\&\quad =\int _{Q_T}I_{0 \mid t}^{1-\alpha }\left( |\psi |^p\right) D_{t \mid T}^{1-\alpha }\omega dxdt+\int _{Q_T}\psi _0(x)D_{t \mid T}^{\alpha }H dxdt, \end{aligned}$$

(18)

which implies

$$\begin{aligned}&\int _{Q} \psi _0(x)\varphi ^{l}(x)dx+\int _{Q_T}\vert \psi \vert ^{p}\omega d x d t \nonumber \\&\quad =\int _{Q_T}\psi (-\Delta )^{\frac{\beta }{2}}\varphi ^l(x)D_{t \mid T}^{1-\alpha }\omega _1(t)dxdt-\int _{Q_T}\psi \varphi ^l(x)\omega _{1}^{'}(t)dxdt. \end{aligned}$$

(19)

It has

$$\begin{aligned}&\int _{Q} \psi _0(x)\varphi ^{l}(x)dx+\int _{Q_T}\vert \psi \vert ^{p}\omega dxdt \nonumber \\&\quad \le l\int _{Q_T}\psi \varphi ^{l-1}(x) \bigr \vert (-\Delta )^{\frac{\beta }{2}}\varphi (x)\bigr \vert D_{t \mid T}^{1-\alpha }\omega _1(t)dxdt+\int _{Q_T}\psi \varphi ^{l}(x)\bigr \vert \omega _{1}^{'}(t)\bigr \vert dxdt \nonumber \\&\quad \le l\int _{Q_T}\psi \varphi ^{l-1}\omega ^{\frac{1}{p}}\omega ^{\frac{1}{q}}\frac{1}{\omega }\bigr \vert (-\Delta )^{\frac{\beta }{2}}\varphi (x)\bigr \vert D_{t \mid T}^{1-\alpha }\omega _1(t)dxdt+\int _{Q_T}\psi \varphi ^{l}\omega ^{\frac{1}{p}}\omega ^{\frac{1}{q}}\frac{1}{\omega }\bigr \vert \omega _{1}^{'}(t)\bigr \vert dxdt. \end{aligned}$$

where we used the estimate $\ell f^{\ell -1}(-\Delta )^{\beta /2}f \ge (-\Delta )^{\beta /2}f^{\ell }$ for any bounded and continuous function $f\ge 0$ and all $\ell \ge 1$ [20]. Applying Young’s inequality with $\frac{1}{p}+\frac{1}{q}=1, p, q>1$, and taking the weight coefficient $\varepsilon =\frac{1}{4p}$, we consequently get that

$$\begin{aligned}&\int _{Q} \psi _0(x)\varphi ^{l}(x)dx+\int _{Q_T}\vert \psi \vert ^{p}\omega d x d t \nonumber \\&\quad \le \frac{1}{4p}\int _{Q_T}\vert \psi \vert ^p \omega dxdt +\frac{l^{q}4^{q-1}}{q}\int _{Q_T} \omega ^{1-q} \varphi ^{(l-1)q}(x)\{\bigr \vert (-\Delta )^{\frac{\beta }{2}}\varphi (x)\bigr \vert D_{t \mid T}^{1-\alpha }\omega _1(t)\}^{q}dxdt\nonumber \\&\qquad + \frac{1}{4p}\int _{Q_T}|\psi |^p \omega dxdt+\frac{4^{q-1}}{q}\int _{Q_T}\omega ^{1-q} \varphi ^{lq}(x)\bigr \vert \omega _{1}^{'}(t)\bigr \vert ^{q}dxdt. \end{aligned}$$

Similarly, taking $Q_2=[0,1]\times \{y\in \mathbb {R}^N; \vert y \vert \le 2 \}$, and substituting $\tau =\frac{t}{T}$, $y=\frac{x}{T^{\alpha /\beta }}$, $(-\Delta _x)^{\frac{\beta }{2}}\varphi =T^{-\alpha }(-\Delta _y)^{\frac{\beta }{2}}\varphi$, we have

$$\begin{aligned}&\int _{Q} \psi _0(x)\varphi ^{l}(x)dx+\left( 1-\frac{1}{2p}\right) \int _{Q_T}\vert \psi \vert ^{p}\omega dxdt \nonumber \\&\quad \le CT^{\frac{\alpha N}{\beta }+1-q}\int _{Q_2} (1-\tau )^{\gamma +\alpha q-q}\Psi ^{l-q} (\vert y \vert ) \{\bigr \vert (-\Delta _y)^{\frac{\beta }{2}}\varphi (T^{\alpha /\beta }y)\bigr \vert \}^{q} dy d\tau \nonumber \\&\qquad + CT^{\frac{\alpha N}{\beta }+1-q}\int _{Q_2}(1-\tau )^{\gamma -q}\Psi ^{l} (\vert y \vert ) dy d\tau . \end{aligned}$$

(20)

If ${\frac{\alpha N}{\beta }+1-q}<0$, that is $1<p<1+\frac{\beta }{\alpha N}$. When $2\le p<1+\frac{\beta }{\alpha N}$, $\{\bigr \vert (-\Delta _y)^{\frac{\beta }{2}}\varphi \bigr \vert \}^{q}$ is bounded in $Q_2$, the remaining two terms of the integral are still bounded in $Q_2$. Letting $T\rightarrow \infty$ we can obtain that the right terminal term of (20) is zero, while the left terminal term is positive. Therefore, we obtain a contradiction when $T\rightarrow \infty$. While $1<p\le 2$, $\varphi _2 \in H^s$, let $s\ge N(\frac{1}{2}-\frac{1}{q}+\beta )$, we can get that $\{\bigr \vert (-\Delta _y)^{\frac{\beta }{2}}\varphi \bigr \vert \}^{q}$ is bounded in $Q_2$. Taking T sufficiently large, we obtain a similar result.

$\mathbf {Case~~2:}$ The proof for the case $p<1/\alpha$ is similar to the case $1< p<1+\frac{\beta }{\alpha N}$, we redefine the test function H. Let $H(x,t):=D_{t \mid T}^{1-\alpha }B(x,t)$, where $B=\omega _1(t)B_{1}^{l}(x)$ with $B_{1}(x)=\Psi (\frac{\vert x \vert }{R})$, $R\in (0,T)$, T and R cannot be infinite at the same time. The definition of $\Psi$ is the same as in Case 1.

We set $\bar{Q}_T=[0,T]\times \bar{Q}$, $\bar{Q}=\{x\in \mathbb {R}^N; \vert x\vert \le 2R\}$, $\bar{Q}_2=[0,1]\times \{y\in \mathbb {R}^N; \vert y \vert \le 2 \}$. Repeating our steps in Case 1, $\tau =t/T$, $y=x/R$, $(-\Delta _x)^{{\beta }/{2}}B_1=R^{-\beta }(-\Delta _y)^{{\beta }/{2}}B_1$, with some details omitted, we also have

$$\begin{aligned}&\int _{\bar{Q}} \psi _0(x)B_{1}^{l}(x)dx+\left( 1-\frac{1}{2p}\right) \int _{\bar{Q}_T}\vert \psi \vert ^{p}B dxdt \nonumber \\&\quad \le CR^{N-\beta q}T^{1+(\alpha -1)q}\int _{\bar{Q}_2} (1-\tau )^{\gamma +\alpha q-q}\Psi ^{l-q} (\vert y \vert ) \{\bigr \vert (-\Delta _y)^{\frac{\beta }{2}}B_1(Ry)\bigr \vert \}^{q} dy d\tau \nonumber \\&\qquad + CR^{N}T^{1-q}\int _{\bar{Q}_2}(1-\tau )^{\gamma -q}\Psi ^{l} (\vert y \vert ) dy d\tau \nonumber \\&\le CR^{N-\beta q}T^{1+(\alpha -1)q} + CR^{N}T^{1-q}. \end{aligned}$$

(21)

Since $1+(\alpha -1)q <0$, namely, $p<\frac{1}{\alpha }$, let $T \rightarrow \infty$, we conclude that

$$\begin{aligned} \int _{\bar{Q}_T}\vert \psi \vert ^{p}\omega _1(t)B_{1}^{l}(x)dxdt=\int _{\bar{Q}_T}\vert \psi \vert ^{p}B_{1}^{l}(x)dxdt\le 0. \end{aligned}$$

Moreover, let $R\rightarrow \infty$, then $B_{1}^{l}(x) \rightarrow 1$, we get a contradiction. $\square$

Proof of Theorem 1.6

Let $\eta =\epsilon \int _{\mathbb {R}^N} f(x)\varphi ^{l}(x)dx$, $y=\frac{x}{T^{\alpha /\beta }}$, we have

$$\begin{aligned} \eta&= \epsilon \int _{\mathbb {R}^N} f(x)\varphi ^{l}(x)dx \nonumber \\&\ge \epsilon \int _{\vert x \vert \ge \epsilon _0} f(x)\varphi ^{l}(x)dx\nonumber \\&\gtrsim \epsilon T^{\frac{\alpha (N-\delta )}{\beta }}\int _{2\ge \vert y \vert \ge {\epsilon _0}/{T^{\alpha /\beta }}}\vert y \vert ^{-\delta }\Psi ^{l}(\vert y \vert )dy,\nonumber \\&\gtrsim \epsilon T^{\frac{\alpha (N-\delta )}{\beta }}{\left\{ \begin{array}{ll}2^{N-\delta },&{} N\ge \delta , \\ (\frac{\epsilon _0}{T^{\alpha /\beta }})^{N-\delta },&{} N<\delta .\end{array}\right. }\nonumber \\&\gtrsim \epsilon T^{(\frac{\alpha (N-\delta )}{\beta })_{+}}, \end{aligned}$$

(22)

where $(\frac{\alpha (N-\delta )}{\beta })_{+} = max\{\frac{\alpha (N-\delta )}{\beta }),0\}$. At the same time, we can deduce from (20) that the corresponding inequality holds,

$$\begin{aligned} \eta \le CT^{\frac{\alpha N}{\beta }+1-q}\lesssim T^{\frac{\alpha N}{\beta }+1-q}. \end{aligned}$$

(23)

Consequently, from (22) and (23) we have access to

$$\begin{aligned} \epsilon \lesssim T^{\frac{1}{\kappa }}, \end{aligned}$$

(24)

where $\kappa =\frac{\alpha N}{\beta }+1-q-max\{\frac{\alpha (N-\delta )}{\beta },0\}=\frac{min\{\alpha N,\alpha \delta \}}{\beta }-\frac{1}{p-1}<0$. Thus, it follows

$$\begin{aligned} T \lesssim \epsilon ^{\frac{1}{\kappa }}, \end{aligned}$$

which closes the proof. $\square$

Proof of Corollary 1.7

Let $\eta =\chi \int _{\mathbb {R}^N} f(x)\varphi ^{l}(x)dx$, and $y=\frac{x}{T^{\alpha /\beta }}$, our estimate is as follows:

$$\begin{aligned} \eta&= \chi \int _{\mathbb {R}^N} f(x)\varphi ^{l}(x)dx \nonumber \\&\ge \chi \int _{\vert x \vert \le \chi _0 } f(x)\varphi ^{l}(x)dx\nonumber \\&\gtrsim \chi T^{\frac{\alpha (N-\delta )}{\beta }}\int _{\vert y \vert \le {\chi _0}/{T^{\alpha /\beta }}}\vert y \vert ^{-\delta }\Psi ^{l}(\vert y \vert )dy,\nonumber \\&\gtrsim \chi T^{\frac{\alpha (N-\delta )}{\beta }}\int _{\vert y \vert \le 2}\vert y \vert ^{-\delta }dy,\nonumber \\&\gtrsim \chi T^{\frac{\alpha (N-\delta )}{\beta }}. \end{aligned}$$

(25)

As in Theorem 1.6, we can get the conclusion that completes the proof. $\square$