Global Existence of Small Data Solutions to Semi-linear Fractional $\sigma$-Evolution Equations with Mass and Nonlinear Memory

Abstract

In this paper we study the global (in time) existence of small data solutions to semi-linear fractional $\sigma -$evolution equations with mass and nonlinear memory. Our main goals was to explain on the one hand the influence of the memory term and on the other hand, the influence of higher regularity of the data on qualitative properties of solutions.

Appendix

1.1 Results from Harmonic Analysis

We recall some results from Harmonic Analysis (cf. with [5]).

Proposition 7.1

Let $r\in (1,\infty ), p>1$ and $\sigma \in (0,p)$. Let Q(u) denote one of the functions $|u|^{p}, \pm u|u|^{p-1}$. Then the following inequality holds:

$$\begin{aligned} \Vert Q(u)\Vert _{H^{\sigma }_{r}} \lesssim \Vert u\Vert _{H^{\sigma }_{r}}\Vert u\Vert _{L^{\infty }}^{p-1} \end{aligned}$$

for any $u\in H^{\sigma }_{r}(\mathbb {R}^n)\cap L^{\infty }(\mathbb {R}^n)$. Here we use for $\gamma \ge 0$ and $1< q< \infty$ the fractional Sobolev spaces or Bessel potential spaces

$$\begin{aligned} H^{\gamma }_{q}(\mathbb {R}^n):=\big \{f\in S'(\mathbb {R}^{n}):\Vert f\Vert _{H^{\gamma }_{q}}:=\Vert F^{-1}(\langle \xi \rangle ^{\gamma }F(f))\Vert _{L^{q}}<\infty \big \}. \end{aligned}$$

Moreover, $\langle D\rangle ^{\gamma }$ stands for the pseudo-differential operator with symbol $\langle \xi \rangle ^{\gamma }$ and it is defined by $\langle D\rangle ^{\gamma }u=F^{-1}(\langle \xi \rangle ^{\gamma }F(u))$.

Proof

This result is a special case of the following more general inequality for Triebel-Lizorkin spaces $F^\sigma _{r,q}$:

$$\begin{aligned} \Vert Q(u)\Vert _{F^\sigma _{r,q} }\lesssim \Vert u\Vert _{ F^\sigma _{r,q} }\Vert u\Vert _{ L^\infty }^{p-1} \quad \text{ for } \text{ any } \,\, u\in F^\sigma _{r,q} \cap L^\infty , \end{aligned}$$

where $q>0$, whose proof may be found in [6, Theorem 1 in Section 5.4.3]. $\square$

Proposition 7.2

Let $r\in (1,\infty ), p>1$ and $\sigma \in (0,p)$. Let Q(u) denote one of the functions $|u|^{p}, \pm u|u|^{p-1}$. Then the following inequality holds:

$$\begin{aligned} \Vert Q(u)\Vert _{\dot{H}^{\sigma }_{r}} \lesssim \Vert u\Vert _{\dot{H}^{\sigma }_{r}}\Vert u\Vert _{L^{\infty }}^{p-1} \end{aligned}$$

for any $u\in \dot{H}^{\sigma }_{r}(\mathbb {R}^n)\cap L^{\infty }(\mathbb {R}^n)$, where

$$\begin{aligned} \dot{H}^{\gamma }_{q}(\mathbb {R}^{n}):=\{f\in S'(\mathbb {R}^{n}):\Vert f\Vert _{\dot{H}^{\gamma }_{q}}:=\Vert F^{-1}(|\xi |^{\gamma }F(f))\Vert _{L^{q}}<\infty \}. \end{aligned}$$

Here $|D|^{\gamma }$ stands for the pseudo-differential operator with symbol $|\xi |^{\gamma }$ and it is defined by $|D|^{\gamma }u=F^{-1}(|\xi |^{\gamma }F(u))$.

Proof

We will use a homogeneity argument. For any positive $\lambda$ we define $u_\lambda (x)=u(\lambda x)$. Applying Proposition 7.1 to $u_\lambda$ we get

$$\begin{aligned} \Vert Q(u_\lambda )\Vert _{ H^\sigma _r }\lesssim \Vert u_\lambda \Vert _{ H^\sigma _r }\Vert u_\lambda \Vert _{ L^\infty }^{p-1}. \end{aligned}$$

(7.1)

Since for $r\in (1,\infty )$ we have the decomposition

$$\begin{aligned} \Vert v\Vert _{ H^\sigma _r }\approx \Vert v\Vert _{ \dot{H}^\sigma _r }+\Vert v\Vert _{L^r } \qquad \text{ for } \text{ any } \,\, v\in H^\sigma _r \end{aligned}$$

and the scaling properties

$$\begin{aligned} \Vert u_\lambda \Vert _{ \dot{H}^\sigma _r }=\lambda ^{\sigma -\frac{n}{r}}\Vert u\Vert _{ \dot{H}^\sigma _r }, \,\, \Vert u_\lambda \Vert _{ L^r }=\lambda ^{-\frac{n}{r}}\Vert u\Vert _{ L^r } \,\,\text{ and } \,\, \Vert u_\lambda \Vert _{ L^\infty }=\Vert u\Vert _{ L^\infty } \end{aligned}$$

diving both sides of (7.1) by $\lambda ^{\sigma -\frac{n}{r}}$ and taking the limit as $\lambda \rightarrow \infty$ we obtain the desired inequality. $\square$

Proposition 7.3

Let $r\in (1,\infty )$ and $\sigma >0$. Then the following inequality holds:

$$\begin{aligned} \Vert uv\Vert _{ H^\sigma _r }\lesssim \Vert u\Vert _{ H^\sigma _r }\Vert v\Vert _{ L^\infty }+\Vert u\Vert _{ L^\infty }\Vert v\Vert _{ H^\sigma _r } \end{aligned}$$

for any $u,v\in H^\sigma _r \cap L^\infty$.

Proof

The result that we want to prove is a special case of the following inequality for Triebel–Lizorkin spaces $F^\sigma _{r,q}$:

$$\begin{aligned} \Vert uv\Vert _{ F^\sigma _{r,q}}\lesssim \Vert u\Vert _{ F^\sigma _{r,q}}\Vert v\Vert _{ L^\infty }+\Vert u\Vert _{ L^\infty }\Vert v\Vert _{ F^\sigma _{r,q} } \end{aligned}$$

for any $u,v\in F^\sigma _{r,q} \cap L^\infty$, where $q>0$, whose proof can be found in [6, Theorem 2 in Section 4.6.4]. $\square$

Finally, let us state the corresponding inequality in homogeneous spaces $\dot{H}^\sigma _r$. For the proof it is possible to follow the same strategy as in the proof of Proposition 7.2.

Proposition 7.4

(Fractional Leibniz formula) Let $r\in (1,\infty )$ and $\sigma >0$. Then the following inequality holds:

$$\begin{aligned} \Vert uv\Vert _{\dot{H}^{\sigma }_{r}} \lesssim \Vert u\Vert _{\dot{H}^{\sigma }_{r}}\Vert v\Vert _{L^{\infty }}+\Vert u\Vert _{L^{\infty }}\Vert v\Vert _{\dot{H}^{\sigma }_{r}} \end{aligned}$$

for any $u,v\in \dot{H}^{\sigma }_{r}(\mathbb {R}^n)\cap L^{\infty }(\mathbb {R}^n).$

1.2 Inequalities

First we recall Young’s inequality.

Lemma 7.5

Let $u \in L^{p}(\mathbb {R}^{n})$ and $v \in L^{r}(\mathbb {R}^{n})$ with $1\le p,r\le \infty$. Then $u*v \in L^{q}(\mathbb {R}^{n})$, where $1+\frac{1}{q}=\frac{1}{p}+\frac{1}{r}$ and

$$\begin{aligned} \Vert u*v\Vert _{L^{q}}\lesssim \Vert u\Vert _{L^{p}}\Vert v\Vert _{L^{r}}. \end{aligned}$$

Finally, we recall the following Lemma from [1]:

Lemma 7.6

Suppose that $\theta \in [0,1), a\ge 0$ and $b\ge 0$. Then there exists a constant $C=C(a,b,\theta )>0$ such that for all $t>0$ the following estimate holds:

$$\begin{aligned} \begin{array}{ll} \int _{0}^{t}(t-\tau )^{-\theta } (1+t-\tau )^{-a}(1+\tau )^{-b} \,\mathrm{d}\tau \\ \qquad \le \left\{ \begin{array}{ll} C(1+t)^{-\min \{a+\theta ,b\}}\qquad \quad &{}\text {if}\quad \max \{a+\theta ,b\}>1, \\ C(1+t)^{-\min \{a+\theta ,b\}}\ln (2+t) \qquad \quad &{}\text {if}\quad \max \{a+\theta ,b\}=1,\\ C(1+t)^{1-a-\theta -b}\qquad \quad &{}\text {if}\quad \max \{a+\theta ,b\}<1. \end{array} \right. \end{array} \end{aligned}$$