Global existence and blow-up for space and time nonlocal reaction-diffusion equation

TL;DR

This paper investigates a fractional reaction-diffusion equation, demonstrating conditions for finite-time blow-up versus global existence, and analyzing the long-term behavior of solutions.

Contribution

It provides new insights into the conditions leading to blow-up or global solutions for space-time fractional reaction-diffusion equations.

Findings

Solutions can blow up in finite time under certain initial conditions.

For realistic initial data, solutions exist globally in time.

The asymptotic behavior of bounded solutions is characterized.

Abstract

A time-space fractional reaction-diffusion equation in a bounded domain is considered. Under some conditions on the initial data, we show that solutions may experience blow-up in a finite time. However, for realistic initial conditions, solutions are global in time. Moreover, the asymptotic behavior of bounded solutions is analysed.

Equations

$${\partial }_t^{\alpha }u+(-\Delta {)}_{\Omega }^{s}u=-u(1-u),x\in \Omega ,t>0,$$

$$u=0,x\in {\mathbb{R}}^n\setminus \Omega ,t>0,$$

$$u(x,0)=u_0(x),x\in \Omega ,$$

$${\partial }_t^{\alpha }u(x,t)=\frac{1}{\Gamma (1-\alpha )}\underset{0}{\overset{t}{\int }}(t-\tau {)}^{-\alpha }{\partial }_{\tau }u(x,\tau )d\tau ,$$

$${\left(-\Delta \right)}_{\Omega }^{s}u(x,t)$$

$$=\underset{\epsilon \downarrow 0}{\lim }{\left(-\Delta \right)}_{\Omega ,\epsilon }^{s}u(x,t),x\in \Omega ,$$

$${\int }_{\Omega }\frac{|u(x,t)|}{(1+|x|{)}^{n+2s}}dx<\infty ,$$

$${\left(-\Delta \right)}_{\Omega ,\epsilon }^{s}u(x,t)=C_{n,s}{\int }_{\left\{\xi \in \Omega :|x-\xi |>\epsilon \right\}}\frac{u(x,t)-u(\xi ,t)}{|x-\xi {|}^{n+2s}}d\xi ,x\in {\mathbb{R}}^n.$$

$$D((-\Delta {)}_{\Omega }^s)=\{u\in H_0^s(\bar{\Omega }),(-\Delta {)}_{\Omega }^{s}u\in L^2(\Omega )\}.$$

$$D(A_{\infty })=\{u\in D((-\Delta {)}_{\Omega }^s)\cap L^{\infty }(\Omega );Au\in L^{\infty }(\Omega )\},A_{\infty }=Au.$$

$$H^s(\Omega )=\{u\in L^2(\Omega ):|u{|}_{H^s(\Omega )}<\infty \},$$

$$|u{|}_{H^s(\Omega )}={\left(\int \underset{\Omega \times \Omega }{\int }\frac{|u(x)-u(\xi ){|}^2}{|x-\xi {|}^{n+2s}}dxd\xi \right)}^{\frac{1}{2}}$$

$$\Vert u{\Vert }_{H^s(\Omega )}=\Vert u{\Vert }_{L^2(\Omega )}+|u{|}_{H^s(\Omega )}.$$

$$\begin{array}{rl}& (-\Delta {)}_{\Omega }^s{e}_n(x)={\lambda }_n{e}_n(x),x\in \Omega ,\\ & e_n(x)=0,x\in {\mathbb{R}}^n\setminus \Omega ,\end{array}$$

$$u\in C^{0,\kappa }((0,\infty ),L^{\infty }(\Omega )),\text{for some}\kappa >0;$$

$$u(.,t)\in D(A_{\infty }),\text{for all}t>0,\text{and}{\partial }_t^{\alpha }u\in C((0,\infty ),L^{\infty }(\Omega )).$$

$$\Vert u(t,\cdot ){\Vert }_{L^2(\Omega )}^2\le \frac{C}{1+\nu t^{\alpha }}\Vert u_0{\Vert }_{L^2(\Omega )}^2\le C{t}^{-\alpha },t>0,$$

$${\left(\frac{\Gamma (\alpha +1)}{4(H_0+1/2)}\right)}^{\frac{1}{\alpha }}\le T^{\ast }\le {\left(\frac{\Gamma (\alpha +1)}{H_0}\right)}^{\frac{1}{\alpha }}.$$

$$\underset{\Omega }{\int }{\partial }_t^{\alpha }\tilde{u}\cdot \tilde{u}dx+\underset{\Omega }{\int }(-\Delta {)}_{\Omega }^s\tilde{u}\cdot \tilde{u}dx=\underset{\Omega }{\int }{\tilde{u}}^2\left(\tilde{u}-1\right)dx.$$

$$0\le \Vert u{\Vert }_{L^2(\Omega )}^2\lesssim \underset{\Omega }{\int }\underset{\Omega }{\int }\frac{|u(x)-u(y){|}^2}{|x-y{|}^{n+2s}}dxdy\lesssim \left((-\Delta {)}_{\Omega }^{s}u,u\right),u\in H^s(\Omega ),$$

$$-\underset{\Omega }{\int }(-\Delta {)}_{\Omega }^s\tilde{u}\cdot \tilde{u}dx\le 0.$$

$$2w(t){\partial }^{\alpha }w(t)\ge {\partial }^{\alpha }{w}^2(t),w\in C^1(0,\infty ),$$

$${\partial }_t^{\alpha }\underset{\Omega }{\int }{\tilde{u}}^2\left(x,t\right)dx\lesssim \underset{\Omega }{\int }{\tilde{u}}^2\left(x,t\right)dx.$$

$$\left\{\begin{array}[]{l}\partial^{\alpha}\varphi(t)\lesssim\varphi(t),\\ \varphi(0)=0,\end{array}\right.$$

$$\underset{\Omega }{\int }{\partial }_t^{\alpha }\hat{u}\cdot \hat{u}dx+\underset{\Omega }{\int }(-\Delta {)}_{\Omega }^s\hat{u}\cdot \hat{u}dx=\underset{\Omega }{\int }{\hat{u}}^2\left(\hat{u}-1\right)dx.$$

$${\partial }_t^{\alpha }\underset{\Omega }{\int }{\hat{u}}^2\left(x,t\right)dx\lesssim \underset{\Omega }{\int }{\hat{u}}^2\left(x,t\right)dx.$$

$${\partial }_t^{\alpha }u+(-\Delta {)}_{\Omega }^{s}u\le 0,x\in \Omega ,t>0,$$

$${\partial }^{\alpha }E(t)+\nu E(t)\le 0,t>0,$$

$${\partial }^{\alpha }\bar{E}(t)+\nu \bar{E}(t)=0,t>0,$$

$$\bar{E}(0)=E_0=\underset{\Omega }{\int }{u}_0^2(x)dx,$$

$$\bar{E}(t)=E_0{E}_{\alpha }(-\nu t^{\alpha }),$$

Full text

1. Introduction

In this paper, we consider the time and space fractional equation

[TABLE]

$\Omega\subset\mathbb{R}^{n}$ is a bounded domain, supplemented with the homogeneous boundary condition

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and with the initial condition

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where $\partial_{t}^{\alpha}$ is the Caputo fractional derivative (see [KST06, page 90])

[TABLE]

of order $\alpha\in(0,1],$ defined for a differentiable function $u,$ and $\left(-\Delta\right)_{\Omega}^{s}$ is the regional fractional Laplacian

[TABLE]

defined for $u(x,t)$ such that

[TABLE]

where $s\in(0,1)$ and $C_{n,s}$ is a normalizing constant (whose value is not important here), and

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The domain of the operator $(-\Delta)^{s}_{\Omega}$ is

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Set $\mathcal{L}^{\infty}(\Omega):=\overline{D(A_{\infty})}^{L^{\infty}(\Omega)},\,A_{\infty}=(-\Delta)^{s}_{\Omega},$ where

[TABLE]

The fractional Sobolev space $H^{s}(\Omega)$ is defined by [AB17, Section 2]

[TABLE]

where

[TABLE]

denotes the Gagliardo (Aronszajn-Slobodeckij) seminorm. The space $H^{s}(\Omega)$ is a Hilbert space; it is equipped with the norm

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The space $H_{0}^{s}(\bar{\Omega})$ consists of functions of $H^{s}(\bar{\Omega})$ vanishing on the boundary of $\Omega.$

Our paper is motivated by the recent one of [AAAKT15] in which global solutions and blowing-up solutions to the equation (1.1), when $s=1$ are proved. Our problem (1.1)-(1.3) is a natural generalization of results in [AAAKT15]. We will prove the existence of globally bounded solutions as well as blowing-up solutions under some condition imposed on the initial data. Note that, similar studies for time-fractional and time-space-fractional reaction-diffusion equations were considered in [AA18, AGKW19, DCCM15, CSWSS18, KSZ17, KLW16, VZ15, VZ17, GW2017].

We will use first eigenfunction $e_{1}(x)>0$ associated to the first eigenvalue $\lambda_{1}>0$ that satisfies the fractional eigenvalue problem [BP2016, Theorem 2.8]

[TABLE]

is normalized such that $\displaystyle\int\limits_{\Omega}e_{1}(x)dx=1.$

The space $C^{0,\kappa}((0,T),L^{\infty}(\Omega))$ consists of functions that are continuous and $\kappa$-Hölderian with respect to $t.$

2. Main results

Theorem 2.1**.**

**(i): **

Let $u_{0}(x)\in\mathcal{L}^{\infty}(\Omega)$ be such that $0\leq u_{0}(x)\leq 1.$ Then, problem (1.1)-(1.3) admits a global strong solution on $[T_{0},\infty),$ for every $T_{0}>0,$ i.e., $u$ is a mild solution, and

[TABLE]

[TABLE]

The equation (1.1) is satisfied for $t>0.$ It satisfies $0\leq u(x,t)\leq 1$ for all $x\in\Omega,\,t>0$ and

[TABLE]

for some positive constant $C.$

**(ii): **

If $1+\lambda_{1}\leq\displaystyle\int\limits_{\Omega}u_{0}(x)e_{1}(x)dx=H_{0},$ then the solution of problem (1.1)-(1.3) blows-up in a finite time $T^{*}$ that satisfies the bi-lateral estimate

[TABLE]

Proof.

The proof is divided in steps.

Global existence. Firstly, we show that $u\geq 0.$ Multiplying scalarly in ${{L}^{2}}(\Omega)$ equation (1.1) by $\tilde{u}:=\min\left(u,0\right),$ we obtain

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Using the estimate [AB17, Proposition 2.4]

[TABLE]

follows

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Using the inequality [AAK17, Remark 3.1]

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via an approximating process, we obtain

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Setting $\varphi(t)=\int\limits_{\Omega}\tilde{u}^{2}\left(x,t\right)dx$ in (2.3), we get

[TABLE]

which implies $\displaystyle\int\limits_{\Omega}\tilde{u}^{2}\left(x,t\right)dx=0$ as $\tilde{u}\left(x,0\right)=0$, hence $\tilde{u}=0$; whereupon $u\geq 0.$

Now, we show the upper estimate $u\leq 1.$ Multiplying scalarly in ${{L}^{2}}(\Omega)$ equation (1.1) by $\hat{u}:=\min\left(1-u,0\right),$ we obtain

[TABLE]

Repeating the above calculations for $\hat{u}:=\min\left(1-u,0\right)$, we get

[TABLE]

This implies that $\int\limits_{\Omega}\hat{u}^{2}\left(x,t\right)dx=0$ as $\hat{u}\left(x,0\right)=0$, hence $\hat{u}=0$; whereupon $u\leq 1.$ The result follows as $0\leq u\leq 1.$ Consequently, the solution exists on $[0,+\infty)$.

Decay estimate. Since $0\leq u\leq 1$, then $-u+u^{2}\leq 0.$ So, $u$ satisfies to the equation

[TABLE]

with initial and boundary conditions (1.2), (1.3).

Multiplying scalarly equation (2.4) by $u$ and using Poincaré-type inequality (2.2), we obtain

[TABLE]

where we have set $E(t)=\displaystyle\int\limits_{\Omega}u^{2}dx.$

By comparison, we have $E(t)\leq\bar{E}(t),$ where $\bar{E}(t)$ is the solution of the problem

[TABLE]

[TABLE]

whose unique solution is

[TABLE]

where $E_{\alpha}(z)$ is the Mittag-Leffler function [KST06, page 40]:

[TABLE]

The following Mittag-Leffler function’s estimate is known [Sim14, Theorem 4]:

[TABLE]

Then by comparison principle [JL2016] with the equation

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with initial-boundary conditions (1.2), (1.3), the estimate (2.1) then follows.

Blow-up. Multiplying equations (1.1) by $e_{1}(x)$ and integrating over $\Omega,$ we obtain

[TABLE]

Since

[TABLE]

as $u=0,\,e_{1}(x)=0,\,\,\,x\in\mathbb{R}^{n}\setminus\Omega$ and

[TABLE]

the function $H(t)=\displaystyle\int\limits_{\Omega}u(x,t)e_{1}(x)\,dx$ then satisfies

[TABLE]

Let $\tilde{H}(t)=H(t)-(1+\lambda_{1}),$ then from (2.6), we get

[TABLE]

As $0\leq\tilde{H}_{0}=\tilde{H}(0),$ then from the results in [DFR14, Formula (8)], the solution of inequality (2.7) blows-up in a finite time.

The proof of the Theorem is complete. ∎

Acknowledgements

The research of Alsaedi and Kirane is supported by NAAM research group, University of King Abdulaziz, Jeddah. The research of Torebek is financially supported in parts by the FWO Odysseus 1 grant G.0H94.18N: Analysis and Partial Differential Equations and by a grant No.AP08052046 from the Ministry of Science and Education of the Republic of Kazakhstan. No new data was collected or generated during the course of research