Decay estimates in time for classical and anomalous diffusion

TL;DR

This paper investigates how solutions to classical and anomalous diffusion equations decay over time within bounded domains, considering fractional and nonlinear operators and their impact on long-term behavior.

Contribution

It provides new decay estimates for solutions of fractional and nonlinear diffusion equations, highlighting the influence of fractional operators on asymptotic behavior.

Findings

Decay rates for classical diffusion solutions

Impact of fractional operators on long-time asymptotics

Decay estimates for nonlinear diffusion equations

Abstract

We present a series of results focused on the decay in time of solutions of classical and anomalous diffusive equations in a bounded domain. The size of the solution is measured in a Lebesgue space, and the setting comprises time-fractional and space-fractional equations and operators of nonlinear type. We also discuss how fractional operators may affect long-time asymptotics.

Tables

Table 1. Table 1: Results from  vespri .

	Operator $𝒩$	Values of ${\lambda }_1$, ${\lambda }_2$	Range of $\ell$	Decay rate $\Theta$	Reference
Nonlinear classical diffusion	${\Delta }_p{u}^m$	${\lambda }_1\in (0,1]$, ${\lambda }_2\in [0,1)$	$\ell \in [1,+\infty )$	$\Theta \left(t\right)=\frac{1}{t^{\frac{\alpha }{m\left(p-1\right)}}}$	Thm 1.2 vespri
Nonlinear classical diffusion	${\Delta }_p{u}^m$ with $(m,p)\ne (1,2)$	${\lambda }_1=0$, ${\lambda }_2=1$	$\ell \in [1,+\infty )$	$\Theta \left(t\right)=\frac{1}{t^{\frac{1}{m\left(p-1\right)-1}}}$	Thm 1.2 vespri
Bi-Laplacian	${\Delta }_{2}u$	${\lambda }_1=0$, ${\lambda }_2=1$	$\ell \in [1,+\infty )$	$\Theta \left(t\right)=e^{-\frac{t}{C}}$	Thm 1.2 vespri
Graphical mean curvature	$\mathrm{div}\left(\frac{\nabla u}{\sqrt{1+{\left|\nabla u\right|}^2}}\right)$	${\lambda }_1\in (0,1]$, ${\lambda }_2\in [0,1)$	$\ell \in [1,+\infty )$	$\Theta \left(t\right)=\frac{1}{t^{\alpha }}$	Thm 1.5 vespri
Graphical mean curvature	$\mathrm{div}\left(\frac{\nabla u}{\sqrt{1+{\left|\nabla u\right|}^2}}\right)$	${\lambda }_1=0$, ${\lambda }_2=1$	$\ell \in [1,+\infty )$	$\Theta \left(t\right)=e^{-\frac{t}{C}}$	Thm 1.5 vespri
Fractional $p\text{-}$Laplacian	${\left(-\Delta \right)}_p^{s}u$	${\lambda }_1\in (0,1]$, ${\lambda }_2\in [0,1)$	$\ell \in [1,+\infty )$	$\Theta \left(t\right)=\frac{1}{t^{\frac{\alpha }{p-1}}}$	Thm 1.6 vespri
Fractional $p\text{-}$Laplacian	${\left(-\Delta \right)}_p^{s}u$, $p>2$	${\lambda }_1=0$, ${\lambda }_2=1$	$\ell \in [1,+\infty )$	$\Theta \left(t\right)=\frac{1}{t^{\frac{1}{p-2}}}$	Thm 1.6 vespri
Fractional $p\text{-}$Laplacian	${\left(-\Delta \right)}_p^{s}u$, $p⩽2$	${\lambda }_1=0$, ${\lambda }_2=1$	$\ell \in [1,+\infty )$	$\Theta \left(t\right)=e^{-\frac{t}{C}}$	Thm 1.6 vespri
Superposition of fractional $p\text{-}$Laplacians	${\sum }_{j=1}^N{\beta }_j{\left(-\Delta \right)}_{p_j}^{s_j}u$, ${\beta }_j>0$	${\lambda }_1\in (0,1]$, ${\lambda }_2\in [0,1)$	$\ell \in [1,+\infty )$	$\Theta \left(t\right)=\frac{1}{t^{\frac{\alpha }{p_{\max }-1}}}$	Thm 1.7 vespri
Superposition of fractional $p\text{-}$Laplacians	${\sum }_{j=1}^N{\beta }_j{\left(-\Delta \right)}_{p_j}^{s_j}u$, with ${\beta }_j>0$ and $p_{\max }>2$	${\lambda }_1=0$, ${\lambda }_2=1$	$\ell \in [1,+\infty )$	$\Theta \left(t\right)=\frac{1}{t^{\frac{1}{p_{\max }-2}}}$	Thm 1.7 vespri
Superposition of fractional $p\text{-}$Laplacians	${\sum }_{j=1}^N{\beta }_j{\left(-\Delta \right)}_{p_j}^{s_j}u$, with ${\beta }_j>0$ and $p_{\max }⩽2$	${\lambda }_1=0$, ${\lambda }_2=1$	$\ell \in [1,+\infty )$	$\Theta \left(t\right)=e^{-\frac{t}{C}}$	Thm 1.7 vespri
Superposition of anisotropic fractional Laplacians	${\sum }_{j=1}^N{\beta }_j{\left(-{\partial }_{x_j}^2\right)}^{s_j}u$, ${\beta }_j>0$	${\lambda }_1\in (0,1]$, ${\lambda }_2\in [0,1)$	$\ell \in [1,+\infty )$	$\Theta \left(t\right)=\frac{1}{t^{\alpha }}$	Thm 1.8 vespri
Superposition of anisotropic fractional Laplacians	${\sum }_{j=1}^N{\beta }_j{\left(-{\partial }_{x_j}^2\right)}^{s_j}u$, ${\beta }_j>0$	${\lambda }_1=0$, ${\lambda }_2=1$	$\ell \in [1,+\infty )$	$\Theta \left(t\right)=e^{-\frac{t}{C}}$	Thm 1.8 vespri
Fractional porous media I	${𝒫}_{1,s}\left(u\right)$	${\lambda }_1\in (0,1]$, ${\lambda }_2\in [0,1)$	$\ell \in [1,+\infty )$	$\Theta \left(t\right)=\frac{1}{t^{\frac{\alpha }{m}}}$	Thm 1.9 vespri
Fractional porous media I	${𝒫}_{1,s}\left(u\right)$, $m>1$	${\lambda }_1=0$, ${\lambda }_2=1$	$\ell \in [1,+\infty )$	$\Theta \left(t\right)=\frac{1}{t^{\frac{1}{m-1}}}$	Thm 1.9 vespri
Fractional porous media I	${𝒫}_{1,s}\left(u\right)$, $m⩽1$	${\lambda }_1=0$, ${\lambda }_2=1$	$\ell \in [1,+\infty )$	$\Theta \left(t\right)=e^{-\frac{t}{C}}$	Thm 1.9 vespri
Fractional graphical mean curvature	${\mathscr{H}}^s\left(u\right)$	${\lambda }_1\in (0,1]$, ${\lambda }_2\in [0,1)$	$\ell \in [1,+\infty )$	$\Theta \left(t\right)=\frac{1}{t^{\alpha }}$	Thm 1.10 vespri
Fractional graphical mean curvature	${\mathscr{H}}^s\left(u\right)$	${\lambda }_1=0$, ${\lambda }_2=1$	$\ell \in [1,+\infty )$	$\Theta \left(t\right)=e^{-\frac{t}{C}}$	Thm 1.10 vespri

Table 2. Table 2: Results from  elisa .

	Operator $𝒩$	Values of ${\lambda }_1$, ${\lambda }_2$	Range of $\ell$	Decay rate $\Theta$	Reference
Fractional porous media II	${𝒫}_{2,s}$	${\lambda }_1\in (0,1]$, ${\lambda }_2\in [0,1)$	$\ell \in [1,+\infty )$	$\Theta \left(t\right)=\frac{1}{t^{\frac{\alpha }{2}}}$	Thm 1.3 elisa
Fractional porous media II	${𝒫}_{2,s}$	${\lambda }_1=0$, ${\lambda }_2=1$	$\ell \in [1,+\infty )$	$\Theta \left(t\right)=\frac{1}{t}$	Thm 1.3 elisa
Classical Kirchhoff operator	$𝒦\left(u\right)$ with $M\left(0\right)>0$	${\lambda }_1\in (0,1]$, ${\lambda }_2\in [0,1)$	$\ell \in [1,+\infty )$	$\Theta \left(t\right)=\frac{1}{t^{\alpha }}$	Thm 1.4 elisa
Classical Kirchhoff operator	$\begin{array}{c}𝒦\left(u\right)\text{ with }M\left(t\right)=bt,\\ b>0\text{ and }n⩽4\end{array}$	${\lambda }_1\in (0,1]$, ${\lambda }_2\in [0,1)$	$\ell \in [1,+\infty )$	$\Theta \left(t\right)=\frac{1}{t^{\frac{\alpha }{3}}}$	Thm 1.4 elisa
Classical Kirchhoff operator	$\begin{array}{c}𝒦\left(u\right)\text{ with }M\left(t\right)=bt,\\ b>0\text{ and }n⩾5\end{array}$	${\lambda }_1\in (0,1]$, ${\lambda }_2\in [0,1)$	$\ell \in [1,\frac{2n}{n-4})$	$\Theta \left(t\right)=\frac{1}{t^{\frac{\alpha }{3}}}$	Thm 1.4 elisa
Classical Kirchhoff operator	$𝒦\left(u\right)$ with $M\left(0\right)>0$	${\lambda }_1=0$, ${\lambda }_2=1$	$\ell \in [1,+\infty )$	$\Theta \left(t\right)=e^{-\frac{t}{C}}$	Thm 1.4 elisa
Classical Kirchhoff operator	$\begin{array}{c}𝒦\left(u\right)\text{ with }M\left(t\right)=bt,\\ b>0\end{array}$	${\lambda }_1=0$, ${\lambda }_2=1$	$\ell \in [1,+\infty )$	$\Theta \left(t\right)=\frac{1}{\sqrt{t}}$	Thm 1.4 elisa
Fractional Kirchhoff operator	${𝒦}_s\left(u\right)$ with $M\left(0\right)>0$	${\lambda }_1\in (0,1]$, ${\lambda }_2\in [0,1)$	$\ell \in [1,+\infty )$	$\Theta \left(t\right)=\frac{1}{t^{\alpha }}$	Thm 1.5 elisa
Fractional Kirchhoff operator	$\begin{array}{c}𝒦\left(u\right)\text{ with }M\left(t\right)=bt,\\ b>0\text{ and }n⩽4s\end{array}$	${\lambda }_1\in (0,1]$, ${\lambda }_2\in [0,1)$	$\ell \in [1,+\infty )$	$\Theta \left(t\right)=\frac{1}{t^{\frac{\alpha }{3}}}$	Thm 1.5 elisa
Fractional Kirchhoff operator	$\begin{array}{c}𝒦\left(u\right)\text{ with }M\left(t\right)=bt,\\ b>0\text{ and }n>4s\end{array}$	${\lambda }_1\in (0,1]$, ${\lambda }_2\in [0,1)$	$\ell \in [1,\frac{2n}{n-4s})$	$\Theta \left(t\right)=\frac{1}{t^{\frac{\alpha }{3}}}$	Thm 1.5 elisa
Fractional Kirchhoff operator	${𝒦}_s\left(u\right)$ with $M\left(0\right)>0$	${\lambda }_1=0$, ${\lambda }_2=1$	$\ell \in [1,+\infty )$	$\Theta \left(t\right)=e^{-\frac{t}{C}}$	Thm 1.5 elisa
Fractional Kirchhoff operator	$\begin{array}{c}{𝒦}_s\left(u\right)\text{ with }M\left(t\right)=bt,\\ b>0\end{array}$	${\lambda }_1=0$, ${\lambda }_2=1$	$\ell \in [1,+\infty )$	$\Theta \left(t\right)=\frac{1}{\sqrt{t}}$	Thm 1.5 elisa
Classical magnetic operator	$ℳ\left(u\right)$	${\lambda }_1\in (0,1]$, ${\lambda }_2\in [0,1)$	$\ell \in [1,+\infty )$	$\Theta \left(t\right)=\frac{1}{t^{\alpha }}$	Thm 1.6 elisa
Classical magnetic operator	$ℳ\left(u\right)$	${\lambda }_1=0$, ${\lambda }_2=1$	$\ell \in [1,+\infty )$	$\Theta \left(t\right)=e^{-\frac{t}{C}}$	Thm 1.6 elisa
Fractional magnetic operator	${ℳ}_s\left(u\right)$	${\lambda }_1\in (0,1]$, ${\lambda }_2\in [0,1)$	$\ell \in [1,+\infty )$	$\Theta \left(t\right)=\frac{1}{t^{\alpha }}$	Thm 1.7 elisa
Fractional magnetic operator	${ℳ}_s\left(u\right)$	${\lambda }_1=0$, ${\lambda }_2=1$	$\ell \in [1,+\infty )$	$\Theta \left(t\right)=e^{-\frac{t}{C}}$	Thm 1.7 elisa

Equations

$$\{\begin{array}{ll}{\lambda }_1{\partial }_t^{\alpha }u+{\lambda }_2{\partial }_{t}u+\mathcal{N}[u]=0& \text{mbox}in\Omega \times (0,+\infty ),\\ u=0& \text{mbox}in({\mathbb{R}}^n\setminus \Omega )\times (0,+\infty ),\\ u(\cdot ,0)=u_0(\cdot )& \text{mbox}in\Omega .\end{array}$$

$${\partial }_t^{\alpha }v(t):=\frac{d}{dt}{\int }_0^t\frac{v(\tau )-v(0)}{(t-\tau {)}^{\alpha }}d\tau .$$

$$(-\Delta {)}^{s}u(x):={\int }_{{\mathbb{R}}^n}\frac{u(x)-u(y)}{|x-y{|}^{n+2s}}dy,\text{with}s\in (0,1)$$

$$\Delta_{p}u^{m}:={\rm div}(|\nabla u^{m}|^{p-2}\nabla u^{m}),\qquad{\mbox{with $p\in(1,+\infty)$ and $m\in(0,+\infty)$,}}$$

$$\mathrm{div}\left(\frac{\nabla u}{\sqrt{1+|\nabla u{|}^2}}\right),$$

$$(-\Delta {)}_p^{s}u(x):={\int }_{{\mathbb{R}}^n}\frac{|u(x)-u(y){|}^{p-2}(u(x)-u(y))}{|x-y{|}^{n+sp}}dy,$$

$$p\in (1,+\infty )$$

$$\sum _{j=1}^N{\beta }_j(-\Delta {)}_{p_j}^{s_j}u,\text{with}{p}_j\in (1,+\infty )\text{and}{s}_j\in (0,1),$$

$$(-{\Delta }_{\beta }{)}^{\sigma }u(x)=\sum _{j=1}^n{\beta }_j(-{\partial }_{x_j}^2{)}^{{\sigma }_j}u(x)$$

$$(-{\partial }_{x_j}^2{)}^{{\sigma }_j}u(x)={\int }_{\mathbb{R}}\frac{u(x)-u(x+\rho e_j)}{{\rho }^{1+2{\sigma }_j}}d\rho ,$$

$$\displaystyle{\mathcal{P}}_{1,s}(u):=(-\Delta)^{s}u^{m}\qquad{\mbox{with $s\in(0,1)$ and $m\in(0,+\infty)$,}}$$

$${\mathcal{P}}_{2,s}(u):=-\mathrm{div}(u\nabla \mathcal{R}(u)),\text{mbox}where\mathcal{R}(u)(x):={\int }_{{\mathbb{R}}^n}\frac{u(y)}{|x-y{|}^{n-2s}}dy$$

$${\mathcal{H}}^s(u)(x):={\int }_{{\mathbb{R}}^n}F\left(\frac{u(x)-u(x+y)}{|y|}\right)\frac{dy}{|y{|}^{n+s}},$$

$$\displaystyle\qquad\qquad{\mbox{with $s\in(0,1)$ and }}F(r):=\int_{0}^{r}\frac{d\tau}{(1+\tau^{2})^{\frac{n+1+s}{2}}},$$

$$\mathcal{K}(u)(x):=-M\left(\Vert \nabla u{\Vert }_{L^2(\Omega )}^2\right)\Delta u(x),$$

$${\mathcal{K}}_s(u)(x):=M\left({\iint }_{{\mathbb{R}}^n\times {\mathbb{R}}^n}\frac{|u(y)-u(Y){|}^2}{|y-Y{|}^{n+2s}}dydY\right)(-\Delta {)}^{s}u(x),$$

$$\mathcal{M}u:=-(\nabla -iA{)}^{2}u,$$

$${\mathcal{M}}_{s}u(x):={\int }_{{\mathbb{R}}^n}\frac{u(x)-e^{i(x-y)A\left(\frac{x+y}{2}\right)}u(y)}{|x-y{|}^{n+2s}}dy,\text{mbox}withs\in (0,1),$$

$$\Vert u(\cdot ,t){\Vert }_{L^{\ell }(\Omega )}⩽C^{\ast }\Theta (t)\text{mbox}forallt⩾1,$$

$$\|u(\cdot,t)\|_{L^{\ell}(\Omega)}^{\ell-1+\gamma}\leqslant C\,\int_{\Omega}|u(x,t)|^{\ell-2}\,{\mbox{\textfrak{Re}}}\Big{(}\overline{u}(x,t)\,{\mathcal{N}}u(x,t)\Big{)}\,dx,$$

$$\{\begin{array}{ll}{\partial }_t{\mathcal{G}}_s+(-\Delta {)}^s{\mathcal{G}}_s=0& \text{mbox}in{\mathbb{R}}^n\times (0,+\infty ),\\ {\mathcal{G}}_s(x,0)={\delta }_0(x).& \end{array}$$

$${\mathcal{G}}_1(x,t)=\frac{1}{(4t{)}^{\frac{n}{2}}}{e}^{-\frac{|x{|}^2}{4t}}.$$

$${\mathcal{G}}_{1/2}(x,t)=\frac{ct}{(t^2+|x{|}^2{)}^{\frac{n+1}{2}}},$$

$${\int }_{{\mathbb{R}}^n}{\mathcal{G}}_s(x,t)dx=1,$$

$${\mathcal{G}}_s(x,t)>0\text{mbox}forall(x,t)\in {\mathbb{R}}^n\times (0,+\infty ),$$

$${\mathcal{G}}_s(x,t)=\frac{1}{t^{\frac{n}{2s}}}{\mathcal{G}}_s\left(\frac{x}{t^{\frac{1}{2s}}},1\right).$$

$$q_k(s,\rho ):={\int }_{{\mathbb{R}}^n\setminus B_{\rho }}{\mathcal{G}}_s(x,k)dx.$$

$$0⩽q_k(s,\rho )⩽{\int }_{{\mathbb{R}}^n}{\mathcal{G}}_s(x,k)dx=1,$$

$$\begin{array}{rl}& q(s,\rho ):=\prod _{k=1}^{+\infty }{q}_k(s,\rho )\in [0,1]\\ \text{mbox}and& q(s):=\underset{\rho \to 0}{\lim }q(s,\rho ).\end{array}$$

$${\int }_A{\mathcal{G}}_s(x,t)dx$$

$$q_k(s,\rho )=\frac{1}{k^{\frac{n}{2s}}}{\int }_{{\mathbb{R}}^n\setminus B_{\rho }}{\mathcal{G}}_s\left(\frac{x}{k^{\frac{1}{2s}}},1\right)dx=\frac{{\rho }^n}{k^{\frac{n}{2s}}}{\int }_{{\mathbb{R}}^n\setminus B_1}{\mathcal{G}}_s\left(\frac{\rho y}{k^{\frac{1}{2s}}},1\right)dy$$

$$=\frac{1}{(k/{\rho }^{2s}{)}^{\frac{n}{2s}}}{\int }_{{\mathbb{R}}^n\setminus B_1}{\mathcal{G}}_s\left(\frac{y}{(k/{\rho }^{2s}{)}^{\frac{1}{2s}}},1\right)dy={\int }_{{\mathbb{R}}^n\setminus B_1}{\mathcal{G}}_s\left(y,\frac{k}{{\rho }^{2s}}\right)dy,$$

$$\underset{x\in B_1}{\inf }{\mathcal{G}}_s(x,1):={\iota }_s>0,$$

Full text

1 Decay estimates, methods, results and perspectives

In this note we present some results, recently obtained in vespri; elisa, focused on the long-time behavior of solutions of evolution equations which may exhibit anomalous diffusion, caused by either time-fractional or space-fractional effects (or both). The case of several nonlinear operators will be also taken into account (and indeed some of the results that we present are new also for classical diffusion run by nonlinear operators).

The results that we establish give quantitative bounds on the decay in time of smooth solutions, confined in a smooth bounded set with Dirichlet data. The size of the solution will be measured in classical Lebesgue spaces, and we will detect different types of decays according to the different cases that we take into consideration (the main order of decay being affected by the structure of the diffusion in time and by the possible nonlinear character of the spatial operator).

The evolution equation that we take into account is very general, and it can be written as an initial datum problem with homogeneous external Dirichlet condition of the type

[TABLE]

In this setting, $u=u(x,t)$ is a smooth solution of (1), $\Omega$ is a bounded set of $\mathbb{R}^{n}$ with smooth boundary (and we are not trying here to optimize the smoothness assumptions on the solution or on the domain), the convex parameters $\lambda_{1}$, $\lambda_{2}\in[0,1]$ are such that $\lambda_{1}+\lambda_{2}=1$, the (possibly nonlinear) operator ${\mathcal{N}}$ acts on the space variable $x$, and the time-fractional parameter $\alpha$ lies in $(0,1)$.

Also, in our setting, the symbol $\partial_{t}^{\alpha}$ stands for the so-called Caputo time-fractional derivative, defined, up to normalizing constants that we omit for simplicity, by

[TABLE]

Such a time-fractional derivative naturally arises in many context, including geophysics MR2379269, neurology MR2139665; MR3814763 (see also MR3071207 and the references therein) and viscoelasticity MR2351653, and can be seen as a natural consequence of classical models of diffusion in highly ramified media such as combs comb. In addition, from the mathematical point of view, equations involving the Caputo derivatives can be framed into the broad line of research devoted to Volterra type integrodifferential operators, see MR2125407; MR3563229.

The operator ${\mathcal{N}}$ in (1) takes into account the diffusion in the space variable, and can be either of classical or of fractional type, and concrete choices will be made in what follows. More precisely, our setting always comprises, as particular situations, the cases of diffusion driven by the Laplacian or by the fractional Laplacian, defined by

[TABLE]

where the integral is taken in the principal value sense (to allow cancellations near the singularity).

In our framework, we also deal with the case in which ${\mathcal{N}}$ is nonlinear, studying the cases of the classical $p$-Laplacian and porous media diffusion (see MR1230384; MR2286292)

[TABLE]

the case of graphical mean curvature, given in formula (13.1) of giusti,

[TABLE]

the case of the fractional $p$-Laplacian (see e.g. MR3861716)

[TABLE]

and possibly even the sum of different nonlinear operator of this type, with coefficients $\beta_{j}>0$,

[TABLE]

the case of the anisotropic fractional Laplacian, that is the sum of fractional directional derivatives in the directions of the space $e_{j}$, given by

[TABLE]

for $\beta_{j}>0$, $\beta=(\beta_{1},\dots,\beta_{n})$ and $\sigma=(\sigma_{1},\dots,\sigma_{n})$, where

[TABLE]

considered for example in anisotropic. The list of possible diffusion operators continues with two fractional porous media operators (see MR2773189; MR2737788)

[TABLE]

the graphical fractional mean curvature operator (see MR3331523)

[TABLE]

the classical Kirchhoff operator for vibrating strings

[TABLE]

and the fractional Kirchhoff operator (see MR3120682)

[TABLE]

with $M:[0,+\infty)\to[0,+\infty)$ nondecreasing and $s\in(0,1)$.

The case of complex valued operators is also considered, in view of a classical (see MR0142894) and fractional (see MR3794886) magnetic settings, in which we took into account the operators

[TABLE]

where $A:\mathbb{R}^{n}\to\mathbb{R}^{n}$ represents the magnetic field.

For further motivations and additional details on these operators, we refer to vespri; elisa: here we just mention that, given the general assumptions that we take, the operator ${\mathcal{N}}$ in (1) comprises many cases of interest in both pure and applied mathematics, with applications in several disciplines, see for instance MR1809268; MR3469920; MR3645874 for detailed discussions on anomalous diffusion with several applications in different contexts.

In our setting, we will obtain decay estimates in suitable Lebesgue spaces $L^{\ell}(\Omega)$, for some appropriate exponent $\ell\geqslant 1$. The typical estimate that we establish is that all solutions $u$ of (1) satisfy

[TABLE]

where $C^{*}>0$ depends on the structural assumptions of the problem (namely on $\Omega$, $\lambda_{1}$, $\lambda_{2}$, $\alpha$, $\mathcal{N}$, $u_{0}$ and $\ell$), and $\Theta:[1,+\infty)\to(0,+\infty)$ is an appropriate decay function, described here below in concrete situations, possibly depending on another constant $C>0$. The proof of the decay in (2) relies on energy estimates, which are in turn based on suitable Sobolev embeddings that employ the “parabolic” structure of the problem, leading to an appropriate ordinary differential inequality (if $\lambda_{1}$ in (1) is equal to zero), or an appropriate integral inequality (if $\lambda_{1}=1$), or a mixed differential/integrodifferential inequality (if $\lambda_{1}\in(0,1)$), for the norm-map $t\mapsto\|u(\cdot,t)\|_{L^{\ell}(\Omega)}$.The solutions of the equations related to those inequalities are used as barriers and compared to the function $\|u(\cdot,t)\|_{L^{\ell}(\Omega)}$, as presented in Theorem 1.1 of vespri and Theorems 1.1 and 1.2 of elisa.

More precisely, the “elliptic” character of the spatial diffusive operator is encoded in an inequality of the type

[TABLE]

where $\gamma$ and $C$ are positive structural constants, “\textfrakRe” denotes the real part and $\overline{u}$ is the complex conjugate of $u$ (in case the problem is set in the reals, the inequality in (3) obviously simplifies).

We observe that (3) becomes more transparent when $\ell=2$ and ${\mathcal{N}}u=-\Delta u$, with $u$ real valued: in such a case, after an integration by parts which takes into account the Dirichlet datum of $u$, the inequality in (3) boils down to the classical Sobolev-Poincaré inequality with $\gamma=1$.

Once the inequality in (3) is established for the operator $\mathcal{N}$ under consideration, one obtains a bound in terms of an ordinary differential equation, or more generally of a nonlinear integral equation on the variable $t$: depending on $\gamma$ and on the type of time-derivative, this provides an estimate on the decay in $t$ of the norm-map $t\mapsto\|u(\cdot,t)\|_{L^{\ell}(\Omega)}$, which can be either polynomial or exponential (in particular, different operators $\mathcal{N}$ can lead to different values of $\gamma$ and therefore to different asymptotics in time for the solution $u$).

This strategy, suitably adapted to the different situations, applies to many operators: the concrete cases that we comprise are listed explicitly in the Tables 1 and 2, which present the main results achieved in vespri; elisa. For the first table, the theorems cited in the last column are the ones proving (3) and a decay estimate in the case $\lambda_{1}=1$ and $\lambda_{2}=0$ for the operators in their row. Then, combining these results with Theorem 1.1 and 1.2 of elisa, the declared estimates trivially follow. However, in Table 1 for the first time we apply the estimates for the case $\lambda_{2}=1$ of elisa to the operators analyzed in vespri, stating the expected decays in a quantitative way.

It would be interesting to detect the optimality of the estimates listed in Tables 1 and 2, and to investigate other cases of interest as well. For related decay estimates, see MR2454809; MR3294409; MR3563229. As a matter of fact, decay estimates for evolutionary problems are a classical topic of research that has produced a very abundant, and extremely interesting, literature. Without aiming at providing an exhaustive list of all the important contributions on this topic, we mention that:

• •

the classical doubly-nonlinear operator $\Delta_{p}u^{m}$ with $\lambda_{2}=1$ has been addressed in MR2268115,

• •

the classical $1$-Laplace operator has been dealt with in MR2033382; KURD; MAZON,

• •

decay estimates for the fractional $p$-Laplacian $(-\Delta_{p})^{s}$ with $s\in(0,1)$ and $p>1$ with $\lambda_{2}:=1$ have been first established in Section 6 of BCM (see also BCM0),

• •

the case of the fractional $1$-Laplacian, namely $(-\Delta_{p})^{s}$ with $s\in(0,1)$ and $p:=1$ has been treated in MAZON,

• •

the porous medium equation for $\lambda_{2}=1$ has been deeply analyzed in MR3294409,

• •

some interesting estimates for the Kirchoff equation are given in MR2454809,

• •

see also MR2529737, where several decay estimates have been obtained by using integral inequalities.

We also remark that the interplay between time derivatives and fractional diffusion produces interesting decay patterns also in nonlinear equations, see e.g. MR3511786. Furthermore, in general, the fractional aspect of the problem can cause significant differences, as can be observed also from the time decays of Tables 1 and 2. For instance, one may notice that the decay switch from polynomial to exponential in the Kirchhoff equations when the time-diffusion changes from fractional to classical, and this independently on the fact that the space diffusion is of classical or fractional type (roughly speaking, in this context, it is just the character of the diffusion in time which detects the time decay, regardless the character of the diffusion in space).

We also point out that classical and fractional operators share several common properties, but they also exhibit structural differences at a fundamental level. For instance, to exhibit an elementary but very interesting feature in which long-time behaviors are affected by fractional environments, we recall that fractional diffusion in space, as modeled by the fractional Laplacian $(-\Delta)^{s}$ with $s\in(0,1)$, is related to Lévy-type and $2s$-stable stochastic processes, and in such case the long jumps of the underlying random walk causes significant differences with respect to the classical Brownian motion. In particular, fractional processes are typically recurrent only in dimension $1$ and for values of $s$ greater or equal to $1/2$ (being transient in dimension $2$ and higher, and also in dimension $1$ for values of $s$ smaller than $1/2$), and this is an important difference with respect to the case of Gaussian processes, which are recurrent in dimensions $1$ and $2$ (and transient in dimension $3$ and higher). See Example 3.5 in MR3529858 and the references therein for a detailed treatment of recurrence and transiency for Lévy-type processes. In this note, in § 2, we present a very simple, and somewhat heuristic, discussion of the recurrence and transiency properties related to the long jump random walks, based on PDE methods and completely accessible to a broad audience.

For a detailed list of other elementary structural differences between classical and fractional diffusion see also § 2.1 in abatangelo.

2 Recurrence and transiency of long jump random processes

In this section, we discuss a simple PDE approach to the recurrence of the long jump random walk related to $(-\Delta)^{s}$ in dimension $1$ and for values of $s$ greater or equal to $1/2$ and to its transiency in dimension $2$ and higher, and also in dimension $1$ for values of $s$ smaller than $1/2$. The treatment will comprise the classical case $s=1$ as well, showing how the structural differences between the different regimes naturally arise from a PDE analysis.

To this end, for $s\in(0,1]$, we denote by ${\mathcal{G}}_{s}(x,t)$ the solution of the (possibly fractional) heat equation with initial datum given by the Dirac’s Delta, namely

[TABLE]

When $s=1$, we have that such function reduces to the classical Gauss kernel for the heat flow, namely

[TABLE]

In general, when $s\in(0,1)$, the expression of ${\mathcal{G}}_{s}$ is less explicit, except when $s=1/2$; in the latter case, it holds that

[TABLE]

where $c>0$ is a normalizing constant – the need of which lying in the general mass conservation law

[TABLE]

see also formula (2.29) in abatangelo and page 1363 in MR2588782. Furthermore, see again page 1363 in MR2588782, we have that

[TABLE]

and it enjoys the natural scaling property

[TABLE]

See MR2283957, MR2588782, formulas (2.41)–(2.45) in abatangelo and the references therein for a discussion about the fractional heat kernel and its differences with the classical case.

For every $k\in\{1,2,3,\dots\}$ and $\rho>0$, we define

[TABLE]

We observe that

[TABLE]

thanks to (4). Let also

[TABLE]

In view of (8), we can consider $q(s)$ as related to the probability of the stochastic process associated with the operator $(-\Delta)^{s}$ of “drifting away without coming back”. Namely (see SEMA), we know that

[TABLE]

represents the probability that a particle starting at the origin at time [math] and following the stochastic process producing $(-\Delta)^{s}$ ends up in the region $A\subseteq\mathbb{R}^{n}$ at time $t$. In this sense, the quantity $q_{k}(s,\rho)$ in (7) represents the probability that this particle lies outside $B_{\rho}$ at time $k$.

Roughly speaking, for small $\rho$, a natural Ansatz is to assume these events to be more or less independent from each other: indeed, in view of (6), using the substitution $y:=x/\rho$ we have that

[TABLE]

representing the probability of a particle to lie outside $B_{1}$ at time $k/\rho^{2s}$. In view of this, since the time steps $k/\rho^{2s}$ are very separated from each other when $\rho$ is small, we may think that the quantity $q(s,\rho)$ in (9) is a good approximation of the probability that the particle does not lie in $B_{1}$ in all the time steps $k/\rho^{2s}$, as well as a good approximation of the probability that the particle does not lie in $B_{\rho}$ in all the time steps $k\in\{1,2,3,\dots\}$. In this heuristics, the case in which the quantity $q(s)$ in (9) is equal to [math] indicates that the particle will come back infinitely often to its original position at the origin in integer times (with probability $1$); conversely, the case in which the quantity $q(s)$ in (9) is equal to $1$ indicates that the particle will return to its original position at the origin in integer times only with probability zero.

In this sense, computing $q(s)$ gives an interesting indication of the recurrence properties of the associated stochastic process, and, in our case, this calculation can be performed as follows. First of all, we notice that

[TABLE]

thanks to (5), and

[TABLE]

As a consequence, recalling (4) and (6), if $\rho\in(0,1]$

[TABLE]

This gives that

[TABLE]

for some $C>c>0$, depending only on $n$ and $s$, and accordingly

[TABLE]

Also, for a fixed $C_{0}\in(0,+\infty)$, the convergence of the series

[TABLE]

can be reduced to that of the series

[TABLE]

and consequently

[TABLE]

for some $C_{1}>0$ depending only on $n$, $s$ and $C_{0}$. This and (10) lead to

[TABLE]

for some $C_{2}>C_{3}>0$ depending only on $n$ and $s$.

Therefore,

[TABLE]

Taking the limit as $\rho\to 0$, we thereby find that

[TABLE]

When $s=1$, we can write (11) as

[TABLE]

that is, in our framework, the classical random walk “comes back to the initial” point in dimensions $1$ and $2$, and it “drifts away forever” in dimension $3$ and higher.

When $s\in(0,1)$, the situation is different, since (11) produces the alternative

[TABLE]

That is, in our setting, the fractional random walk “comes back to the initial” point only in dimensions $1$ and only if the fractional parameter is above a certain threshold (namely $s\geqslant 1/2$). Conversely, the fractional random walk “drifts away for ever” already in dimension $2$, and even in dimension $1$ if the fractional parameter is too small (namely $s<1/2$).

See e.g. MR3738798 and the references therein for a comprehensive treatment of recurrence and transiency of general stochastic processes.

Acknowledgements.

This work has been supported by the Australian Research Council Discovery Project 170104880 NEW “Nonlocal Equations at Work”. Part of this work was carried out on the occasion of a very pleasant visit of the first author to the University of Western Australia, which we thank for the warm hospitality. The authors are members of INdAM/GNAMPA.