A simple flattening lower bound for solutions to some linear integrodifferential equations

TL;DR

This paper introduces a simple PDE-based method to establish lower bounds on the asymptotic behavior of solutions to certain linear integro-differential equations, aiding the understanding of nonlocal evolution dynamics.

Contribution

It presents a novel, straightforward PDE approach to derive lower bounds for solutions, complementing traditional heat kernel decay estimates in nonlocal equations.

Findings

Lower bounds for solutions established using PDE arguments

Applicable to jump diffusion processes with specific initial data

Provides a foundational step for studying invasion phenomena in nonlinear problems

Abstract

Estimates on the asymptotic behaviour of solution to linear integro-differential equations are fundamental in understanding the dynamics occuring in many nonlocal evolution problems. They are usually derived by using precise decay estimates on the heat kernel of the considered diffusion process. In this note, we show that for some generic jump diffusion and particular initial data, one can derive a lower bound of the asymptotic behaviour of the solution using a simple PDE argument. This is viewed as an independant preliminary brick to study invasion phenomena in nonlinear reaction diffusion problems.

Equations

$$\frac{\partial u}{\partial t}(t,x)=\mathcal{D}[u](t,x),t\in (0,+\infty ),x\in \mathbb{R},$$

$$\mathcal{D}[u](t,x):=\text{P.V.}\left({\int }_{\mathbb{R}}[u(t,x-z)-u(t,x)]J(z)dy\right).$$

$$u(0,x)=u_0(x),x\in \mathbb{R},$$

$${\int }_{|z|\le 1}J(z)|z{|}^{2}dz\le 2{\mathcal{J}}_1\text{ and }\frac{{\mathcal{J}}_0}{|z{|}^{1+2s}}\ge J(z)\text{mathds}{1}_{|z|>1}(z)\ge \frac{{\mathcal{J}}_0^{-1}}{|z{|}^{1+2s}}\text{mathds}{1}_{\{|z|\ge R_0\}}.$$

$$\forall t\in \in [0,+\infty ),\forall x\in \mathbb{R},u(t,x)={\int }_{\mathbb{R}}p(t,y)u_0(x-y)dy,$$

$$\frac{\partial p}{\partial t}(t,x)=\mathcal{D}[p](t,x),t\in (0,+\infty ),x\in \mathbb{R},$$

$$p(0,x)={\delta }_0(x),x\in \mathbb{R},$$

$$\forall t\in (0,+\infty ),\forall x\in \mathbb{R},\frac{{C_1}^{-1}}{t^{\frac{1}{2s}}[1+|t^{-\frac{1}{2s}}x{|}^{1+2s}]}\le p_s(t,x)\le \frac{C_1}{t^{\frac{1}{2s}}[1+|t^{-\frac{1}{2s}}x{|}^{1+2s}]}.$$

$$u(t,x)\ge {\int }_{\mathbb{R}}{u}_0(x-y)\frac{{C_1}^{-1}}{t^{\frac{1}{2s}}[1+|t^{-\frac{1}{2s}}y{|}^{1+2s}]}dy$$

$$={\int }_{t^{-\frac{1}{2s}}(x-b)}^{+\infty }\frac{a{C_1}^{-1}}{1+|z{|}^{1+2s}}dz$$

$$={\int }_{t^{-\frac{1}{2s}}(x-b)}^{+\infty }\left(\frac{a{C}_1^{-1}}{|z{|}^{1+2s}}-\frac{a{C_1}^{-1}}{(1+|z{|}^{1+2s})|z{|}^{1+2s}}\right)dz,$$

$$\forall t\in (0,+\infty ),\underset{x\to +\infty }{\lim }\frac{x^{2s}}{t}u(t,x)\ge \underset{x\to +\infty }{\lim }\frac{x^{2s}}{t}\left(\frac{ta{C_1}^{-1}}{(x-b{)}^{2s}}-{\int }_{t^{-\frac{1}{2s}}(x-b)}^{+\infty }\frac{a{C_1}^{-1}}{(1+|z{|}^{1+2s})|z{|}^{1+2s}}\right)=a{C_1}^{-1}.$$

$$\forall t\in (0,+\infty ),\underset{x\to +\infty }{\lim }{x}^{2s}u(t,x)\ge \kappa t.$$

$$\frac{\partial u}{\partial t}(t,x)=\mathcal{D}[u](t,x)+f(u(t,x)),t\in (0,+\infty ),x\in \mathbb{R},$$

$$\forall \theta \in (0,+\infty ),\forall t\in (0,+\infty ),\underset{x\to +\infty }{\lim }\frac{x^{2s}}{\theta t}\underline{u}(t,x)<+\infty .$$

$$\underset{x\to +\infty }{\lim }\frac{x^{2s}}{{\theta }_0{t}_0}\underline{u}(t_0,x)\le C_0.$$

$$\forall t\in (0,+\infty ),\underset{x\to +\infty }{\lim }\frac{x^{2s}}{t}u(t,x)\ge \kappa a,$$

$$\frac{\partial u}{\partial t}(t,x)\ge \mathcal{D}[u](t,x),$$

$$\frac{\partial v}{\partial t}(t,x)\le \mathcal{D}[v](t,x),$$

$$u(0,x)\ge v(0,x),$$

$$\frac{\partial u}{\partial t}(t,x)\ge \mathcal{D}[u](t,x),$$

$$\frac{\partial v}{\partial t}(t,x)\le \mathcal{D}[v](t,x),$$

$$u(t,x)>v(t,x),$$

$$u(t_0,x)\ge v(t_0,x),$$

$$\forall t\in [0,+\infty ),\forall x\in (-\infty ,b),u(t,x)>\frac{a}{2}.$$

$$\frac{\partial v}{\partial t}(t,x)=\mathcal{D}[v](t,x),t\in (0,+\infty ),x\in \mathbb{R},$$

$$v(0,x)=v_0(x):=a\rho \star \text{mathds}{1}_{(-\infty ,b]}(x),x\in \mathbb{R}.$$

$$\forall x\in \mathbb{R},v_0(x+b)+v_0(b-x)$$

$$=a\left({\int }_{-\infty }^0\rho (x-y)dy+{\int }_{-\infty }^0\rho (x+y)dy\right)$$

$$=a.$$

$$w(t,x)=\{\begin{array}{ll}\frac{1}{2}& t\in (0,+\infty ),x\in (-\infty ,0],\\ \frac{\kappa t}{x^{2s}+2\kappa t}& t\in (0,+\infty ),x\in (0,+\infty ),\end{array}$$

$$\mathcal{D}[w](t,x)$$

$$+{\int }_R^{+\infty }[w(t,x+z)-w(t,x)]J(z)dz$$

$$\ge {\int }_{-\infty }^{-R}[w(t,x+z)-w(t,x)]J(z)dz+{\int }_{-R}^R[w(t,x+z)-w(t,x)]J(z)dz-w(t,x){\int }_R^{+\infty }J(z)dz$$

$$={\int }_{-R}^R[w(t,x+z)-w(t,x)]J(z)dz+{\int }_{-\infty }^{-x}[w(t,x+z)-w(t,x)]J(z)dz$$

$$+{\int }_{-x}^{-R}[w(t,x+z)-w(t,x)]J(z)dz-w(t,x){\int }_R^{+\infty }J(z)dz$$

$$\ge {\int }_{-R}^R[w(t,x+z)-w(t,x)]J(z)dz+\left[\frac{1}{2}-w(t,x)\right]{\int }_x^{+\infty }J(z)dz-w(t,x){\int }_R^{+\infty }J(z)dz$$

$$\ge {\int }_{-R}^R[w(t,x+z)-w(t,x)]J(z)dz+\frac{{{\mathcal{J}}_0}^{-1}}{2s}\left[\frac{1}{2}-w(t,x)\right]\frac{1}{x^{2s}}-\frac{{\mathcal{J}}_0}{2s{R}^{2s}}w(t,x).$$

$$\forall t\in (0,+\infty ),\forall x\in [R_0+R,+\infty ),w(t,x+z)-w(t,x)=z{\int }_0^1{\partial }_{x}w(t,x+\tau z)d\tau ,$$

Full text

Keywords: integro-differential operators, fractional Laplace operator, acceleration, spreading.

1 Introduction

In this note, we are interested in asymptotic behaviour of the solution to a generic nonlocal integro-differential equation of the form:

[TABLE]

where ${\mathcal{D}}[\,{\cdot}\,]$ is a diffusion operator with a kernel of convolution type $J$, that is

[TABLE]

This equation is complemented by an initial condition

[TABLE]

to form an evolution problem. We assume that the initial data $u_{0}$ belongs to $\mathscr{C}(\mathbb{R},\mathbb{R}_{+})\cap L^{\infty}(\mathbb{R})$ and satisfies the following hypothesis.

Hypothesis 1.1**.**

There exist $a>0$ and $b\in\mathbb{R}$ such that $u_{0}\geq a\mathds{1}_{(-\infty,b]}$.

Morever, we assume the kernel $J$ is a nonnegative function, satisfying the following properties.

Hypothesis 1.2**.**

Let $s$ be a positive real number. The kernel $J$ is symmetric and such that there exist positive constants ${\mathcal{J}}_{0},{\mathcal{J}}_{1}$ and $R_{0}$, the latter being larger than $1$, such that

[TABLE]

The operator ${\mathcal{D}}[\,{\cdot}\,]$ can be seen as the infinitesimal generator of a generic symmetric jump process [3], and appears, for instance, in population dynamics where it describes the dispersion of individuals of a population modelled by the density $u$. Roughly speaking, the value $J(z)$ represents the probability of a jump of size $z$, which makes the tails of the convolution kernel of crucial importance in quantifying the dynamics. One may readily notice that the assumptions on $J$ allow to cover the two broad types of integro-differential operators usually considered in the literature, which are the fractional Laplace operator $(-\Delta)^{s}u$ and a standard convolution operator with integrable kernel, often written $J\star u-u$, respectively.

The characterisation of the asymptotic behaviour of solutions to linear diffusion problems such as (1.1)-(1.2) is a classical question, which can be answered for particular classes of Levy processes through the time and space scaling properties of their associated heat kernel, see for instance [13, 2] for the fractional Laplace operator and [1, 3, 7, 8, 9, 10, 11] in the case of more general Levy processes. Indeed, when such a heat kernel exists, the solution $u$ to (1.1)-(1.2) is given by

[TABLE]

where $p$ is the solution to

[TABLE]

where $\delta_{0}$ is the Dirac delta distribution. It then follows that the asymptotics of $u$ can be derived from the time and space scaling properties of $p$. For example, for the fractional Laplace operator $(-\Delta)^{s}$, it is well known ([2, 6]) that the heat kernel $p_{s}$ satisfies, for some positive constant $C_{1}$, the following scalings

[TABLE]

As a consequence, a solution to (1.1)-(1.2) with a fractional Laplace operator satisfies

[TABLE]

which, in particular, provides the following estimate:

[TABLE]

It is expected that an analogous estimate holds for solutions of (1.1)-(1.2) when ${\mathcal{D}}[\,{\cdot}\,]$ is a generic operator whose kernel satisfies Hypothesis 1.2, and we shall here obtain such a flattening estimate directly from the problem, without any further restriction on the considered Lévy measure other than those given in Hypothesis 1.2 or any knowledge of the associated heat kernel. Let us state precisely this result.

Theorem 1.3**.**

Assume that the kernel $J$ and the initial datum $u_{0}$ satisfy Hypothesis 1.2 and 1.1, respectively. Then, there exists a constant $\kappa$, depending on $J$ and $u_{0}$, such that the solution $u$ to (1.1)-(1.2) has the following asymptotic flattening behaviour at infinity:

[TABLE]

One of the main applications of the above type of estimate arises naturally in the study of propagation phenomena described by some semi-linear equation of the form

[TABLE]

where $f$ is a nonlinearity describing the local dynamics of the modelled system. On way to capture both the evolution and speed of transition in the resulting problem is by means of the construction of super- and sub-solution that mimic the essential features (reaction and dispersal) of the system. The natural time and space scalings of the equation play an important role in these constructions. When the nonlinearity $f$ is nonnegative, the solution to (1.1)-(1.2) is a trivial sub-solution to the problem, giving rise to a first lower bound on the decay of the solution. In order to achieve a more detailed description of the dynamics, a more sophisticated sub-solution needs to be constructed and the estimate obtained in Theorem 1.3 simplifies such a construction by first deriving the natural time and space scalings of the solution of the semi-linear equation and next allowing to compare this solution with a sub-solution via the use of a parabolic comparison principle.

For instance, when the non-linearity is monostable with, for instance, [math] and $1$ as equilibria, i.e., $f(0)=f(1)=0$, one can try to understand the dynamics of the solution to (1.3)-(1.2) by considering a level set of height $\lambda$ in $(0,1)$ and a sub-solution $\underline{u}$ such that $\underline{u}\leq\lambda$ and satisfying the following decay at infinity:

[TABLE]

If such a sub-solution exists, then, for a positive real number $\theta_{0}$ and a positive time $t_{0}$, there exists a positive constant $C_{0}$ such that

[TABLE]

From Theorem 1.3, one also has

[TABLE]

so that there exists a positive real number $t^{\prime}$ such that $\kappa at^{\prime}>2\theta_{0}t_{0}C_{0}$. It follows that $\underline{u}(t_{0},x)\leq u(t^{\prime},x)$ for $x$ large enough, say $x>x_{0}>0$ and, using Theorem 1.3, one has $\underline{u}(t_{0},x)\leq u(t,x)$ for $t\geq t^{\prime}$ and $x>x_{0}>0$. Combining this with the invasion property usually satisfied by the solution $u$ in such context, i.e. a property asserting that $u(t,x)$ tends to $1$ as $t$ tends to infinity and uniformly in $x$ in $(-\infty,A]$ for any real number $A$, we may find a time $t^{\prime\prime}>t^{\prime}$ such that $u(t^{\prime\prime},x)\geq\underline{u}(t_{0},x)$ for $x\in\mathbb{R}$. Due to a parabolic comparison principle, it follows that $u(t^{\prime\prime}+t-t_{0},x)\geq\underline{u}(t,x)$ for $t\geq t_{0}$ and $x\in\mathbb{R}$, implying that the level set of height $\lambda$ of the solution travels at a speed at least equal to that of the sub-solution.

The present note is organised as follows. First, some comparison principles are recalled and a useful a priori bound is derived. The argument needed to prove Theorem 1.3 is next developed.

2 Preliminaries

Let us start by recalling the different comparison principles that we will use throughout this note.

Theorem 2.1** (standard comparison principle).**

Assume that the kernel $J$ satisfies Hypothesis 1.2 and let $u$ and $v$ be two functions in $\mathscr{C}^{1}(\mathbb{R}_{+}^{*},\mathscr{C}(\mathbb{R}))\cap\mathscr{C}(\mathbb{R}_{+},\mathscr{C}(\mathbb{R}))$, satisfying, for some positive real number $T$,

[TABLE]

Then, one has $u(t,x)\geq v(t,x)$ for $t\in(0,T)$, $x\in\mathbb{R}$.

The second comparison principle works with a subset in space.

Theorem 2.2** (adapted comparison principle).**

Assume that the kernel $J$ satisfies Hypothesis 1.2 and let $u$ and $v$ be two functions in $\mathscr{C}^{1}(\mathbb{R}_{+}^{*},\mathscr{C}(\mathbb{R}))\cap\mathscr{C}(\mathbb{R}_{+}^{*},\mathscr{C}(\mathbb{R}))$, satisfying, for some $0\leq t_{0}<t_{1}$ and $R_{0}\in\mathbb{R}$,

[TABLE]

Then, one has $u(t,x)\geq v(t,x)$ for $t\in(t_{0},t_{1})$, $x\in\mathbb{R}$.

The proofs of Theorem 2.1 and Theorem 2.2 are rather standard and will be omitted here, but the reader can refer to [5, 14] for some ideas.

Some a priori estimates on the solution to (1.1)-(1.2) are also needed.

Proposition 2.3**.**

Assume that the kernel $J$ and the initial datum $u_{0}$ satisfy Hypothesis 1.2 and 1.1, respectively. Let $u$ be a positive solution to (1.1)-(1.2). Then, one has

[TABLE]

Proof.

Let $\rho$ be a smooth symmetric mollifier of unit mass, i.e., a nonnegative even function $\rho$ in $\mathscr{C}_{c}^{\infty}(\mathbb{R})$ such that $\int_{\mathbb{R}}\rho(z)\,dz=1$, and consider the solution $v$ to the problem:

[TABLE]

Since by construction $u_{0}\geq a\mathds{1}_{(-\infty,b]}$ and is continuous, we may assume that, up to a rescaling of $\rho$, $u_{0}\geq v_{0}$. Therefore, by the comparison principle in Theorem 2.1, we have $u(t,x)\geq v(t,x)$ for $t\geq 0$ and $x\in\mathbb{R}$. Moreover since $v_{0}$ is monotone non increasing and equation (2.5) is invariant under translation in space, we deduce that, for $t>0$, the function $v(t,\cdot)$ is monotone decreasing.

Let us now observe that the function $\bar{v}(t,x):=v(t,x+b)+v(t,b-x)$ satisfies equation (2.5), with the initial condition $\bar{v}(0,\cdot)=v_{0}(x+b)+v_{0}(b-x)$. By a straightforward change of variables, we have

[TABLE]

By the uniqueness of the mild solution of the initial value problem (see for instance Theorem 4.3 in [12]), this implies that $\bar{v}\equiv a$. As a consequence, we deduce that $2\,v(t,b)=a$ for $t>0$ and, since $v(t,\cdot)$ is decreasing, we have $v(t,x)>\frac{a}{2}$ for $t>0$ and $x<b$, thus ending the proof. ∎

Remark 2.4**.**

When the semigroup generated by ${\mathcal{D}}[\,{\cdot}\,]$ is regularising, i.e. the solution $u$ belongs to $\mathscr{C}^{1}(\mathbb{R}_{+}^{*},\mathscr{C}(\mathbb{R}_{+}))$ whenever $u_{0}$ is in $L^{\infty}(\mathbb{R})$, the above argument holds for initial date that are step functions, e.g. $v(0,\cdot)=a\mathds{1}_{(-\infty,b]}$. This is not necessarily the case for non-regularising semigroups such as those related to a convolution operator with a continuous integrable kernel. In such situations, the regularity of $v$ is the same as of $u_{0}$ and $v$ has a jump discontinuity at the point $b$, preventing an evaluation $\bar{v}$ at this point.

Remark 2.5**.**

Note that the proof of the above estimate relies solely on an elementary use of the comparison principle for the evolution problem. It is valid, in full generality, as soon as the considered semigroup possesses some basic properties, such as mapping a continuous function to another one, satisfying a comparison principle and having a translation invariant infinitesimal generator. In particular, it holds true for semigroups generated by operators satisfying the Bony–Courrège–Priouret maximum principle [4], characterised by an elliptic part and a Lévy-type part, the latter being associated with a symmetric Lévy measure, that is a nonnegative, nonzero measure $\nu$ on $\mathbb{R}$, satisfying $\nu(\{0\})=0$, $\nu(-A)=\nu(A)$ for every Borel set in $\mathbb{R}$, and $\int_{\mathbb{R}}\min\{1,z^{2}\}\nu(dz)<+\infty$.

3 Proof of Theorem 1.3

Having this preliminary estimate at hand, we can now prove Theorem 1.3. Our strategy is to construct an adequate subsolution. Let $w$ be the parametric function defined by

[TABLE]

with $\kappa=\frac{{{\mathcal{J}}_{0}}^{-1}}{8s}$.

Let us estimate ${\mathcal{D}}[\,{w}\,]$. Let $R>1$ to be chosen later. Since $J$ satisfies Hypothesis 1.2, we have by a direct computation, for $t\in(0,+\infty)$ and $x\in[R_{0}+R,+\infty)$,

[TABLE]

The remaining integral can be estimated using the regularity and the convexity with respect to space of $w$, together with the symmetry of $J$. Indeed, since $w(t,\cdot)$ belongs to $\mathscr{C}^{1}(\mathbb{R}_{+})$ for $t>0$, we have, for $x\geq R_{0}+R$,

[TABLE]

and thus

[TABLE]

The kernel $J$ being symmetric, we have

[TABLE]

which can be rewritten as

[TABLE]

using that $w(t,\cdot)$ belongs to $\mathscr{C}^{2}(\mathbb{R}_{+})$ for $t>0$. Since $w$ is convex with respect to space in $\mathbb{R}_{+}$, the latter integral is positive and we get

[TABLE]

Altogether, we then have for $t>0$ and $x\geq R_{0}+R$,

[TABLE]

which, by using the definition of $w$, yields

[TABLE]

Since $\kappa=\frac{{{\mathcal{J}}_{0}}^{-1}}{8s}$, we end up with

[TABLE]

For any $C>0$, let us define $t^{*}:=\frac{2C}{\kappa}$ and choose $R$ large enough says $R\geq R_{C}:=\left(8C{{\mathcal{J}}_{0}}^{2}\right)^{\frac{1}{2s}}$. From the above computations, it then follows that

[TABLE]

Equipped with this subsolution, let us now conclude. Using Proposition 2.3, there exists $a\in\mathbb{R}_{+}^{*}$ and $b\in\mathbb{R}$ such that $u(t,x)>\frac{a}{2}$ for $t>0$ and $x<b$. By definition of $w$, we thus have $u(t,x-R_{0}-R_{C}-b)>\frac{a}{2}\geq aw(t,x)$ for $t>0$ and $x\leq R_{0}+R_{C}$. Therefore, setting $\tilde{u}(t,x):=u(t,x-R_{0}-R_{C}-b)$, we have

[TABLE]

Using Theorem 2.2, it follows that for $t\in(0,t^{*})$ and $x\in\mathbb{R}$, one has $\tilde{u}(t,x)\geq aw(t,x)$ and therefore, by the definition of $t^{*}$,

[TABLE]

It follows that

[TABLE]

or, equivalently, that, for all positive real number $C$,

[TABLE]

This implies that

[TABLE]

thus ending the proof.