Evolution driven by the infinity fractional Laplacian

Abstract

We consider the evolution problem associated to the infinity fractional Laplacian introduced by Bjorland et al. (Adv Math 230(4–6):1859–1894, 2012) as the infinitesimal generator of a non-Brownian tug-of-war game. We first construct a class of viscosity solutions of the initial-value problem for bounded and uniformly continuous data. An important result is the equivalence of the nonlinear operator in higher dimensions with the one-dimensional fractional Laplacian when it is applied to radially symmetric and monotone functions. Thanks to this and a comparison theorem between classical and viscosity solutions, we are able to establish a global Harnack inequality that, in particular, explains the long-time behavior of the solutions.

1 Introduction

In this paper we study a parabolic equation associated to the (normalized) infinity fractional Laplacian operator. We recall that the local version of the game had been introduced by Peres et al. in 2009 [33] where it is shown that the standard infinity Laplace equation is solved by the value function for a two-players random turn “tug-of-war” game. The game is as follows: a token is initially placed at a position $x_0\in \Omega$ and every turn a fair coin is tossed to choose which of the players plays. This player moves the token to any point in the ball of radius $\varepsilon >0$ around the current position. If, eventually, iterating this process, the token reaches a point $x_e\in \partial \Omega$, the players are awarded (or penalized) $f(x_e)$ (payoff function). For a PDE overview of the infinity Laplacian operator and its role as an absolute minimizer for the $L^\infty$ norm of the gradient, see [26, 27].

In 2012 Bjorland et al. [6] introduced equations involving the so-called infinity fractional Laplacian as a model for a nonlocal version of the “tug-of-war” game. Following their explantation, instead of flipping a coin at every step, every player chooses a direction and it is an s-stable Lévy process that chooses both the active player and the distance to travel. The infinity fractional Laplacian, with symbol $\Delta _{\infty }^{s}$, is a nonlinear integro-differential operator, the original definition is given in Lemma 2.1 below. However, for the purpose of this paper, we also consider the alternative equivalent definition introduced in [6] (see also [16]) given by

$$\begin{aligned} \Delta _{\infty }^{s}\phi (x):= C_s \sup _{|y|=1}\inf _{|\tilde{y}|=1} \int _{0}^\infty \left( \phi (x+\eta y)+\phi (x-\eta \tilde{y})-2\phi (x)\right) \frac{\mathop {}\!\textrm{d}{\eta }}{\eta ^{1+2s}}\qquad \text {where}\; s\in (1/2,1).\nonumber \\ \end{aligned}$$

(1.1)

The constant is usually taken as $C_s=(4^ss\Gamma (\frac{1}{2}+s))/(\pi ^{\frac{1}{2}}\Gamma (1-s))$ but the value is irrelevant for our results. In their paper [6] the authors study two stationary problems involving the infinity fractional Laplacian posed in bounded space domains, namely, a Dirichlet problem and a double-obstacle problem.

Here, we consider the evolution problem

with $s\in (1/2,1)$ and $n\ge 2$. When $n=1$ the operator $- \Delta _{\infty }^{s}$ is just the usual linear fractional Laplacian operator $(-\Delta )^s$ of order s, and Eq. (1.2) is just the well-known fractional heat equation [7, 23]. See also a detailed study of that equation using PDE techniques in [3, 8, 20, 38]. Note that for $n\ge 2$ the operator is nonlinear so a new theory is needed. A non-normalized version is introduced in [12] along with a well-posedness theory for the corresponding equations of the type (1.2)–(1.3). However, the two problems are not equivalent nor closely related.

The local counterpart of (1.2), given by $\partial _t u= \Delta _\infty u$, is studied in [21], where the authors present several applications such as image processing [10] or tug-of-war games over a finite time horizon [4]. In this case, the game finishes when the terminal time is reached, instead of when the boundary of the domain is reached. In the nonlocal case treated in the present paper, the application to image processing has not been described to our knowledge. However, a derivation similar to the one in [4], based on dynamic programming, seems to be reasonable.

Here, we develop an existence theory of suitable viscosity solutions for the parabolic problem (1.2)–(1.3), based on approximation with monotone schemes. We show that the obtained class of solutions enjoys a number of good properties. As in the elliptic case [6], we lack a uniqueness result in the context of viscosity solutions. However, we are able to prove an important comparison theorem relating two types of solutions, classical and viscosity ones, see Theorem 2.6. As a counterpart, we also obtain uniqueness and comparison of classical solutions. Moreover, we show that for smooth, radially symmetric functions which are nonincreasing along the radius in ${\mathbb {R}^n}$ with $n\ge 2$, the operator $-\Delta _{\infty }^{s}$ reduces to the classical fractional Laplacian $(-\Delta )^s$ in dimension $n=1$ (Theorem 6.1). A similar example regarding nondecreasing one-dimensional profiles can be found in Lemma 6.3. In this way we may construct a large class of classical solutions that make the comparison theorem relevant (Theorems 2.10, 2.12). Note that no similar reduction to a lower dimensional problem applies in general, even in the radial case (see Sect. 6.2 for a counterexample).

Using the developed tools, we study the asymptotic behavior of the constructed solutions, and obtain a global Harnack type principle, see Theorem 2.13.

1.1 Related literature

It is interesting to compare the nonlocal model (1.2) with the local version of the infinity Laplacian that has been studied by many authors, both in the stationary and evolution cases, cf. [1, 2, 21, 26, 27, 33,34,35]. Asymptotic expansions for the game theoretical p-Laplacian in the local case and related approximation schemes in the elliptic case are studied in [19, 29, 30] and in the parabolic case in [28]. For the variational version of the p-Laplacian operator see [17].

There exist in the literature other nonlocal generalizations of the p-Laplacian and the infinity Laplacian. Let us mention (i) the normalized version [5, 6] with asymptotic expansions and game theoretical approach [9, 16, 25]; (ii) nonnormalized version [12] both elliptic and parabolic; (iii) Hölder infinity Laplacian [11]; and (iv) the (variational) fractional p-Laplacian [13, 14, 31, 32, 36, 37, 39].

2 Preliminaries and statement of main results

First let us fix some notation that we will use along the paper.

For given $\delta >0$, standard mollifiers are denoted by $\rho _\delta$. Following [6], we say that $\phi \in C^{1,1}(x)$ at some $x\in {\mathbb {R}^n}$ if there exists $p_x\in {\mathbb {R}^n}$ and $C_x,\eta _x>0$ such that

$$\begin{aligned} |\phi (x+y)-\phi (x)-p_x\cdot y| \le C_x |y|^2\qquad \text {for all }\; |y|<\eta _x. \end{aligned}$$

(2.1)

Note that $C_b ^2(B_R(x))\subset C^{1,1}(x)$. Here $C_{b }^k(U)$ is the space of functions on the set U with bounded continuous derivatives of all orders in [0, k]. Let us also define:

$$\begin{aligned} B({\mathbb {R}^n})&:=\{\phi : {\mathbb {R}^n}\rightarrow \mathbb {R}\,|\, \phi \text {is pointwisely bounded}\},\\ UC({\mathbb {R}^n})&:=\{\phi : {\mathbb {R}^n}\rightarrow \mathbb {R}\,|\, \phi \text {is uniformly continuous}\},\\ BUC({\mathbb {R}^n})&:=B({\mathbb {R}^n})\cap UC({\mathbb {R}^n})\qquad with \qquad \Vert \phi \Vert _{C_b ({\mathbb {R}^n})}:=\sup _{x\in {\mathbb {R}^n}}|\phi (x)|, \end{aligned}$$

and for $\beta \in (0,1]$, we define $|\phi |_{_{C^{0,\beta }}({\mathbb {R}^n})}=\sup _{x,y\in {\mathbb {R}^n}}|\phi (x)-\phi (y)|/|x-y|^\beta$ and

$$\begin{aligned} C^{0,\beta }({\mathbb {R}^n})&:=\{\phi \in C_{b }({\mathbb {R}^n}) \,|\, \Vert \phi \Vert _{C^{0,\beta }}<\infty \} \qquad \text {where}\qquad \Vert \phi \Vert _{C^{0,\beta }}=\Vert \phi \Vert _{C_{b }}+|\phi |_{C^{0,\beta }}. \end{aligned}$$

A modulus of continuity is a nondecreasing function $\omega :\mathbb {R}^+\rightarrow \mathbb {R}^+$ such that $\lim _{r\rightarrow 0^+}\omega (r)=0$. For a function $f\in BUC({\mathbb {R}^n})$, we define the corresponding modulus of continuity as follows:

$$\begin{aligned} \omega _f(r)= \sup _{|y|\le r}\Vert f(\cdot +y)-f\Vert _{C_b ({\mathbb {R}^n})}. \end{aligned}$$

For a Hölder continuous function $f\in C^{0,\beta }({\mathbb {R}^n})$, $\omega _f(r)\le |f|_{C^{0,\beta }}r^\beta$.

We will also need $e_i:=(0,0,\ldots ,0,1,0,\ldots ,0)\in {\mathbb {R}^n}$, where 1 is at the ith component, and denote average integrals by

2.1 Alternative characterization of the infinity fractional Laplacian

We have the following alternative characterization of operator $\Delta _{\infty }^{s}$ that we will use throughout:

Lemma 2.1

(Alternative characterization) Assume $\phi \in C^{1,1}(x)\cap B({\mathbb {R}^n})$. Then:

• If $\nabla \phi (x)\not =0$, then

  $$\begin{aligned} \Delta _{\infty }^{s}\phi (x)=C_s\int _0^\infty \big (\phi \left( x+ \eta \zeta \right) +\phi \left( x- \eta \zeta \right) -2\phi (x)\big )\frac{\mathop {}\!\textrm{d}{\eta }}{ \eta ^{1+2s}}\qquad \text {where}\; \zeta :=\nabla \phi (x)/|\nabla \phi (x)|. \end{aligned}$$

• If $\nabla \phi (x)=0$, then

  $$\begin{aligned} \Delta _{\infty }^{s}\phi (x)= & {} C_s\sup _{|y|=1} \int _{0}^\infty \big (\phi (x+ \eta y)-\phi (x)\big ) \frac{\mathop {}\!\textrm{d}{\eta }}{\eta ^{1+2s}}\\{} & {} + C_s\inf _{|y|=1} \int _{0}^\infty \big (\phi (x-\eta y)-\phi (x)\big ) \frac{\mathop {}\!\textrm{d}{\eta }}{\eta ^{1+2s}}. \end{aligned}$$

The equivalence when $\nabla \phi (x)=0$ follows from the fact that the integrals in this case are well-defined and can be combined to get (1.1). When $\nabla \phi (x)\ne 0$, it can be shown that the supremum and infimum of (1.1) is actually taken at $\zeta$, see Proposition 2.2 in [16]. To sketch the proof, assume for simplicity that the supremum in (1.1) is taken at y, and let us argue that $y=\zeta$. Indeed, by splitting the integral and using the definitions of $C^{1,1}$ and the infimum,

$$\begin{aligned} \Delta _{\infty }^{s}\phi (x)\le & {} C_s\int _0^\infty \big (\phi \left( x+\eta y\right) +\phi \left( x-\eta \zeta \right) -2\phi (x)\big )\frac{\mathop {}\!\textrm{d}{\eta }}{\eta ^{1+2s}}\\\le & {} C_s\big (\nabla \phi (x)\cdot (y-\zeta )\big )\int _0^{\eta _x}\eta \frac{\mathop {}\!\textrm{d}{\eta }}{\eta ^{1+2s}}+C. \end{aligned}$$

Now, since $\Delta _{\infty }^{s}\phi (x)$ is well-defined and the integral diverges if $y\not =\zeta$, we must have $y=\zeta$. A similar argument holds for the infimum.

2.2 Existence of solutions and basic properties

We are able to construct a suitable class of viscosity solutions of (1.2)–(1.3). The two steps are as follows:

(i) Approximating $\Delta _{\infty }^{s}$ by removing the singularity, i.e., we introduce

$$\begin{aligned} \mathcal {L}_\varepsilon [\phi ](x):=C_s \sup _{|y|=1}\inf _{|\tilde{y}|=1}\int _\varepsilon ^\infty \big (\phi (x+\eta y)+\phi (x-\eta \tilde{y})-2\phi (x)\big )\frac{\mathop {}\!\textrm{d}{\eta }}{\eta ^{1+2s}}. \end{aligned}$$

(ii) Discretizing in time by letting $\tau >0$ and $t_j:=j\tau$ for $j\in {{\,\mathrm{{{\mathbb {N}}}}\,}}$, i.e., $t_j\in \tau {{\,\mathrm{{{\mathbb {N}}}}\,}}$, and then considering the semidiscrete problem

We study the properties of (2.2)–(2.3) in Sect. 3. Existence of viscosity solutions follows by taking the limit in this approximate scheme, as well as properties inherited from the approximations.

Theorem 2.2

(Existence and a priori results) If $u_0\in BUC({\mathbb {R}^n})$, then there is at least one viscosity solution $u\in C_{b }({\mathbb {R}^n}\times [0,\infty ))$ of (1.2)–(1.3). Moreover:

• (a) ($C_b$-bound) For all $t>0$, $\Vert u(\cdot ,t)\Vert _{C_b ({\mathbb {R}^n})}\le \Vert u_0\Vert _{C_b ({\mathbb {R}^n})}$.

• (b) (Uniform continuity in space) For all $y\in {\mathbb {R}^n}$ and all $t>0$,    $\Vert u(\cdot +y,t)-u(\cdot ,t)\Vert _{C_b ({\mathbb {R}^n})} \le \omega _{u_0}(|y|).$

• (c) (Uniform continuity in time) For all $t,\tilde{t}>0$,    $\Vert u(\cdot ,t)-u(\cdot ,\tilde{t})\Vert _{C_b ({\mathbb {R}^n})}\le \tilde{\omega }(|t-{\tilde{t}}|)\qquad \text {where}\; \tilde{\omega }(r):=\inf _{\delta>0}\Big \{\omega _{u_0}(\delta )+r\sup _{\varepsilon >0}\Vert \mathcal {L}_\varepsilon [u_{0,\delta }]\Vert _{C_b ({\mathbb {R}^n})}\Big \}$ is a modulus satisfying $\tilde{\omega }(r)\le \omega _{u_0}(r^{1/3}) +C\big (r^{1/3}+r\big )$, $C:=c_s\Vert u_0\Vert _{C_b ({\mathbb {R}^n})}\Vert \nabla \rho \Vert ^{2-2s}_{L^1({\mathbb {R}^n})}\Vert D^2\rho \Vert _{L^1({\mathbb {R}^n})}^{2s-1}$, and $\rho$ is a standard mollifier.

Remark 2.3

The definition of viscosity solutions is given Sect. 4 (Definition 4.3). We obtain viscosity solution as limits of monotone approximations of the problem in Sect. 3.

Note that if $u_0$ is Hölder continuous and $s\in (1/2,1)$, then the above modulii will be (more) explicit.

Lemma 2.4

If $u_0\in C^{0,\beta }({\mathbb {R}^n})$ for $\beta \in (0,1]$, then

$$\begin{aligned} \omega _{u_0}(\delta )=|u_0|_{C^{0,\beta }}\delta ^\beta \qquad \text {and}\qquad \Vert \mathcal {L}_\varepsilon [u_{0,\delta }]\Vert _{C_b ({\mathbb {R}^n})}\le c(s,\rho ) |u_0|_{C^{0,\beta }} \delta ^{\beta -2s}. \end{aligned}$$

The above result will be proved at the end of Sect. 4.

It follows after a minimization in $\delta$ that $\tilde{\omega }(r)=c(s,\rho )|u_0|_{C^{0,\beta }}r^{\frac{1}{2s}}$, and the solution u will be Hölder continuous with the correct parabolic regularity.

Corollary 2.5

(Existence and a priori results) If $u_0\in C^{0,\beta }({\mathbb {R}^n})$ for $\beta \in (0,1]$, then there is at least one viscosity solution $u\in C_{b }({\mathbb {R}^n}\times [0,\infty ))$ of (1.2)–(1.3). Moreover:

• (a) ($C_b$-bound) For all $t>0$ $\Vert u(\cdot ,t)\Vert _{C_b ({\mathbb {R}^n})}\le \Vert u_0\Vert _{C_b ({\mathbb {R}^n})}$.

• (b) (Hölder in space) For all $y\in {\mathbb {R}^n}$ and all $t>0$,    $\Vert u(\cdot +y,t)-u(\cdot ,t)\Vert _{C_b ({\mathbb {R}^n})}\le |u_0|_{C^{0,\beta }}|y|^\beta .$

• (c) (Hölder in time) There is a constant $c(s,\rho )$ only depending on s and $\rho$ such that for all $t,\tilde{t}>0$,    $\Vert u(\cdot ,t)-u(\cdot ,\tilde{t})\Vert _{C_b ({\mathbb {R}^n})}\le C |u_0|_{C^{0,\beta }}|t-{\tilde{t}}|^{\frac{\beta }{2\,s}}$.

2.3 Classical solutions, radial solutions, comparison, and uniqueness

There could be other ways of obtaining viscosity solutions, and unfortunately, we lack general comparison and uniqueness results. Nevertheless, we can obtain that classical solutions are unique and we can compare our constructed viscosity solutions with classical sub- and supersolutions of (1.2)–(1.3).¹

Theorem 2.6

(Comparison between classical and viscosity solutions) Assume $u_0\in BUC({\mathbb {R}^n})$. Let $\underline{u},\overline{u}\in C^2_b ({\mathbb {R}^n}\times [0,\infty ))$ be respective classical sub- and supersolution of (1.2)–(1.3), and let $u\in BUC({\mathbb {R}^n}\times [0,\infty )$ be a viscosity solution of (1.2)–(1.3) as constructed in Theorem 2.2. Then $\underline{u} \le u \le \overline{u}$ in ${\mathbb {R}^n}\times (0,\infty )$.

The above result is proved in Sect. 7. We want to emphasize that it is done in a rather nonstandard way, since we inherit the comparison from the approximation scheme when the solution is classical. In general, this cannot be done in the context of viscosity solutions since the approximation scheme only converges up to a subsequence.

Remark 2.7

By Theorem 2.6, we can in addition get comparison of constructed viscosity solutions as long as the initial datas are separated by an initial data which produces a classical solution.

An immediate consequence of Theorem 2.6:

Corollary 2.8

(Comparison of classical sub- and supersolutions) Let $u,v\in C^2_b ({\mathbb {R}^n}\times [0,\infty ))$ be respective classical sub- and supersolutions of (1.2)–(1.3) with initial data $u_0,v_0$. If $u_0\le v_0$, then $u\le v$.

Corollary 2.9

(Uniqueness of solutions) Classical solutions of (1.2)–(1.3) in $C^2_b ({\mathbb {R}^n}\times [0,\infty ))$ are unique.

Theorem 2.6 might be an empty statement unless we provide a class of classical solutions of (1.2)–(1.3). The following result, proved in Sect. 6, solves this issue.

Theorem 2.10

(Existence of classical radial solutions) Assume that $u_0\in C_b ^\infty ({\mathbb {R}^n})$ is radial and radially nonincreasing. Then there exists a classical and radial solution $u\in C^\infty _b ({\mathbb {R}^n}\times [0,\infty ))$ of (1.2)–(1.3). Moreover, if $U_0(r):=u_0(|x|)$ and $U_0(-r):=U_0(r)$ for $r=|x|\ge 0$, then

$$\begin{aligned} u(x,t)= (P_{s}(\cdot ,t)*U_0)(r)= \int _{-\infty }^\infty P_{s}(r-s,t) U_0(s)\mathop {}\!\textrm{d}{s} \qquad \text {for all}\; |x|=r, \end{aligned}$$

where $P_{s}$ is the fundamental solution of the one-dimensional fractional heat equation (cf. 5.3).

Remark 2.11

• (a) The idea in the above result is that, for radially nonincreasing radial functions, the operators $-\Delta _{\infty ,\mathbb {R}^n}^s$ and $(-\Delta )^s_{\mathbb {R}^1}$ coincide (Proposition 6.1), and (1.2) then reduces to the one-dimensional fractional heat equation.

• (b) In view of Theorem 2.6, this classical solution is also a viscosity solution in our sense.

Another class of classical solutions are:

Theorem 2.12

(Existence of classical solutions with one-dimensional profiles) Assume that $U_0\in C^\infty _{b }(\mathbb {R})$ is nondecreasing, and let $u_0\in C^\infty _{b }({\mathbb {R}^n})$ be defined as

$$\begin{aligned} u_0(x):=U_0(x_1). \end{aligned}$$

Then there exists a classical solution $u\in C^\infty _b ({\mathbb {R}^n}\times [0,\infty ))$ of (1.2)–(1.3). Moreover,

$$\begin{aligned} u(x,t)= (P_{s}(\cdot ,t)*U_0)(x_1)= \int _{-\infty }^\infty P_{s}(x_1-s,t) U_0(s)\mathop {}\!\textrm{d}{s}, \end{aligned}$$

where $P_{s}$ is the fundamental solution of the one-dimensional fractional heat equation (cf. (5.3)).

The proof is similar to the one of Theorem 2.10, and we will omit it. One just needs to note that $P_{s}(\cdot ,t)*U_0$ is nondecreasing.

2.4 Asymptotic behavior and Harnack inequality

Having established Theorems 2.6 and 2.10, we can prove that solutions of (1.2)–(1.3) behave like solutions of the one-dimensional fractional heat equation, up to suitable constants. In Sect. 5, we recall some results on that equation and its fundamental solution denoted by $P_s$. In Sect. 8, we prove the following result.

Theorem 2.13

(Global Harnack principle) Let $u\in BUC({\mathbb {R}^n}\times [0,\infty ))$ be a viscosity solution of (1.2)–(1.3), as constructed in Theorem 2.2, with initial data $u_0\in BUC({\mathbb {R}^n})$ such that $u_0\not \equiv 0$ and

$$\begin{aligned} 0\le u_0(x)\le (1+|x|^2)^{-\frac{1+2s}{2}} \qquad \text {for all}\; |x|\ge R\ge 1. \end{aligned}$$

Then, for all $\tau >0$, there exist constants $C_1,C_2>0$ depending only on s, R, and $u_0$, such that

$$\begin{aligned} C_1 P_s(|x|,t) \le u(x,t) \le C_2 P_s(|x|,t)\qquad \text {for all}\; (x,t)\in {\mathbb {R}^n}\times [\tau ,\infty ). \end{aligned}$$

Moreover, for all $\tau >0$, there exist constants $\tilde{C}_1,\tilde{C}_2>0$ depending only on s, R, and $u_0$, such that

$$\begin{aligned} \tilde{C}_1\frac{t}{(t^{\frac{1}{s}}+|x|^2)^{\frac{1+2s}{2}}} \le u(x,t) \le \tilde{C}_2\frac{t}{(t^{\frac{1}{s}}+|x|^2)^{\frac{1+2s}{2}}}\qquad \text {for all}\; (x,t)\in {\mathbb {R}^n}\times [\tau ,\infty ). \end{aligned}$$

In particular, $u>0$ in ${\mathbb {R}^n}\times [\tau ,\infty )$.

Remark 2.14

• (a) Note that $u_0$ is not necessarily in $L^1({\mathbb {R}^n})$ since the decay required for large x is the one corresponding to the one-dimensional fractional heat kernel $P_s$.

• (b) The above theorem provides a counterexample to conservation of mass for (1.2)–(1.3): For any smooth compactly supported $0\le u_0\in L^1({\mathbb {R}^n})$, the corresponding solution u satisfies

  $$\begin{aligned} \int _{{\mathbb {R}^n}} u(x,1) \mathop {}\!\textrm{d}{x} \ge \tilde{C}_1 \int _{{\mathbb {R}^n}}\frac{1}{(1+|x|^2)^{\frac{1+2s}{2}}} \mathop {}\!\textrm{d}{x}. \end{aligned}$$

  The last integral is infinite if $1+2s\le n$, and hence there is no conservation of mass for $n \ge 3$.

• (c) In Theorem 2.12, we construct other types of special solutions which could also be used to prove the global Harnack principle.

3 Properties of an approximation scheme

We will now start the detailed development of the theory. The basic idea we follow is to discretize explicitly in time and use the asymptotic expansion of $\Delta _{\infty }^{s}$ found in [16] to provide a monotone zero-order approximation of the operator.

We recall that, for $s\in (1/2,1)$,

(3.1)

Lemma 3.1

The operator $\mathcal {L}_\varepsilon :C_{b }({\mathbb {R}^n})\rightarrow C_{b }({\mathbb {R}^n})$ is well-defined and bounded.

Remark 3.2

Note that, in general, $\Delta _{\infty }^{s}:C_{b }^\infty ({\mathbb {R}^n}) \not \rightarrow C_{b }({\mathbb {R}^n})$. See Sect. 9.

Proof of Lemma 3.1

Let $\phi \in C_b ({\mathbb {R}^n})$. Since $\int _{\varepsilon }^\infty \eta ^{-(1+2\,s)} \mathop {}\!\textrm{d}{t}=\frac{1}{2\,s}\varepsilon ^{-2\,s}$, we have $\Vert \mathcal L_{\varepsilon }[\phi ]\Vert _{C_b ({\mathbb {R}^n})}\le \frac{4C_s}{2\,s}\Vert \phi \Vert _{C_b ({\mathbb {R}^n})}\varepsilon ^{-2\,s}$ for any $\varepsilon >0$. It follows that $\mathcal {L}_\varepsilon [\phi ]$ is bounded. If $\mathcal L_{\varepsilon }[\phi ]$ is continuous, it also follows that $\mathcal {L}_\varepsilon$ is a bounded operator on $C_{b }({\mathbb {R}^n})$. To show continuity at an arbitrary point $x_1\in \mathbb {R}^n$, we fix $\epsilon >0$. By the above bound there is (large) $R>0$ such that

$$\begin{aligned} \Vert {\mathcal {L}}_{R}[\phi ]\Vert _{C_b ({\mathbb {R}^n})} < \frac{\epsilon }{4}. \end{aligned}$$

For $x_2\in B(x_1,1)$, we find by the triangle inequality and $\sup \inf (\cdots )-\sup \inf (\cdots )\le \sup \sup (\cdots -\cdots )$,

$$\begin{aligned} \begin{aligned}&\big |\mathcal {L}_\varepsilon [\phi ](x_1)-\mathcal {L}_\varepsilon [\phi ](x_2)\big |\\&\quad \le C_s \sup _{|{\tilde{y}}|=1}\sup _{|y|=1}\int _\varepsilon ^{R} \big |\big (\phi (x_1+\eta y)+ \phi (x_1+\eta {\tilde{y}})\big ) - \big (\phi (x_2+\eta y)+\phi (x_2+\eta {\tilde{y}})\big )\big | \frac{\mathop {}\!\textrm{d}{\eta }}{\eta ^{1+2s}}\\&\qquad +2C_s\int _\varepsilon ^{R}\frac{\mathop {}\!\textrm{d}{\eta }}{\eta ^{1+2s}}|\phi (x_1)-\phi (x_2)|+\frac{\epsilon }{4} +\frac{\epsilon }{4} \\&\quad \le 2C_s \omega _{\phi ,R}(x_1-x_2)\int _\varepsilon ^{R}\frac{\mathop {}\!\textrm{d}{\eta }}{\eta ^{1+2s}}+ \frac{\epsilon }{2}, \end{aligned} \end{aligned}$$

where $\omega _{\phi ,R}$ is the modulus of continuity of $\phi$ in the ball $B(0,|x_1|+1+R)$. Since the integral is finite, the last expression is less than $\varepsilon$ when $|x_2- x_1|$ is small enough and continuity of $\mathcal {L}_\varepsilon [\phi ]$ follows. $\square$

To state the consistency, we introduce admissible test functions $\phi$: There is $\eta _x>0$, such that

$$\begin{aligned} (i) \quad \phi \in C^2({\bar{B}}_{\eta _x})\qquad \text {and}\qquad (ii) \quad \phi \in B({\mathbb {R}^n})\cap UC({\mathbb {R}^n}\setminus {\bar{B}}_{\eta _x}). \end{aligned}$$

Lemma 3.3

(Consistency, Theorem 1.1 in [16]) Under the above assumptions on $\phi$, for every $\varepsilon < \eta _x$,

$$\begin{aligned} \Big |\mathcal {L}_\varepsilon [\phi ](x)-\Delta _{\infty }^{s}\phi (x) \Big |=o_\varepsilon (1), \end{aligned}$$

where the bound $o_\varepsilon (1)$ depends only on $|\nabla \phi (x)|^{-1}$, $\Vert D^2\phi \Vert _{C_{b }({\bar{B}}_{\eta _x})}$, and $\omega _{\phi ,{\bar{B}}_{\eta _x}^c}$.

We also need $\varepsilon$ independent bounds to send $\varepsilon \rightarrow 0$.

Lemma 3.4

(Uniform bound) If $\phi \in C_{b }^2({\mathbb {R}^n})$, then there is a constant c(s) only depending on s such that

$$\begin{aligned} |\mathcal {L}_\varepsilon [\phi ](x)| \le c(s)\Vert \nabla \phi \Vert _{ C_{b }({\mathbb {R}^n})}^{2-2s}\Vert D^2\phi \Vert _{ C_{b }({\mathbb {R}^n})}^{2s-1}. \end{aligned}$$

Proof

We add to $\mathcal {L}_\varepsilon$ the gradient term

$$\begin{aligned} \sup _{|y|=1}\inf _{|\tilde{y}|=1} \int _\varepsilon ^{1} tp_x\cdot (y-\tilde{y}) \frac{\mathop {}\!\textrm{d}{t}}{t^{1+2s}} = \int _\varepsilon ^{1} \eta \frac{\mathop {}\!\textrm{d}{\eta }}{\eta ^{1+2s}}\sup _{|y|=1}\inf _{|\tilde{y}|=1}p_x\cdot (y-\tilde{y})=0, \end{aligned}$$

Since the intgrand is bounded, we then split the resulting integral in two—an integral with the $\inf$ and an integral with the $\sup$. The result for the $\sup$-part is:

$$\begin{aligned} \sup _{|y|=1}\int _\varepsilon ^\infty \big (\phi (x+\eta y)-\phi (x)-tp_x\cdot y \eta 1_{0<\eta <1}\big )\frac{\mathop {}\!\textrm{d}{\eta }}{\eta ^{1+2s}}. \end{aligned}$$

Splitting this integral in two, $\int _\varepsilon ^r+\int _r^\infty$, and Taylor expanding, we find the following upper bound

$$\begin{aligned} \frac{1}{2}\Vert D^2\phi \Vert _{C_{b }}\int _0^r\eta ^2 \frac{\mathop {}\!\textrm{d}{\eta }}{\eta ^{1+2s}} + 2\Vert \nabla \phi \Vert _{C_{b }}\int _r^\infty \eta \frac{\mathop {}\!\textrm{d}{\eta }}{\eta ^{1+2s}}&\le \frac{1}{2}\Vert D^2\phi \Vert _{C_{b }}\frac{1}{2-2s}r^{2-2s}\\&\quad + 2\Vert \nabla \phi \Vert _{C_{b }}\frac{1}{2s-1}r^{1-2s}. \end{aligned}$$

Minimizing with respect to r then proves the result for the $\sup$-part. The $\inf$-part is similar. $\square$

Remark 3.5

Note that $\mathcal {L}_\varepsilon$ is monotone in the following two ways:

• (i) $\mathcal {L}_\varepsilon [\phi ]\le 0$ at any global maximum of $\phi$.

• (ii) In the sense of monotone approximations in viscosity solution theory:

  $$\begin{aligned} \psi _1\le \psi _2\qquad \hbox { in}\ {\mathbb {R}^n}\qquad \implies \qquad L(\varepsilon ,\psi _1,r)\le L(\varepsilon ,\psi _2,r)\qquad \hbox { in}\ {\mathbb {R}^n}, \end{aligned}$$

  where $\mathcal {L}_\varepsilon [\psi ](x)=L(\varepsilon , \psi , \psi (x))$ and $L:\mathbb {R}_+\times BUC({\mathbb {R}^n})\times \mathbb {R}$ is given by

  $$\begin{aligned} L(\varepsilon ,\psi ,r)= C_s \left( \sup _{|y|=1}\int _\varepsilon ^\infty \psi (x+\eta y) \frac{\mathop {}\!\textrm{d}{\eta }}{\eta ^{1+2s}} + \inf _{|y|=1}\int _\varepsilon ^\infty \psi (x+\eta y) \frac{\mathop {}\!\textrm{d}{\eta }}{\eta ^{1+2s}}-\frac{1}{s\varepsilon ^{2s}}r\right) . \end{aligned}$$

These properties are crucial in order to obtain approximation schemes that preserves the properties of the limit problem (1.2)–(1.3).

3.1 Semi-discrete scheme defined on ${\mathbb {R}^n}\times \{\tau {{\,\mathrm{{{\mathbb {N}}}}\,}}\,\cup \,0\}$

We will now study the semi-discrete scheme (2.2)–(2.3).

Proposition 3.6

(Well-posedness and properties) Assume $u_0\in BUC({\mathbb {R}^n})$ and $\varepsilon ,\tau >0$. Then there exists a unique solution $U^j\in C_{b }({\mathbb {R}^n})$ of (2.2)–(2.3). Moreover, if

$$\begin{aligned} \tau \le \frac{s}{C_s}\varepsilon ^{2s}, \end{aligned}$$

(\text {CFL)

then the following properties hold:

• (a) ($C_b$-stability) $\Vert U^j\Vert _{C_b ({\mathbb {R}^n})}\le \Vert u_0\Vert _{C_b ({\mathbb {R}^n})}$.

• (b) (Comparison principle) Let $U^j$ and $V^j$ be sub- and supersolutions of (2.2)–(2.3) with respective initial data $u_0\in BUC({\mathbb {R}^n})$ and $v_0\in BUC({\mathbb {R}^n})$. If $u_0\le v_0$ in ${\mathbb {R}^n}$, then $U^j\le V^j$ in ${\mathbb {R}^n}$ for all $j\in {{\,\mathrm{{{\mathbb {N}}}}\,}}$.

• (c) ($C_b$-contraction) Let $U^j$ and $V^j$ be solutions of (2.2)–(2.3) with respective initial data $u_0\in BUC({\mathbb {R}^n})$ and $v_0\in BUC({\mathbb {R}^n})$. Then

  $$\begin{aligned} \Vert U^j-V^j\Vert _{C_b ({\mathbb {R}^n})} \le \Vert u_0-v_0\Vert _{C_b ({\mathbb {R}^n})} \qquad \hbox { for all}\ j\in {{\,\mathrm{{{\mathbb {N}}}}\,}}. \end{aligned}$$

• (d) (Equicontinuity in space) For all $y\in {\mathbb {R}^n}$ and all $j\in {{\,\mathrm{{{\mathbb {N}}}}\,}}$,       $\Vert U^j(\cdot +y)-U^j\Vert _{C_b ({\mathbb {R}^n})}\le \omega _{u_0}(|y|).$

• (e) (Equicontinuity in time) For all $j,k\in {{\,\mathrm{{{\mathbb {N}}}}\,}}$ and all $0<\varepsilon <1$,       $\Vert U^{j+k}-U^j\Vert _{C_b ({\mathbb {R}^n})}\le \tilde{\omega }(|t_{j+k}-t_j|)$,      where      $\tilde{\omega }$ is defined in Theorem 2.2 (c).

Proof

Since (2.2)–(2.3) is explicit and $\mathcal {L}_\varepsilon :C_b ({\mathbb {R}^n})\rightarrow C_b ({\mathbb {R}^n})$ is well-defined and bounded by Lemma 3.1, existence and uniqueness follows by construction.

Let us then show the different a priori estimates:

(b) Since $u_0\le v_0$, we have $U^0\le V^0$. Then, by induction assume that $U^j\le V^j$. By (2.2), we get

$$\begin{aligned} \begin{aligned} U^{j+1}(x)-V^{j+1}(x)&\le U^{j}(x)-V^{j}(x) + \tau \left( \mathcal {L}_\varepsilon [U^j](x)-\mathcal {L}_\varepsilon [V^j](x)\right) \\&=\left( U^{j}(x)-V^{j}(x)\right) \left( 1-\tau \frac{C_s}{s\varepsilon ^{2s}}\right) \\&\quad + \tau C_s \left( \sup _{|y|=1}\int _\varepsilon ^\infty U^j(x+\eta y) \frac{\mathop {}\!\textrm{d}{t}}{t^{1+2s}} - \sup _{|y|=1}\int _\varepsilon ^\infty V^j(x+\eta y) \frac{\mathop {}\!\textrm{d}{\eta }}{\eta ^{1+2s}} \right) \\&\quad +\tau C_s \left( \inf _{|y|=1}\int _\varepsilon ^\infty U^j(x+\eta y) \frac{\mathop {}\!\textrm{d}{\eta }}{\eta ^{1+2s}} - \inf _{|y|=1}\int _\varepsilon ^\infty V^j(x+\eta y) \frac{\mathop {}\!\textrm{d}{\eta }}{\eta ^{1+2s}} \right) \\&\le 0, \end{aligned} \end{aligned}$$

where the last inequality follows from the induction hypothesis $U^j\le V^j$ and (CFL).

(a) Note that

$$\begin{aligned} V^j:=\inf _{x\in {\mathbb {R}^n}} \{u_0(x)\} \quad and \quad W^j:=\sup _{x\in {\mathbb {R}^n}} \{u_0(x)\} \qquad for all j\in {{\,\mathrm{{{\mathbb {N}}}}\,}}\end{aligned}$$

are solutions of (2.2)–(2.3). Since $\inf _{x\in {\mathbb {R}^n}} \{u_0(x)\} \le u_0 \le \sup _{x\in {\mathbb {R}^n}} \{u_0(x)\}$, we have by (b) that

$$\begin{aligned} \inf _{x\in {\mathbb {R}^n}} \{u_0(x)\}=V^j \le U^j \le W^j=\sup _{x\in {\mathbb {R}^n}} \{u_0(x)\}. \end{aligned}$$

(c) By the proof of (b) and the fundamental inequalities $|\sup (\cdots )-\sup (\cdots )|\le \sup (|\cdots -\cdots |)$ and $|\inf (\cdots )-\inf (\cdots )|\le \sup (|\cdots -\cdots |)$, we can also get that

$$\begin{aligned} \begin{aligned}&|U^{j+1}(x)-V^{j+1}(x)|\\&\quad \le \left| U^{j}(x)-V^{j}(x)\right| \left( 1-\tau \frac{C_s}{s\varepsilon ^{2s}}\right) + \tau C_s \left| \sup _{|y|=1}\int _\varepsilon ^\infty U^j(x+\eta y) \frac{\mathop {}\!\textrm{d}{\eta }}{\eta ^{1+2s}} \right. \\ {}&\left. \qquad - \sup _{|y|=1}\int _\varepsilon ^\infty V^j(x+\eta y) \frac{\mathop {}\!\textrm{d}{\eta }}{\eta ^{1+2s}} \right| \\&\qquad +\tau C_s \left| \inf _{|y|=1}\int _\varepsilon ^\infty U^j(x+\eta y) \frac{\mathop {}\!\textrm{d}{\eta }}{\eta ^{1+2s}} - \inf _{|y|=1}\int _\varepsilon ^\infty V^j(x+\eta y) \frac{\mathop {}\!\textrm{d}{\eta }}{\eta ^{1+2s}} \right| \\&\quad \le \left| U^{j}(x)-V^{j}(x)\right| \left( 1-\tau \frac{C_s}{s\varepsilon ^{2s}}\right) + 2\tau C_s \sup _{|y|=1}\int _\varepsilon ^\infty \left| U^j(x+\eta y)- V^j(x+\eta y) \right| \frac{\mathop {}\!\textrm{d}{\eta }}{\eta ^{1+2s}}\\&\quad \le \left\| U^{j}-V^{j}\right\| _{C_b ({\mathbb {R}^n})}\left( 1-\tau \frac{C_s}{s\varepsilon ^{2s}}\right) + 2 \tau C_s \left\| U^{j}-V^{j}\right\| _{C_b ({\mathbb {R}^n})} \int _\varepsilon ^\infty \frac{\mathop {}\!\textrm{d}{\eta }}{\eta ^{1+2s}}\\&\quad = \left\| U^{j}-V^{j}\right\| _{C_b ({\mathbb {R}^n})}. \end{aligned} \end{aligned}$$

In this way we have proved that

$$\begin{aligned} \left\| U^{j+1}-V^{j+1}\right\| _{C_b ({\mathbb {R}^n})}\le \left\| U^{j}-V^{j}\right\| _{C_b ({\mathbb {R}^n})} \qquad for all j\in {{\,\mathrm{{{\mathbb {N}}}}\,}}. \end{aligned}$$

An iteration then concludes the proof.

(d) This follows by using the translation invariant properties of (2.2)–(2.3) and part (c). More precisely, $W^j:=U^j(\cdot +y)$ is the unique solution of (2.2)–(2.3) with initial data $w_0:= u_0(\cdot +y)$ for all $y\in {\mathbb {R}^n}$. Part (c) then yields

$$\begin{aligned} \Vert U^j(\cdot +y)-U^j\Vert _{C_b ({\mathbb {R}^n})}=\Vert W^j-U^j\Vert _{C_b ({\mathbb {R}^n})}\le \Vert w_0-u_0\Vert _{C_{b }({\mathbb {R}^n})} =\Vert u_0(\cdot +y)-u_0\Vert _{C_b ({\mathbb {R}^n})}. \end{aligned}$$

(e) Consider a mollification of the initial data $u_{0,\delta }:=u_0*\rho _\delta$, and denote the corresponding solution by $U_\delta ^j$. Choose $j=1$ in (2.2)–(2.3) to get

$$\begin{aligned} \Vert U_\delta ^1-U_\delta ^0\Vert _{C_b ({\mathbb {R}^n})}\le \tau \Vert \mathcal {L}_\varepsilon [U_\delta ^0]\Vert _{C_b ({\mathbb {R}^n})}=\tau \Vert \mathcal {L}_\varepsilon [u_\delta ^0]\Vert _{C_b ({\mathbb {R}^n})}:= \tau K({u_{0,\delta }}). \end{aligned}$$

Now, define

$$\begin{aligned} V_\delta ^{j}:=U_\delta ^{j+1} \qquad for all j\in {{\,\mathrm{{{\mathbb {N}}}}\,}}. \end{aligned}$$

Then $V_\delta ^j$ is the unique solution of (2.2)–(2.3) with initial data $V_\delta ^0=U_\delta ^1$. By (c),

$$\begin{aligned} \begin{aligned} \Vert U_\delta ^{j+1}-U_\delta ^j\Vert _{C_b ({\mathbb {R}^n})}&= \Vert V_\delta ^{j}-U_\delta ^j\Vert _{C_b ({\mathbb {R}^n})} \le \Vert V_\delta ^{0}-U_\delta ^0\Vert _{C_b ({\mathbb {R}^n})}\\&=\Vert U_\delta ^{1}-U_\delta ^0\Vert _{C_b ({\mathbb {R}^n})}\le \tau K({u_{0,\delta }}). \end{aligned} \end{aligned}$$

(3.2)

Repeated use of the triangle inequality then yields

$$\begin{aligned} \Vert U_\delta ^{j+k}-U_\delta ^j\Vert _{C_b ({\mathbb {R}^n})}\le \sum _{i=0}^{k-1}\Vert U_\delta ^{(j+i)+1}-U_\delta ^{j+i}\Vert _{C_b ({\mathbb {R}^n})}\le k\tau K({u_{0,\delta }})=(t_{j+k}-t_j)K({u_{0,\delta }}). \end{aligned}$$

Then by (c),

$$\begin{aligned}&\Vert U^{j+k}-U^j\Vert _{C_b ({\mathbb {R}^n})}\le \Vert U^{j+k}-U_\delta ^{j+k}\Vert _{C_b ({\mathbb {R}^n})}+\Vert U_\delta ^{j+k}-U_\delta ^j\Vert _{C_b ({\mathbb {R}^n})} +\Vert U_\delta ^{j}-U^j\Vert _{C_b ({\mathbb {R}^n})}\\&\quad \le 2\Vert u_0-u_{0,\delta }\Vert _{C_{b }}+(t_{j+k}-t_j)K({u_{0,\delta }}) \le 2\omega _{u_0}(\delta )+(t_{j+k}-t_j)K({u_{0,\delta }}), \end{aligned}$$

where we used that by properties of mollifiers, $\Vert u_0-u_{0,\delta }\Vert _{C_{b }}\le \sup _{|y|\le \delta }\Vert u_0(\cdot +y)-u_0\Vert _{C_b ({\mathbb {R}^n})}\le \omega _{u_0}(\delta )$. Hence the result follows by the definition of $\tilde{\omega }$. $\square$

3.2 Semi-discrete scheme defined on ${\mathbb {R}^n}\times [0,\infty )$

In order to get uniform convergence of our approximation scheme, we need to define it on ${\mathbb {R}^n}\times [0,\infty )$. Let us therefore consider the solution of (2.2)–(2.3) $U_\varepsilon :{\mathbb {R}^n}\times \{\tau {{\,\mathrm{{{\mathbb {N}}}}\,}}\,\cup \,0\}\rightarrow \mathbb {R}$ and the function $u_\varepsilon :{\mathbb {R}^n}\times [0,\infty )\rightarrow \mathbb {R}$ defined as:

$$\begin{aligned} {\left\{ \begin{array}{ll} u_\varepsilon (x,0):=U^{0}_\varepsilon (x)=u_0(x),\\ u_\varepsilon (x,t):= \frac{t_{j+1}-t}{\tau }U^{j}_\varepsilon (x)+ \frac{t-t_{j}}{\tau } U^{j+1}_\varepsilon (x)\quad \text {if}\; t\in (t_j, t_{j+1}] \text { with}\; j\in {{\,\mathrm{{{\mathbb {N}}}}\,}}. \end{array}\right. } \end{aligned}$$

Corollary 3.7

(Well-posedness and properties) Under the assumptions of Proposition 3.6, there exists a unique pointwise solution $u_\varepsilon \in BUC({\mathbb {R}^n}\times [0,\infty ))$ of (2.2)–(2.3) with initial data $u_0\in BUC({\mathbb {R}^n})$. The solution, moreover, enjoys $C_b$-stability, comparison principle, $C_b$-contraction, continuity in space, and continuity in time in form of $\Vert u_\varepsilon (\cdot ,t)-u_\varepsilon (\cdot ,\tilde{t})\Vert _{C_b ({\mathbb {R}^n})}\le \tilde{\omega }(|t-{\tilde{t}}|)$ for all $t,\tilde{t}\in [0,\infty ]$.

Proof

We easily inherit all properties from $U_\varepsilon$ to $u_\varepsilon$, e.g.

$$\begin{aligned} \Vert u_\varepsilon (\cdot ,t)\Vert _{C_b }\le & {} \frac{(t_{j+1}-t)}{\tau } \left\| U^{j}_\varepsilon \right\| _{C_b }+ \frac{(t-t_{j})}{\tau } \left\| U^{j+1}_\varepsilon \right\| _{C_b } \\\le & {} \Big (\frac{(t_{j+1}-t)}{\tau }+ \frac{(t-t_{j})}{\tau }\Big ) \left\| u_0\right\| _{C_b }= \left\| u_0\right\| _{C_b }. \end{aligned}$$

The other properties follows in a similar way, and we only explain the most difficult one, the continuity in time. Repeating the steps of the proof of Proposition 3.6(e), for

$\tilde{t}\in (t_{j+k},t_{j+k+1}]$ and $t\in (t_{j},t_{j+1}]$,

$$\begin{aligned} \begin{aligned}&|(u_\varepsilon )_{\delta }(x,\tilde{t})-(u_\varepsilon )_{\delta }(x,t)|\\&\quad \le |(u_\varepsilon )_{\delta }(x,\tilde{t})-(u_\varepsilon )_{\delta }(x,t_{j+k})|+| (u_\varepsilon )_{\delta }(x,t_{j+1})-(u_\varepsilon )_{\delta }(x,t)|\\&\qquad + \sum _{l={j+1}}^{j+k-1}|(u_\varepsilon )_{\delta }(x,t_{l+1}) - (u_\varepsilon )_{\delta }(x,t_{l})|\\&\quad =|(u_\varepsilon )_{\delta }(x,\tilde{t})-(U_\varepsilon ^{j+k})_{\delta }(x)|+|(U_\varepsilon ^{j+1})_{\delta }(x)-(u_\varepsilon )_{\delta }(x,t)\\&\qquad + \sum _{l={j+1}}^{j+k-1}|(U_\varepsilon ^{l+1})_{\delta }(x) - (U_\varepsilon ^{l})_{\delta }(x)|. \end{aligned} \end{aligned}$$

By the definition of linear interpolation

$$\begin{aligned} \Vert (u_\varepsilon )_{\delta }(\cdot ,\tilde{t}) -(U_\varepsilon ^{j+k})_{\delta }\Vert _{C_b ({\mathbb {R}^n})}&\le \frac{(\tilde{t} -t_{j+k})}{\tau }\Vert (U_\varepsilon ^{j+k+1})_{\delta }-(U_\varepsilon ^{j+k})_{\delta }\Vert _{C_b ({\mathbb {R}^n})},\\ \Vert (U_\varepsilon ^{j+1})_{\delta }-(u_\varepsilon )_{\delta }(\cdot ,t)\Vert _{C_b ({\mathbb {R}^n})}&\le \frac{(t_{j+1}-t)}{\tau }\Vert (U_\varepsilon ^{j+1})_{\delta }-(U_\varepsilon ^j)_{\delta }\Vert _{C_b ({\mathbb {R}^n})}, \end{aligned}$$

and then by repeated use of (3.2),

$$\begin{aligned} \Vert (u_\varepsilon )_{\delta }(\cdot ,\tilde{t})-(u_\varepsilon )_{\delta }(\cdot ,t)\Vert _{C_b ({\mathbb {R}^n})}\le & {} \left( (\tilde{t}-t_{j+k})+\sum _{l={j+1}}^{j+k-1}\tau +(t_{j+1}-t)\right) K(u_{0,\delta })\\= & {} (\tilde{t}-t)K(u_{0,\delta }). \end{aligned}$$

We can then conclude the proof continuing as in the proof of Proposition 3.6(e). $\square$

3.3 Compactness in $UC_{\text {loc} }({\mathbb {R}^n}\times [0,\infty ))$

Proposition 3.8

(Compactness) Under the assumptions of Proposition 3.6, there exists a subsequence $\{u_{\varepsilon _k}\}_{k\in {{\,\mathrm{{{\mathbb {N}}}}\,}}}$ and a $u\in C_{b }({\mathbb {R}^n}\times [0,\infty ))$

such that

$$\begin{aligned} u_{\varepsilon _k}\rightarrow u\qquad \text {locally uniformly in}\; {\mathbb {R}^n}\times [0,\infty ) \text {as }\; k\rightarrow \infty . \end{aligned}$$

Proof

The sequence $\{u_{\varepsilon }\}_\varepsilon$ is equibounded and equicontinuous by Corollary 3.7 (see also Proposition 3.6). The result then follows from the Arzelà-Ascoli compactness theorem. $\square$

Taking pointwise limits in the a priori estimates of Corollary 3.7 (see also Propostion 3.6), the limit u immediately inherits these estimates.

Corollary 3.9

(A priori estimates) Assume $u_0\in BUC({\mathbb {R}^n})$. Then the limit u from Propostion 3.8 enjoys the following properties:

• (a) ($C_b$-stability) For all $t>0$, $\Vert u(\cdot ,t)\Vert _{C_b ({\mathbb {R}^n})}\le \Vert u_0\Vert _{C_b ({\mathbb {R}^n})}$.

• (b) (Uniform continuity in space) For all $y\in {\mathbb {R}^n}$ and all $t>0$,       $\Vert u(\cdot +y,t)-u(\cdot ,t)\Vert _{C_b ({\mathbb {R}^n})} \le \omega _{u_0}(|y|).$

• (c) (Uniform continuity in time) For all $t,\tilde{t}>0$,       $\Vert u(\cdot ,t)-u(\cdot ,\tilde{t})\Vert _{C_b ({\mathbb {R}^n})}\le \tilde{\omega }(t-{\tilde{t}})$      where $\tilde{\omega }$ is defined in Theorem 2.2.

4 Definitions, existence and properties of viscosity solutions

In this section we define the concept of viscosity solution. Before doing so we need to introduce two new operators that will be used when testing at zero gradient points.

Definition 4.1

For $\phi \in C^{1,1}(x)\cap B({\mathbb {R}^n})$,

$$\begin{aligned} \Delta _{\infty }^{s,+}\phi (x)&:=C_s \sup _{|y|=1}\int _0^\infty \big (\phi \left( x+ \eta y\right) +\phi \left( x- \eta y\right) -2\phi (x)\big )\frac{\mathop {}\!\textrm{d}{\eta }}{\eta ^{1+2s}},\\ \Delta _{\infty }^{s,-}\phi (x)&:=C_s \inf _{|y|=1}\int _0^\infty \big (\phi \left( x+ \eta y\right) +\phi \left( x- \eta y\right) -2\phi (x)\big )\frac{\mathop {}\!\textrm{d}{\eta }}{\eta ^{1+2s}}. \end{aligned}$$

We immediately have:

Lemma 4.2

For $\phi \in C^{1,1}(x)\cap B({\mathbb {R}^n})$,

$$\begin{aligned} \Delta _{\infty }^{s,-}\phi (x) \le \Delta _{\infty }^{s}\phi (x) \le \Delta _{\infty }^{s,+}\phi (x). \end{aligned}$$

Proof

Recall Lemma 2.1. The result is trivial unless $\nabla \phi (x)=0$. In that case,

$$\begin{aligned}\begin{aligned} \Delta _{\infty }^{s}\phi (x)&=C_s\sup _{|y|=1} \int _{0}^\infty \big (\phi (x+\eta y)-\phi (x)\big ) \frac{\mathop {}\!\textrm{d}{\eta }}{\eta ^{1+2s}}\\&\quad + C_s\inf _{|y|=1} \int _{0}^\infty \big (\phi (x-\eta y)-\phi (x)\big ) \frac{\mathop {}\!\textrm{d}{\eta }}{\eta ^{1+2s}}\\&=C_s\sup _{|y|=1} \int _{0}^\infty \big (\phi (x+\eta y)-\phi (x)\big ) \frac{\mathop {}\!\textrm{d}{\eta }}{\eta ^{1+2s}}\\&\quad - C_s\sup _{|y|=1} \left( - \int _{0}^\infty \big (\phi (x-\eta y)-\phi (x)\big ) \frac{\mathop {}\!\textrm{d}{\eta }}{\eta ^{1+2s}} \right) \\&\le C_s \sup _{|y|=1} \left( \Big ( \int _{0}^\infty \big (\phi (x+\eta y)-\phi (x)\big ) \frac{\mathop {}\!\textrm{d}{\eta }}{\eta ^{1+2s}}\Big )\right. \\&\quad \left. - \Big (- \int _{0}^\infty \big (\phi (x-\eta y)-\phi (x)\big ) \frac{\mathop {}\!\textrm{d}{\eta }}{\eta ^{1+2s}} \Big )\right) \\&= \Delta _{\infty }^{s,+}\phi (x). \end{aligned} \end{aligned}$$

The result $\Delta _{\infty }^{s,-}\phi (x) \le \Delta _{\infty }^{s}\phi (x)$ follows in a similar way. $\square$

We are now ready to define the concept of viscosity solution.

Definition 4.3

(Viscosity solution)

• (a) A globally bounded upper semicontinuous function $u:{\mathbb {R}^n}\times (0,\infty )\rightarrow \mathbb {R}$ is a viscosity subsolution of (1.2) if, for all $(x_0,t_0)\in {\mathbb {R}^n}\times (0,\infty )$, all $\phi \in C^2(B_R(x_0,t_0))\cap BUC({\mathbb {R}^n}\times (0,\infty ){\setminus } B_R(x_0,t_0))$ for some $R>0$ and such that

  • (i) $u(x_0,t_0)-\phi (x_0,t_0)=\sup _{(x,t)\in B_R(x_0,t_0)}\big (u(x,t)-\phi (x,t)\big )$,

  • (ii) $u(x_0,t_0)-\phi (x_0,t_0)>u(x,t)-\phi (x,t)$ for all $(x,t)\in B_R(x_0,t_0){\setminus } (x_0,t_0)$,

  • (iii) $u(x_0,t_0)-\phi (x_0,t_0)\ge u(x,t)-\phi (x,t)$ for all $(x,t)\in {\mathbb {R}^n}\times (0,\infty ) {\setminus } B_R(x_0,t_0)$,

  then

  

• (b) A globally bounded lower semicontinuous function $u:{\mathbb {R}^n}\times (0,\infty )\rightarrow \mathbb {R}$ is a viscosity supersolution of (1.2) if, for all $(x_0,t_0)\in {\mathbb {R}^n}\times (0,\infty )$, all $\phi \in C^2(B_R(x_0,t_0))\cap BUC({\mathbb {R}^n}\times (0,\infty ){\setminus } B_R(x_0,t_0))$ for some $R>0$ and such that

  • (i) $u(x_0,t_0)-\phi (x_0,t_0)=\inf _{(x,t)\in B_R(x_0,t_0)}(u(x,t)-\phi (x,t))$,

  • (ii) $u(x_0,t_0)-\phi (x_0,t_0)<u(x,t)-\phi (x,t)$ for all $(x,t)\in B_R(x_0,t_0){\setminus } (x_0,t_0)$,

  • (iii) $u(x_0,t_0)-\phi (x_0,t_0)\le u(x,t)-\phi (x,t)$ for all $(x,t)\in {\mathbb {R}^n}\times (0,\infty ) \setminus B_R(x_0,t_0)$,

  then

  

• (c) A function $u\in C_{b }({\mathbb {R}^n}\times [0,\infty ))$ is a viscosity solution of (1.2) if it is both a viscosity subsolution and a viscosity supersolution.

• (d) The viscosity solution u takes the initial data in a pointwise way: $u(x,0)=u_0(x)$ for all $x\in {\mathbb {R}^n}$.

Remark 4.4

• (a) In points where the gradient of u is zero, we only require $\partial _tu\in [ \Delta _{\infty }^{s,-}u, \Delta _{\infty }^{s,+}u].$

• (b) In the local elliptic and homogeneous case [22], comparison follows without any condition at points where $\nabla \phi =0$. In more general cases conditions are needed. We impose conditions (4.1) and (4.2) which are generalisations of the conditions introduced in the local parabolic case [1]. It is easy to show that comparison and uniqueness cannot hold without such conditions: E.g. $u(x,t)=1$ and $v(x,t)=2\sin (t)$ would then both be viscosity solutions of (1.2) since $\nabla u=\nabla v=0$ at every point. However comparison does not hold since $u(x,0)=1 \ge 0 =v(x,0)$ while $u(x,\pi /2)=1\le 2 = v(x,\pi /2).$ Let us check that v is not longer a viscosity solution when we impose (4.1). Let $K>0$ and $0\le \psi \in C^2_{b }(\mathbb {R})$ be radial with $\psi (r)=r^2$ for $|r|<1$ and $\psi (r)=0$ for $|r|>2$. We define

  $$\begin{aligned} \phi (x,t)= v(t)+K\psi (|x|)+ \psi (t-t_0)\qquad \text {for some}\; t_0\in (0,\infty ). \end{aligned}$$

  It is then immediate that $v-\phi$ has a strict local max at $(0,t_0)$ and $\nabla \phi =\nabla v=0$. Now let $t_0=2\pi$, then $\partial _{t}\phi (0,2\pi )= \partial _{t}v(0,2\pi )=2\cos (2\pi )=2$, and by radial symmetry and followed by compact support of $\phi$ leads to

  $$\begin{aligned} \begin{aligned} \Delta _{\infty }^{s,+}\phi (0,2\pi )&= KC_s\int _{0}^\infty \big ( \phi (\eta e_1, 2\pi )+\phi (-\eta e_1, 2\pi )-2\phi (0,2\pi )\big )\frac{\mathop {}\!\textrm{d}{\eta }}{\eta ^{1+2s}}\\&\le KC_s \int _{0}^2 \Vert \psi ''\Vert _{C_{b }} \eta ^2 \frac{\mathop {}\!\textrm{d}{\eta }}{\eta ^{1+2s}} =KC_s\Vert \psi ''\Vert _{C_{b }}\frac{2^{2-2s}}{(2-2s)}\le 1 \end{aligned} \end{aligned}$$

  if K is small enough. This contradicts (4.1) since

  $$\begin{aligned} \partial _{t}\phi (0,2\pi )=2 \ge 1\ge \Delta _{\infty }^{s,+}\phi (0,2\pi ). \end{aligned}$$

  We conclude that $v(x,t)=2\sin (t)$ is not a viscosity subsolution in the sense of Definition 4.3.

• (c) Since $\partial _t$, $\Delta _{\infty }^{s}$, $\Delta _{\infty }^{s,+}$ and $\Delta _{\infty }^{s,-}$ are invariant under translations of $\phi$ by constants, without loss of generality, we can replace the conditions on the test function in Definition 4.3 by

  • (i’) $\phi (x_0,t_0)=u(x_0,t_0)$,

  • (ii’) $\phi >u$ (resp. $\phi <u$) in $B_R(x_0,t_0)\setminus (x_0,t_0)$,

  • (iii’) $\phi \ge u$ (resp. $\phi \le u$) in ${\mathbb {R}^n}\times (0,\infty ) \setminus B_R(x_0,t_0)$.

• (d) We can also assume that the max is globally strict by adding a small in $C_{b }^2$ perturbation to $\phi$ supported in $B_R^c$, e.g. replacing $\phi$ by $\phi +\delta \psi$ where $\psi \in C_{b }^2$ is such that $0\le \psi \le 1$, $\psi =0$ in $B_R$ and $\psi >0$ in $B_R^c$. This new test function also satisfies (c), but with a strict inequality in part (iii’). Moreover, at the max point local derivatives up to order 2 coinside with those of $\phi$, while nonlocal derivatives differ by an $O(\delta )$ term since $|\Delta _{\infty }^{s}\psi |+|\Delta _{\infty }^{s,+}\psi |+|\Delta _{\infty }^{s,-}\psi |\le C \Vert \psi \Vert _{C^2_{b }}$. Before concluding the proof we then need to send $\delta \rightarrow 0$. Since this is never a problem, we will omit this modification in some proofs and simply assume globally strict max in the definition of viscosity solutions.

4.1 Existence and properties of solutions

We are now in a position to prove Theorem 2.2.

Proof of Theorem 2.2

By Proposition 3.8 there is a subsequence $u_\varepsilon$ such that

$$\begin{aligned} u_\varepsilon \rightarrow u \qquad locally uniformly in {\mathbb {R}^n}\times [0,\infty )\ as \ \varepsilon \rightarrow 0^+, \end{aligned}$$

and by Corollary 3.9, $u\in BUC({\mathbb {R}^n}\times [0,\infty ))$ and satisfies the a priori estimates in Theorem 2.2. Let us prove that $\tilde{\omega }$ has the property claimed in part (c). To do this we use two basic facts about mollifiers: Since $u_0\in BUC({\mathbb {R}^n})$, it follows that (differentiate $\rho$, use Young for convolutions)

$$\begin{aligned} \Vert D^ku_{0,\delta }\Vert _{C_{b }}\le \Vert D^k\rho \Vert _{L^1}\Vert u_0\Vert _{C_{b }}\delta ^{-k}\qquad \text {for}\; k\in {{\,\mathrm{{{\mathbb {N}}}}\,}}. \end{aligned}$$

Then by Lemma 3.4, we have the following bound:

$$\begin{aligned} \Vert \mathcal {L}_\varepsilon [u_{0,\delta }]\Vert _{C_{b }}\le c(s) \Vert \nabla \rho \Vert _{L^1}^{2-2s}\Vert D^2\rho \Vert _{L^1}^{2s-1}\Vert u_0\Vert _{C_{b }}\delta ^{-2s}. \end{aligned}$$

Since $\delta ^{-2s}\le 1+\delta ^{-2}$, the estimate on $\tilde{\omega }$ follows after taking $\delta =r^{1/3}$.

It remains to check that u is a viscosity solution according to Definition 4.3. By Remark 4.4(d), consider $(x_0,t_0)\in {\mathbb {R}^n}\times (0,\infty )$ and $\phi \in C^2(B_R(x_0,t_0))\cap BUC({\mathbb {R}^n}\times (0,\infty ){\setminus } B_R(x_0,t_0))$ such that

• (i) $u(x_0,t_0)-\phi (x_0,t_0)=\sup _{(x,t)\in B_R(x_0,t_0)}(u(x,t)-\phi (x,t))$,

• (ii) $u(x_0,t_0)-\phi (x_0,t_0)>u(x,t)-\phi (x,t)$ for all $(x,t)\in {\mathbb {R}^n}\times (0,\infty ){\setminus } (x_0,t_0)$.

Local uniform convergence ensures that there exists a sequence $\{(x_\varepsilon ,t_\varepsilon )\}_{\varepsilon >0}$ such that

• (i) $u_\varepsilon (x_\varepsilon ,t_\varepsilon )-\phi (x_\varepsilon ,t_\varepsilon )=\sup _{(x,t)\in B_R(x_\varepsilon ,t_\varepsilon )}(u(x,t)-\phi (x,t)):=M_\varepsilon$,

• (ii) $u_\varepsilon (x_\varepsilon ,t_\varepsilon )-\phi (x_\varepsilon ,t_\varepsilon )>u_\varepsilon (x,t)-\phi (x,t)$ for all $(x,t)\in {\mathbb {R}^n}\times (0,\infty )\setminus (x_\varepsilon ,t_\varepsilon )$,

and

$$\begin{aligned} (x_\varepsilon ,t_\varepsilon )\rightarrow (x_0,t_0) \qquad as \varepsilon \rightarrow 0. \end{aligned}$$

Recall that Corollary 3.7 ensures that $u_\varepsilon$ solves the semidiscrete scheme. For simplicity, we use the notation in Remark 3.5. Let $t_j$ be such that $t_\varepsilon \in (t_j,t_{j+1}]$. It is standard to check that

$$\begin{aligned} \frac{u_\varepsilon (x_\varepsilon ,t_\varepsilon )-u_\varepsilon (x_\varepsilon ,t_j)}{t_\varepsilon -t_j} = L(\varepsilon , u_\varepsilon , u_\varepsilon (x_\varepsilon ,t_j)). \end{aligned}$$

or equivalently

$$\begin{aligned} \frac{(u_\varepsilon (x_\varepsilon ,t_\varepsilon )-M_\varepsilon )-(u_\varepsilon (x_\varepsilon ,t_j)- M_\varepsilon )}{t_\varepsilon -t_j} = L(\varepsilon , u_\varepsilon -M_\varepsilon , u_\varepsilon (x_\varepsilon ,t_j)-M_\varepsilon ) \end{aligned}$$

By defining $\tilde{u}_\varepsilon :=u_\varepsilon -M_\varepsilon$, we have that $\tilde{u}_\varepsilon (x_\varepsilon ,t_\varepsilon )=\phi (x_\varepsilon ,t_\varepsilon )$ and $\phi > \tilde{u}_\varepsilon$ in ${\mathbb {R}^n}\times [0,\infty ){\setminus } (x_\varepsilon ,t_\varepsilon )$. Let us then rewrite the scheme to obtain

$$\begin{aligned} \begin{aligned} \phi (x_\varepsilon ,t_\varepsilon )&=\tilde{u}_\varepsilon (x_\varepsilon ,t_\varepsilon )\\&= \tilde{u}_\varepsilon (x_\varepsilon ,t_j)\left( 1- (t_\varepsilon -t_j) \frac{C_s}{s \varepsilon ^{2s}}\right) \\&\quad + (t_\varepsilon -t_j) C_s \left( \sup _{|y|=1}\int _\varepsilon ^\infty \tilde{u}_\varepsilon (x_\varepsilon +\eta y,t_j) \frac{\mathop {}\!\textrm{d}{\eta }}{\eta ^{1+2s}} + \inf _{|y|=1}\int _\varepsilon ^\infty \tilde{u}_\varepsilon (x_\varepsilon +\eta y,t_j) \frac{\mathop {}\!\textrm{d}{\eta }}{\eta ^{1+2s}}\right) \\&< \phi (x_\varepsilon ,t_j)\left( 1- (t_\varepsilon -t_j) \frac{C_s}{s \varepsilon ^{2s}}\right) \\&\quad + (t_\varepsilon -t_j) C_s \left( \sup _{|y|=1}\int _\varepsilon ^\infty \phi (x_\varepsilon +\eta y,t_ju) \frac{\mathop {}\!\textrm{d}{\eta }}{\eta ^{1+2s}} + \inf _{|y|=1}\int _\varepsilon ^\infty \phi (x_\varepsilon +\eta y,t_j) \frac{\mathop {}\!\textrm{d}{\eta }}{\eta ^{1+2s}}\right) \end{aligned} \end{aligned}$$

that is,

$$\begin{aligned} \frac{\phi (x_\varepsilon ,t_\varepsilon )-\phi (x_\varepsilon ,t_j)}{t_\varepsilon -t_j}< L(\varepsilon , \phi , \phi (x_\varepsilon ,t_j)). \end{aligned}$$

(4.3)

Assume $\nabla \phi (x_0,t_0) >0$ (the $\nabla \phi (x_0,t_0) <0$ case is similar). Then for $\varepsilon _0>0$ small enough, $\nabla \phi (x_\varepsilon ,t_j) >0$ for $\varepsilon \le \varepsilon _0$, and we use (4.3) og Lemma 3.3 to find that

$$\begin{aligned} \partial _t\phi (x_\varepsilon ,t_\varepsilon )\le \Delta _{\infty }^{s}\phi (x_\varepsilon ,t_j ) + o_\varepsilon (1)+ O(\tau ), \end{aligned}$$

(4.4)

where $o_\varepsilon$ depends on $\sup _{\varepsilon \le \varepsilon _0}|\nabla \phi (x_\varepsilon ,t_j)|^{-1}$ which is uniformly bounded by the above discusion. Since $\phi$ is smooth, for every $\eta \in [0,\infty )$,

$$\begin{aligned} \phi \left( x_\varepsilon \pm \frac{\nabla \phi (x_\varepsilon )}{|\nabla \phi (x_\varepsilon )|} \eta ,t_j\right) {\mathop {\longrightarrow }\limits ^{\varepsilon \rightarrow 0^+}} \phi \left( x_0 \pm \frac{\nabla \phi (x_0)}{|\nabla \phi (x_0)|} \eta ,t_0\right) . \end{aligned}$$

The dominated convergence theorem then ensures that

$$\begin{aligned} \Delta _{\infty }^{s}\phi (x_\varepsilon ,t_j)\rightarrow \Delta _{\infty }^{s}\phi (x_0,t_0) \quad as \varepsilon \rightarrow 0^+. \end{aligned}$$

We thus pass to the limit in (4.4) and get the correct viscosity subsolution inequality.

When $\nabla \phi (x_0,t_0)=0$ we have (see proof of Lemma 4.2)

$$\begin{aligned} \begin{aligned} L(\varepsilon , \phi , \phi (x_\varepsilon ,t_j))&\le \sup _{|y|=1}\int _{\varepsilon }^\infty \big (\phi (x_\varepsilon +\eta y, t_j) + \phi (x_\varepsilon -\eta y,t_j) -2\phi (x_\varepsilon ,t_j)\big ) \frac{\mathop {}\!\textrm{d}{\eta }}{\eta ^{1+2s}}\\&= \Delta _{\infty }^{s,+}\phi (x_\varepsilon , t_j)+ o_\varepsilon (1), \end{aligned} \end{aligned}$$

and it only remains to check $\Delta _{\infty }^{s,+}\phi (x_\varepsilon ,t_j) {\mathop {\longrightarrow }\limits ^{\varepsilon \rightarrow 0^+}} \Delta _{\infty }^{s,+}\phi (x_0,t_0).$ To do that, note that

$$\begin{aligned} \begin{aligned}&C_s^{-1}|\Delta _{\infty }^{s,+}\phi (x_\varepsilon ,t_j)-\Delta _{\infty }^{s,+}\phi (x_0,t_0)|\\&\quad \le \sup _{|y|=1} \bigg |\int _{0}^\infty \bigg ( \Big (\phi (x_\varepsilon +\eta y,t_j) - \phi (x_0+\eta y,t_0) \Big ) + \Big (\phi (x_\varepsilon -\eta y,t_j) - \phi (x_0-\eta y,t_0) \Big )\\&\qquad - 2\Big (\phi (x_\varepsilon ,t_j) - \phi (x_0,t_0)\Big ) \bigg )\frac{\mathop {}\!\textrm{d}{\eta }}{\eta ^{1+2s}}\bigg |\\&\quad \le \sup _{|y|=1} \bigg |\int _{0}^{\frac{R}{4}}\bigg ( \Big (\phi (x_\varepsilon +\eta y,t_j) - \phi (x_0+\eta y,t_0) \Big ) + \Big (\phi (x_\varepsilon -\eta y,t_j) - \phi (x_0-\eta y,t_0) \Big )\\&\qquad - 2\Big (\phi (x_\varepsilon ,t_j) - \phi (x_0,t_0)\Big ) \bigg )\frac{\mathop {}\!\textrm{d}{\eta }}{\eta ^{1+2s}}\bigg |\\&\qquad + \sup _{|y|=1} \bigg |\int _{\frac{R}{4}}^\infty \bigg ( \Big (\phi (x_\varepsilon +\eta y,t_j) - \phi (x_0+\eta y,t_0) \Big ) + \Big (\phi (x_\varepsilon -\eta y,t_j) - \phi (x_0-\eta y,t_0) \Big )\\&\qquad - 2\Big (\phi (x_\varepsilon ,t_j) - \phi (x_0,t_0)\Big ) \bigg )\frac{\mathop {}\!\textrm{d}{\eta }}{\eta ^{1+2s}}\bigg |=:I_{\varepsilon }^1+I_{\varepsilon }^2. \end{aligned} \end{aligned}$$

Since $\phi \in BUC({\mathbb {R}^n}\times (0,\infty ))$,

$$\begin{aligned} I_\varepsilon ^2\le 4\, \omega _{\phi } ((x_\varepsilon -x_0, t_j-t_0)) \int _{\frac{R}{4}}^\infty \frac{\mathop {}\!\textrm{d}{\eta }}{\eta ^{1+2s}} \rightarrow 0 \qquad \text {as}\; \varepsilon \rightarrow 0^+. \end{aligned}$$

Then note that $I^1_\varepsilon \le \int _0^{\frac{R}{4}} F_\varepsilon (\eta ) \mathop {}\!\textrm{d}{\eta }$ where

$$\begin{aligned} \begin{aligned} F_\varepsilon (\eta )&=\frac{1}{{\eta ^{1+2s}}}\sup _{|y|=1}\Big | \Big (\phi (x_\varepsilon +\eta y,t_j) - \phi (x_0+\eta y,t_0) \Big )\\&\quad + \Big (\phi (x_\varepsilon -\eta y,t_j) - \phi (x_0-\eta y,t_0) \Big )\\&\quad - 2\Big (\phi (x_\varepsilon ,t_j) - \phi (x_0,t_0)\Big )\Big |. \end{aligned} \end{aligned}$$

By a second order Taylor expansion and continuity of all involved functions,

$$\begin{aligned} |F_\varepsilon (\eta )|\le \Vert D^2 \phi \Vert _{C_b (B_{R/2}(x_0,t_0))} \eta ^{1-2s}\quad \text {for}\; \varepsilon \text {small,}\quad \text {and}\quad F_\varepsilon (\eta )\rightarrow 0 \quad \text {pointwise as}\quad \varepsilon \rightarrow 0. \end{aligned}$$

By the dominated convergence theorem it follows that $I^1_\varepsilon \rightarrow 0$. Finally, the initial condition trivially holds since $u(x,0)=\lim _{\varepsilon \rightarrow 0^+} u_\varepsilon (x,0)=u_0(x).$ $\square$

It remains to proof Lemma 2.4 and then also Corollary 2.5 is proved.

Proof of Lemma 2.4

Since $u_0\in C^{0,\beta }({\mathbb {R}^n})$, $\omega _{u_0}(\delta )\le |u_0|_{C^{0,\beta }}\delta ^\beta$, and basic facts about mollifiers yields

$$\begin{aligned} \Vert D^ku_{0,\delta }\Vert _{C_{b }}\le c(\rho )|u_0|_{C^{0,\beta }}\delta ^{-k+\beta }\quad \text {for}\quad k\in {{\,\mathrm{{{\mathbb {N}}}}\,}}, \end{aligned}$$

see e.g. [18, Appendix A] and [24]. Then, by Lemma 3.4, we have $\Vert \mathcal {L}_\varepsilon [u_{0,\delta }]\Vert _{C_{b }}\le c(s,\rho )\delta ^{-2\,s+\beta }$. $\square$

5 Review of basic results on the fractional heat equation

Here we collect some well-known results on the fractional heat equation that we will need, see e.g. [3, 7, 8, 20]. The one-dimensional problem we consider is

The fundamental solution of (5.1) is given by

$$\begin{aligned} P_{s}(x,t)= \mathcal {F}^{-1}(e ^{-|\xi |^{2s}t})(x) \end{aligned}$$

where $\mathcal {F}$ denotes the Fourier transform and $\mathcal {F}^{-1}$ its inverse. Since the Fourier symbol $e ^{-|\xi |^{2s}t}$ is a tempered distribution, it follows that

$$\begin{aligned} P_{s}\in C^\infty (\mathbb {R}\times (0,\infty )). \end{aligned}$$

Moreover, it is well-known that

$$\begin{aligned} P_{s}(x,t)=t^{-\frac{1}{2s}}F(|x|t^{-\frac{1}{2s}}), \end{aligned}$$

(5.3)

with a profile F(r) that is a smooth and strictly decreasing function of $r>0$. We can also deduce that, for all $\tau >0$, there exist constants $c_1,c_2>0$ depending only on s, such that

$$\begin{aligned} c_1\frac{t}{(t^{\frac{1}{s}}+|x|^2)^{\frac{1+2s}{2}}}\le P_{s}(x,t)\le c_2\frac{t}{(t^{\frac{1}{s}}+|x|^2)^{\frac{1+2s}{2}}}\qquad \text {for all}\; (x,t)\in \mathbb {R}\times [\tau ,\infty ). \end{aligned}$$

(5.4)

Once the basic properties of the fundamental solution are established, we also recall that given any $0\le v_0\in L^1(\mathbb {R})$ (actually a bigger class can be considered), the unique (very weak) solution of (5.1)–(5.2) is given by convolution as

$$\begin{aligned} v(x,t)=(P_{s}(\cdot ,t)*v_0)(x)= \int _{-\infty }^\infty P_{s}(x-y,t) v_0(y)\mathop {}\!\textrm{d}{y}. \end{aligned}$$

(5.5)

Actually, since it is obtained by convolution with a $C^\infty$ kernel, the solution with nonnegative $L^1$-initial data will be $C^\infty$ smooth in $\mathbb {R}\times (0,\infty )$. We will also need:

Lemma 5.1

(Classical solutions) Let $v_0\in C^\infty _b (\mathbb {R})$ be radially symmetric and radially nonincreasing. Then there exists a unique solution $v \in C^\infty _b (\mathbb {R}\times [0,\infty ))$ of (5.1)–(5.2). Moreover, v is radial, radially nonincreasing, and given by (5.5).

Let us also recall Theorem 8.1 in [8].

Lemma 5.2

(Global Harnack principle) Let v be the very weak solution of (5.1)–(5.2) with initial data $v_0\in L^1(\mathbb {R})$ such that $v_0\not \equiv 0$ and

$$\begin{aligned} 0\le v_0(x)\le (1+|x|^2)^{-\frac{1+2s}{2}} \qquad \text {for all}\; |x|\ge R\ge 1. \end{aligned}$$

Then, for all $\tau >0$, there exist constants $k_1,k_2>0$ depending only on s and R, such that

$$\begin{aligned} k_1 \Vert v_0\Vert _{L^1(\mathbb {R})} P_s(x,t) \le v(x,t) \le k_2 \Vert v_0\Vert _{L^1(\mathbb {R})} P_s(x,t)\qquad \text {for all }\; (x,t)\in \mathbb {R}\times [\tau ,\infty ). \end{aligned}$$

Moreover, by (5.4), for all $\tau >0$, there exist constants $C_1,C_2>0$ depending only on s, R, and $\Vert v_0\Vert _{L^1(\mathbb {R})}$, such that

$$\begin{aligned} C_1\frac{t}{(t^{\frac{1}{s}}+|x|^2)^{\frac{1+2s}{2}}} \le v(x,t) \le C_2\frac{t}{(t^{\frac{1}{s}}+|x|^2)^{\frac{1+2s}{2}}}\qquad \text {for all}\; (x,t)\in \mathbb {R}\times [\tau ,\infty ). \end{aligned}$$

6 Smooth solutions and the 1d fractional heat equation

This section investigates different smooth solutions of (1.2)–(1.3).

6.1 Radial solutions

We will now focus on obtaining Theorem 2.10. To do so, we will demonstrate that for radially symmetric and radially nonincreasing functions, the operator $\Delta _{\infty }^{s}$ reduces to the classical fractional Laplacian.

Proposition 6.1

Assume that $\phi \in C^{1,1}(x)\cap B({\mathbb {R}^n})$ is radial and radially nonincreasing, i.e.,

$$\begin{aligned} \phi (x)=\tilde{\Phi }(|x|) \qquad \text {for all}\; x\in {\mathbb {R}^n}, \end{aligned}$$

where $\tilde{\Phi }:[0,\infty )\rightarrow \mathbb {R}$ is nonincreasing. Then

$$\begin{aligned} \Delta _{\infty }^{s}\phi (x)= -(-\partial _{rr}^2)^s\Phi (|x|), \end{aligned}$$

where $\Phi$ is the even extension of $\tilde{\Phi }$ to $\mathbb {R}$: $\Phi (r)=\tilde{\Phi }(r)$ and $\Phi (-r)=\Phi (r)$ for $r\in [0,\infty )$.

Remark 6.2

• (a) When $\phi$ is radial and $\nabla \phi (x)\ne 0$, a similar observation has been done in the proof of Lemma 3.1 in [6].

• (b) “Radially nonincreasing” is needed only when $\nabla \phi (x)=0$, but it cannot be removed in general; see the Sect. 6.2 below.

Proof of Proposition 6.1

Assume $\nabla \phi (x)\not =0$, and let $r=|x|$. We have that

$$\begin{aligned} \nabla \phi (x)=\frac{\Phi '(r)}{r}x, \qquad |\nabla \phi (x)|=|\Phi '(r)|, \qquad \frac{\nabla \phi (x)}{|\nabla \phi (x)|}=- \frac{x}{|x|}, \end{aligned}$$

Lemma 2.1 then yields

$$\begin{aligned} \begin{aligned} \Delta _{\infty }^{s}\phi (x)&=C_s\int _0^\infty \left( \phi \left( x(1+\frac{\eta }{|x|})\right) +\phi \left( x(1-\frac{\eta }{|x|})\right) -2\phi (x)\right) \frac{\mathop {}\!\textrm{d}{\eta }}{\eta ^{1+2s}}\\&=C_s\int _0^\infty \left( \Phi \left( |x|(1+\frac{\eta }{|x|})\right) +\Phi \left( |x|(1-\frac{\eta }{|x|})\right) -2\Phi (|x|)\right) \frac{\mathop {}\!\textrm{d}{\eta }}{\eta ^{1+2s}}\\&=C_s\int _0^\infty \left( \Phi \left( r+\eta \right) +\Phi \left( r-\eta \right) -2\Phi (r)\right) \frac{\mathop {}\!\textrm{d}{\eta }}{\eta ^{1+2s}}\\&= -(-\partial _{rr}^2)^s\Phi (r). \end{aligned} \end{aligned}$$

(6.1)

Assume $\nabla \phi (x)=0$. Note that

$$\begin{aligned} {{\,\textrm{dist}\,}}(0, \partial B_\eta (x)) = {{\,\textrm{dist}\,}}\left( 0, x- \eta \frac{x}{|x|}\right) , \end{aligned}$$

which implies that

$$\begin{aligned} \sup _{z\in \partial B_\eta (x)}\{\phi (z)\}=\phi \left( x- \eta \frac{x}{|x|}\right) , \end{aligned}$$

since $\phi =\Phi (|\cdot |)$ is radially nonincreasing. Then,

$$\begin{aligned} \begin{aligned} \int _{0}^\infty \left( \phi \left( x- \eta \frac{x}{|x|}\right) -\phi (x) \right) \frac{\mathop {}\!\textrm{d}{\eta }}{\eta ^{1+2s}}&\le \sup _{|y|=1}\int _{0}^\infty \left( \phi \left( x+ \eta y\right) -\phi (x) \right) \frac{\mathop {}\!\textrm{d}{\eta }}{\eta ^{1+2s}}\\&\le \int _{0}^\infty \left( \sup _{z\in \partial B_\eta (x)}\{\phi (z)\}-\phi (x) \right) \frac{\mathop {}\!\textrm{d}{\eta }}{\eta ^{1+2s}}\\&=\int _{0}^\infty \left( \phi \left( x- \eta \frac{x}{|x|}\right) -\phi (x) \right) \frac{\mathop {}\!\textrm{d}{\eta }}{\eta ^{1+2s}}, \end{aligned} \end{aligned}$$

so that

$$\begin{aligned} \sup _{|y|=1}\int _{0}^\infty \left( \phi \left( x+ \eta y\right) -\phi (x) \right) \frac{\mathop {}\!\textrm{d}{\eta }}{\eta ^{1+2s}}=\int _{0}^\infty \left( \phi \left( x- \eta \frac{x}{|x|}\right) -\phi (x) \right) \frac{\mathop {}\!\textrm{d}{\eta }}{\eta ^{1+2s}}. \end{aligned}$$

In the same way,

$$\begin{aligned} \inf _{|y|=1}\int _{0}^\infty \left( \phi \left( x+ \eta y\right) -\phi (x) \right) \frac{\mathop {}\!\textrm{d}{\eta }}{\eta ^{1+2s}}=\int _{0}^\infty \left( \phi \left( x+ \eta \frac{x}{|x|}\right) -\phi (x) \right) \frac{\mathop {}\!\textrm{d}{\eta }}{\eta ^{1+2s}}. \end{aligned}$$

Finally, Lemma 2.1 and the argument in (6.1) gives the result. $\square$

Proof of Theorem 2.10

Define $v_0(r):=U_0(|x|)$ for $x\in {\mathbb {R}^n}$ and $r=|x|$, then $v_0\in C_b ^\infty (\mathbb {R})$ is radial and radially nonincreasing. Let v be the corresponding solution of (5.1)–(5.2). By Lemma 5.1, v is radial, radially nonincreasing, and $C_b ^\infty$ smooth. Then by Proposition 6.1, $u(x,t):=v(|x|,t)$ is a classical solution of (1.2)–(1.3). $\square$

6.2 Counterexample for functions not being radially nonincreasing

We show now an example of a function $\phi$ not satisfying the radially nonincreasing assumption in the zero gradient case, and such that the operators $\Delta _{\infty }^{s}$ and $-(-\partial _{rr}^2)^s$ do not coincide. For $R>0$, consider the radial function $\phi :\mathbb {R}^2\rightarrow \mathbb {R}$ given by $\phi (x) = \textbf{1}_{B_{R+1}\setminus B_{R-1}}(x)$, see Fig. 1.

Fig. 1

Characteristic set for the radial counterexample

We note that at $x_0=(r,0)$, we have that $\phi \in C^{1,1}(x_0)\cap B({\mathbb {R}^n})$ and $\nabla \phi (x_0)=0$. Moreover, we have $a=1$ and $R^2+l^2=(R+1)^2$, so that $l=\sqrt{2R+1}$. We denote by $e_1=(1,0)$ and $e_2=(0,1)$. Then

$$\begin{aligned} \begin{aligned} \sup _{|y|=1} \int _0^\infty \left( \phi (x_0+\eta y)-\phi (x_0)\right) \frac{\mathop {}\!\textrm{d}{\eta }}{\eta ^{1+2s}}&\ge \int _0^\infty \left( \phi (x_0+\eta e_2)-\phi (x_0)\right) \frac{\mathop {}\!\textrm{d}{\eta }}{\eta ^{1+2s}}\\&=-\int _l^\infty \frac{\mathop {}\!\textrm{d}{\eta }}{\eta ^{1+2s}}=-\frac{1}{2s} \frac{1}{(2R+1)^s} \end{aligned} \end{aligned}$$

and

$$\begin{aligned} \begin{aligned} \inf _{|y|=1} \int _0^\infty \left( \phi (x_0+\eta y)-\phi (x_0)\right) \frac{\mathop {}\!\textrm{d}{\eta }}{\eta ^{1+2s}}&= \int _0^\infty \left( \phi (x_0+\eta e_1)-\phi (x_0)\right) \frac{\mathop {}\!\textrm{d}{\eta }}{\eta ^{1+2s}}\\&=-\int _1^\infty \frac{\mathop {}\!\textrm{d}{\eta }}{\eta ^{1+2s}}=-\frac{1}{2s}. \end{aligned} \end{aligned}$$

By Lemma 2.1,

$$\begin{aligned} \Delta _{\infty }^{s}\phi (x)\ge -\frac{C_s}{2s}\left( \frac{1}{(2R+1)^s}+1\right) . \end{aligned}$$

On the other hand, we define $\Phi :\mathbb {R}\rightarrow \mathbb {R}$ by $\Phi (r):=\phi (|x|)$ when $r=|x|$ and $\Phi (-r):=\Phi (r)$, and let $r_0=|x_0|$.

$$\begin{aligned} \begin{aligned} -(-\partial _{rr}^2)^s\Phi (r_0)&= C_s\int _0^\infty \big (\Phi (r_0+\eta )+\Phi (r_0-\eta )-2\Phi (r_0)\big ) \frac{\mathop {}\!\textrm{d}{\eta }}{\eta ^{1+2s}}\\&=C_s\int _0^\infty \left( \phi \left( x_0+\eta \frac{x_0}{|x_0|}\right) -\phi (x_0)\right) \frac{\mathop {}\!\textrm{d}{\eta }}{\eta ^{1+2s}}\\&\quad +C_s\int _0^\infty \left( \phi \left( x_0-\eta \frac{x_0}{|x_0|}\right) -\phi (x_0)\right) \frac{\mathop {}\!\textrm{d}{\eta }}{\eta ^{1+2s}}\\&=-C_s \int _1^\infty \frac{\mathop {}\!\textrm{d}{\eta }}{\eta ^{1+2s}}-C_s \int _1^{2R-1} \frac{\mathop {}\!\textrm{d}{\eta }}{\eta ^{1+2s}}-C_s \int _{2R+1}^\infty \frac{\mathop {}\!\textrm{d}{\eta }}{\eta ^{1+2s}}\\&=-\frac{C_s}{2s}\left( 2+\frac{1}{(2R+1)^{2s}}-\frac{1}{(2R-1)^{2s}}\right) . \end{aligned} \end{aligned}$$

Finally, taking R big enough, we get

$$\begin{aligned} -\Delta _{\infty }^{s}\phi (x_0)-(-\partial _{rr})^s\Phi (r_0)\le -\frac{C_s}{2s}\left( 1+\frac{1}{(2R+1)^{2s}}-\frac{1}{(2R-1)^{2s}}-\frac{1}{(2R+1)^s}\right) <0. \end{aligned}$$

Thus, the operators cannot coincide.

6.3 Another example of smooth solutions

We present here another example of functions for which $\Delta _{\infty }^{s}$ reduces to a one-dimensional fractional Laplacian. As before, this allows to produce smooth solutions of (1.2)–(1.3). We will adopt the notation $x=(x_1,\ldots , x_n)\in {\mathbb {R}^n}$.

Lemma 6.3

Assume that $\Phi \in C^2_{b }(\mathbb {R})$ is nondecreasing, and let $\phi \in C^2_{b }({\mathbb {R}^n})$ be defined as

$$\begin{aligned} \phi (x):=\Phi (x_1). \end{aligned}$$

Then

$$\begin{aligned} \Delta _{\infty }^{s}\phi (x)= -(-\partial _{x_1x_1}^2)^s \Phi (x_1). \end{aligned}$$

Remark 6.4

We could also take $\Phi \in C^2_{b }(\mathbb {R})$ and nonincreasing in the above result.

Proof of Lemma 6.3

Note that $\nabla \phi (x)=\Phi '(x_1)e_1$ and $\Phi '(x_1)\ge 0$. On one hand, if $\Phi '(x_1)=0$, then it is clear that

$$\begin{aligned} \begin{aligned} \Delta _{\infty }^{s}\phi (x)&=C_s\sup _{|y|=1} \int _{0}^\infty \big (\phi (x+ \eta y)-\phi (x)\big ) \frac{\mathop {}\!\textrm{d}{\eta }}{\eta ^{1+2s}} + C_s\inf _{|y|=1} \int _{0}^\infty \big (\phi (x-\eta y)-\phi (x)\big ) \frac{\mathop {}\!\textrm{d}{\eta }}{\eta ^{1+2s}}\\&=C_s\int _{0}^\infty \big (\phi (x+ \eta e_1)-\phi (x)\big ) \frac{\mathop {}\!\textrm{d}{\eta }}{\eta ^{1+2s}} + C_s\int _{0}^\infty \big (\phi (x-\eta e_1)-\phi (x)\big ) \frac{\mathop {}\!\textrm{d}{\eta }}{\eta ^{1+2s}}\\&=C_s\int _{0}^\infty \big (\Phi (x_1+ \eta )+\Phi (x_1-\eta )-2\Phi (x_1)\big ) \frac{\mathop {}\!\textrm{d}{\eta }}{\eta ^{1+2s}}\\&=-(-\partial _{x_1x_1}^2)^s \Phi (x_1). \end{aligned} \end{aligned}$$

On the other hand, if $\Phi '(x_1)>0$, then $\zeta =\nabla \phi (x)/|\nabla \phi (x)|=e_1$ (cf. Lemma 2.1) and

$$\begin{aligned} \begin{aligned} \Delta _{\infty }^{s}\phi (x)&=C_s\int _0^\infty \left( \phi (x+\eta e_1)+\phi (x-\eta e_1)-2\phi (x)\right) \frac{\mathop {}\!\textrm{d}{\eta }}{\eta ^{1+2s}}\\&=-(-\partial _{x_1x_1}^2)^s \Phi (x_1). \end{aligned} \end{aligned}$$

$\square$

7 Comparison and local truncation errors

We are not able to prove comparison (neither uniqueness) for the family of viscosity solutions constructed in Sect. 4. However, we are able to compare any constructed viscosity solution with any classical solution.

The argument is based on the fact that for classical solutions we can get full convergence of the scheme (2.2)–(2.3) (and not just compactness and convergence up to a subsequence). Then we can inherit the comparison result of the scheme to the limit solutions.

7.1 Comparison and convergence estimates under regularity assumptions

Proposition 7.1

Assume (CFL) and $u_0\in BUC({\mathbb {R}^n})$. Let $u,v\in C^2_b ({\mathbb {R}^n}\times [0,\infty ))$ be respective classical sub- and supersolutions of (1.2)–(1.3). Then:

• (a) Let $U_\varepsilon ,V_\varepsilon$ respective super- and subsolutions of the scheme (2.2)–(2.3). Then, for all $T<\infty$,

  $$\begin{aligned} u+o_\varepsilon (1) + O(\tau ) \le U_\varepsilon \quad \text {and}\quad v+o_\varepsilon (1) + O(\tau )\ge V_\varepsilon ,\quad \text {uniformly in }\,{\mathbb {R}^n}\times \{\tau {{\,\mathrm{{{\mathbb {N}}}}\,}}\cup 0\}. \end{aligned}$$

• (b) Let $U_\varepsilon$ be a solution of the scheme (2.2)–(2.3). Then, for all $T<\infty$,

  $$\begin{aligned} u+o_\varepsilon (1)+O(\tau ) \le U_\varepsilon \le v+o_\varepsilon (1)+O(\tau ),\qquad \text {uniformly in}\; {\mathbb {R}^n}\times \{\tau {{\,\mathrm{{{\mathbb {N}}}}\,}}\cup 0\}. \end{aligned}$$

We immediately get:

Corollary 7.2

Assume (CFL). Let $u\in C^2_b ({\mathbb {R}^n}\times [0,\infty ))$ be a classical solution of (1.2)–(1.3), and $U_\varepsilon$ be a solution of the scheme (2.2)–(2.3), both with initial data $u_0$. Then, for all $T<\infty$,

$$\begin{aligned} \max _{t_j\le T}\Vert u(\cdot , t_j)- U_\varepsilon (\cdot ,t_j)\Vert _{C_b ({\mathbb {R}^n})} = o_\varepsilon (1) + O(\tau ). \end{aligned}$$

Proof of Proposition 7.1

(a) Define the local truncation error,

$$\begin{aligned} (R_\varepsilon )^j(x):= \frac{u(x, t_j+\tau )-u(x, t_j)}{\tau }- \mathcal {L}_\varepsilon [u(\cdot , t_j)](x). \end{aligned}$$

(7.1)

Clearly, since u is a classical subsolution of (1.2) we have (from Lemma 3.3)

$$\begin{aligned} \begin{aligned} (R_\varepsilon )^j(x)&\le \left( \frac{u(x, t_j+\tau )-u(x, t_j)}{\tau } - \partial _t u(x,t_j)\right) - \big (\mathcal {L}_\varepsilon [u(\cdot , t_j)](x)- \Delta _{\infty }^{s}[u(\cdot , t_j)](x)\big )\\&\le O(\tau )+ o_\varepsilon (1) \end{aligned} \end{aligned}$$

with uniform bounds in $t_j$ and x.

Define now $e^j(x)=u(x,t_j)-(U_\varepsilon )^j(x)=u(x,t_j)-U_\varepsilon (x,t_j)$. By (2.2) and (7.1), we get

$$\begin{aligned} \begin{aligned} e^{j+1}(x)&=u(x,t_{j+1})-(U_\varepsilon )^{j+1}(x)\\&\le e^{j}(x) + \tau \big ( \mathcal {L}_\varepsilon [u(\cdot , t_j)](x) - \mathcal {L}_\varepsilon [(U_\varepsilon )^j](x) \big ) + \tau (R_\varepsilon )^j(x)\\&= e^j(x) (1- \tau \frac{C_s}{s \varepsilon ^{2s}}) + \tau C_s \left( \sup _{|y|=1}\int _\varepsilon ^\infty u(x+\eta y,t_j) \frac{\mathop {}\!\textrm{d}{\eta }}{\eta ^{1+2s}} - \sup _{|y|=1}\int _\varepsilon ^\infty (U_\varepsilon )^j(x+\eta y) \frac{\mathop {}\!\textrm{d}{\eta }}{\eta ^{1+2s}} \right) \\&\quad + \tau C_s \left( \inf _{|y|=1}\int _\varepsilon ^\infty u(x+\eta y,t_j) \frac{\mathop {}\!\textrm{d}{\eta }}{\eta ^{1+2s}} - \inf _{|y|=1}\int _\varepsilon ^\infty (U_\varepsilon )^j(x+\eta y) \frac{\mathop {}\!\textrm{d}{\eta }}{\eta ^{1+2s}} \right) +\tau (R_\varepsilon )^j(x)\\&\le e^j(x) (1- \tau \frac{C_s}{s \varepsilon ^{2s}}) + 2\tau C_s \sup _{|y|=1}\int _\varepsilon ^\infty e^j(x+\eta y) \frac{\mathop {}\!\textrm{d}{\eta }}{\eta ^{1+2s}} +\tau (R_\varepsilon )^j(x)\\&\le \sup _{x\in {\mathbb {R}^n}}e^j(x) (1- \tau \frac{C_s}{s \varepsilon ^{2s}}) + 2\tau C_s \sup _{|y|=1}\int _\varepsilon ^\infty \sup _{x\in {\mathbb {R}^n}}e^j(x+\eta y) \frac{\mathop {}\!\textrm{d}{\eta }}{\eta ^{1+2s}} +\tau \sup _{t_j\le T}\sup _{x\in {\mathbb {R}^n}}(R_\varepsilon )^j(x)\\&= \sup _{x\in {\mathbb {R}^n}}e^j(x) (1- \tau \frac{C_s}{s \varepsilon ^{2s}}) + \tau \frac{C_s}{s \varepsilon ^{2s}}\sup _{x\in {\mathbb {R}^n}}e^j(x)+\tau \sup _{t_j\le T}\sup _{x\in {\mathbb {R}^n}}(R_\varepsilon )^j(x)\\&= \sup _{x\in {\mathbb {R}^n}}e^j(x)+\tau \sup _{t_j\le T}\sup _{x\in {\mathbb {R}^n}}(R_\varepsilon )^j(x). \end{aligned} \end{aligned}$$

I.e.,

$$\begin{aligned} \sup _{x\in {\mathbb {R}^n}}e^{j+1}(x)\le \sup _{x\in {\mathbb {R}^n}}e^j(x)+\tau \sup _{t_j\le T}\sup _{x\in {\mathbb {R}^n}}(R_\varepsilon )^j(x). \end{aligned}$$

Iterating, we obtain

$$\begin{aligned} \begin{aligned} \sup _{x\in {\mathbb {R}^n}}e^{j}(x)&\le \sup _{x\in {\mathbb {R}^n}}e^0(x)+j\tau \sup _{t_j\le T}\sup _{x\in {\mathbb {R}^n}}(R_\varepsilon )^j(x)\\&\le \sup _{x\in {\mathbb {R}^n}}\big (u(x,0)-(U_\varepsilon )^0(x)\big )+T\big (O(\tau )+ o_\varepsilon (1)\big )\\&\le 0+O(\tau )+ o_\varepsilon (1). \end{aligned} \end{aligned}$$

By changing the roles of $u,U_\varepsilon$ with $-v,-V_\varepsilon$, we obtain the other inequality in a similar way.

(b) Follows directly from part (a). $\square$

7.2 Comparison for classical sub- and supersolutions

In order to continue, we note that Proposition 7.1 and Corollary 7.2 hold exactly as before with the time interpolant $u_\varepsilon$ replacing $U_\varepsilon$ (cf. the proof of Corollary 3.7).

Proof of Theorem 2.6

The proof is similar for $\overline{u},\underline{u}$, and we only provide it for $\underline{u}$. Since u is a constructed viscosity solution in the sense of Theorem 2.2, by Proposition 3.8 there is a sequence $u_{\varepsilon _j}\in BUC({\mathbb {R}^n}\times [0,\infty ))$ of time-interpolated solutions of (2.2)–(2.3) with initial condition $u_0$ such that

$$\begin{aligned} u_{\varepsilon _j} \rightarrow u \qquad locally uniformly in \; {\mathbb {R}^n}\times [0,\infty ) \;as \; \varepsilon _j\rightarrow 0^+. \end{aligned}$$

Then by taking the limit as $\varepsilon _j\rightarrow 0^+$ in Proposition 7.1(b), we get $\underline{u}\le u$. $\square$

8 Global Harnack principle

The proof of Theorem 2.13 is based on the relation between our problem and the smooth solutions of the fractional heat equation, the properties of smooth solutions in 1D for the fractional heat equation presented in the review Sect. 5, and the comparison principle of Theorem 2.6 for viscosity and classical solutions.

Proof of Theorem 2.13

A key point of the proof is the fact that if v is a smooth, radial, and radially nonincreasing solution of the fractional heat equation in one dimension, then $u(x,t):=v(|x|,t)$ is a solution of (1.2)–(1.3). See Theorem 2.10.

(1) Upper bound. Let $\overline{u}_0:{\mathbb {R}^n}\rightarrow \mathbb {R}_+$ be such that (i) $\overline{u}_0(x)=(1+|x|^2)^{-\frac{1+2s}{2}}$ if $|x|\ge R+1$; (ii) $\overline{u}_0$ is radially symmetric and radially nonincreasing; (iii) $\overline{u}_0\in C^\infty _b ({\mathbb {R}^n})$; and (iv) $u_0\le \overline{u}_0$ in ${\mathbb {R}^n}$. Consult Fig. 2.

Moreover, let $\overline{v}_0:\mathbb {R}\rightarrow \mathbb {R}$ be defined by $\overline{v}_0(r):=\overline{u}_0(|x|)$ with $r=|x|$ and $\overline{v}_0(-r):=\overline{v}_0(r)$. Clearly, $\overline{v}_0\in C^\infty _b (\mathbb {R})$ is radially symmetric and radially nonincreasing. Let $\overline{v}$ be the corresponding solution of the fractional heat equation (5.1)–(5.2) and define $\overline{u}(x,t)=\overline{v}(|x|,t)$. By Theorem 2.10, $\overline{u}\in C^\infty _b ({\mathbb {R}^n}\times [0,\infty ))$ is a classical solution of (1.2)–(1.3). Moreover, $\overline{u}$ is radial and radially nonincreasing. Since $\overline{u}_0(x)=(1+|x|^2)^{-\frac{1+2s}{2}}$ if $|x|\ge R+1$, then $\overline{v}_0(r)=(1+|r|^2)^{-\frac{1+2s}{2}}$ if $|r|\ge R+1$, so that, by Lemma 5.2, for all $t>\tau$ we have

$$\begin{aligned} \overline{u}(x,t)=\overline{v}(|x|,t) \le k_2\Vert \overline{v}_0\Vert _{L^1(\mathbb {R})}P_s(|x|,t)\le C_1 \frac{t}{(t^{\frac{1}{s}}+|x|^2)^{\frac{1+2s}{2}}} \end{aligned}$$

Finally, since $\overline{u}\in C^\infty _b ({\mathbb {R}^n}\times [0,\infty ))$ is a classical solution of (1.2)–(1.3) and $u_0\le \overline{u}_0$ we have, by Theorem 2.6, that $u(x,t) \le \overline{u}(x,t)$.

Fig. 2

Upper bound for $u_0$ in the proof of Theorem 2.13

(2) Lower bound. Without loss of generality, assume $u_0(0)=\sup _{x\in {\mathbb {R}^n}} u_0(x)$>0. By continuity of $u_0$, there exists $R_0>0$ such that $u_0(x)\ge u_0(0)/2$ for all $x\in B_{R_0}(0)$. Consider e.g. the scaled standard mollifier

$$\begin{aligned} \underline{u}_0(x)=\frac{u_0(0)}{2}e ^{1-\frac{R_0^2}{(R_0^2-|x|^2)_+}} \end{aligned}$$

Clearly, (i) $\underline{u}_0(x)=0\le (1+|x|^2)^{-\frac{1+2s}{2}}$ if $|x|\ge R_0$; (ii) $\underline{u}_0$ is radially symmetric and radially nonincreasing; (iii) $\underline{u}_0\in C^\infty _b ({\mathbb {R}^n})$; and (iv) $u_0\ge \underline{u}_0$ in ${\mathbb {R}^n}$ since

$$\begin{aligned} \underline{u}_0(x) \le \underline{u}_0(0) = \frac{u_0(0)}{2}e ^{1-\frac{R_0^2}{(R_0^2-0)_+}}= \frac{u_0(0)}{2} \le u_0(x) \quad \hbox { for all}\ x\in B_{R_0}(0) \end{aligned}$$

and $\underline{u}_0(x)=0\le u_0$ for $x\in {\mathbb {R}^n}{\setminus } B_{R_0}(0)$ (see Fig. 3). From here, the proof follows as in Step (1) by using the lower bound in Lemma 5.2. $\square$

Fig. 3

Lower bound for $u_0$ in the proof of Theorem 2.13

Fig. 4

Evolution of the level sets of the solution

9 Extensions and open problems

• There is an important open problem concerning the uniqueness and general comparison principle of viscosity solutions, either defined in our way or another suitable way that includes existence. For the moment we know that the following two classes of BUC viscosity solutions are unique: (i) radial radially nonincreasing solutions and (ii) monotone solutions evolving in one dimension only. Uniqueness in these cases follows by comparison with classical solutions. The problem is also open for elliptic equations of the same type, cf. [6].

• A main question that we deal with here is: how different is the theory and its results from the linear case (fractional heat equation)? The answer seems to be that they are quite different if $n\ge 2$, since then the infinity fractional Laplacian is a heavily nonlinear operator.

• It is not clear whether for $n\ge 2$ the solutions evolve in time towards a radial profile (as in the local case, see below) or preserve a certain distortion. This is an interesting open problem to which we give a partial answer in our Sect. 8 with the global Harnack principle. In Fig. 4 (obtained with a rigorous finite difference scheme taken from the companion paper [15]) the distortion of the initial datum can still be observed for all the computed times.

• In particular, the evolution equation for the local version posed in the whole space has been studied by Portilheiro and the fourth author in [34, 35]. Then, there is a fine asymptotic behaviour as $t\rightarrow \infty$ that implies a sharp convergence rate to radiality. The Aleksandrov Principle is a main ingredient in the proof. On the condition that the Aleksandrov Principle is true for some class of solutions of our Cauchy problem, we could also obtain a similar sharp asymptotic behaviour as $t\rightarrow \infty$ for such solutions. Such discussion is not included here.

• Large part of the concepts and results of this paper can be applied to the more general equation $\partial _tu =\Delta _{\infty }^{s}u+ f(x,t)$. In particular, this could be applied to the stationary equation $\Delta _{\infty }^{s}u = f(x)$, thus relating the present results to the results of [6].

• We end the discussion by including an example demonstrating that the operator $\Delta _{\infty }^{s}$ could indeed be pointwise discontinuous, even when applied to some smooth function $\phi$. Consider $\Phi \in C_{b }^2(\mathbb {R})$ satisfying $\Phi (x_1)=\Phi (-x_1)$ and strictly decreasing for $x_1\ge 0$. As in Lemma 6.3, we define $\phi (x):=\Phi (x_1)$ (see Fig. 5) where, for the sake of simplicity, $x=(x_1,x_2)\in \mathbb {R}^2$. On one hand, when $x_1\ne 0$, we have $\zeta =\pm e_1$ (cf. Lemma 2.1) which yields $\Delta _{\infty }^{s}\phi (x_1,x_2)=-(-\partial _{x_1x_1}^2)^s\Phi (x_1)$. On the other hand, when $x_1=0$, $\nabla \phi (0,x_2)=0$ and by construction,

  $$\begin{aligned} \inf _{|y|=1} \int _{0}^\infty \big (\phi (0,x_2)-\eta y)-\phi (0,x_2)\big ) \frac{\mathop {}\!\textrm{d}{\eta }}{\eta ^{1+2s}}= \int _{0}^\infty \big (\Phi (x_1-\eta )-\Phi (x_1)\big ) \frac{\mathop {}\!\textrm{d}{\eta }}{\eta ^{1+2s}}. \end{aligned}$$

  Since $x_1=0$ is a maximum point and $\phi (0,x_2)=\phi ((0,x_2)+\eta e_2)=\Phi (x_1)$,

  $$\begin{aligned} 0\ge & {} \sup _{|y|=1} \int _{0}^\infty \big (\phi ((0,x_2)+ \eta y)-\phi (0,x_2)\big ) \frac{\mathop {}\!\textrm{d}{\eta }}{\eta ^{1+2s}}\\\ge & {} \int _{0}^\infty \big (\phi ((0,x_2)+ \eta e_2)-\phi (0,x_2)\big ) \frac{\mathop {}\!\textrm{d}{\eta }}{\eta ^{1+2s}}= 0. \end{aligned}$$

  We then conclude by Lemma 2.1 and symmetry of $\Phi$ that

  $$\begin{aligned} \begin{aligned} \Delta _{\infty }^{s}\phi (0,x_2)&=C_s\int _{0}^\infty \big (\Phi (- \eta )-\Phi (0)\big ) \frac{\mathop {}\!\textrm{d}{\eta }}{\eta ^{1+2s}}\\&= \frac{1}{2}C_s\int _{0}^\infty \big (\Phi (\eta )+\Phi (- \eta )-2\Phi (0)\big ) \frac{\mathop {}\!\textrm{d}{\eta }}{\eta ^{1+2s}}\\&=-\frac{1}{2}(-\partial _{x_1x_1}^2)^s\Phi (0).\\ \end{aligned} \end{aligned}$$

  Hence,

  $$\begin{aligned} \Delta _{\infty }^{s}\phi (x_1,x_2)= {\left\{ \begin{array}{ll} -(-\partial _{x_1x_1}^2)^s\Phi (x_1),&{}\qquad \text {if}\qquad x_1\ne 0,\\ -\frac{1}{2}(-\partial _{x_1x_1}^2)^s\Phi (x_1),&{}\qquad \text {if}\qquad x_1=0. \end{array}\right. } \end{aligned}$$

Fig. 5

Example of discontinuity of the operator $\Delta _{\infty }^{s}$