Solutions of the space-fractional heat equation with polynomial growth: Liouville theorems and Schauder estimates

Introduction

In this paper, we extend the definition of viscosity solutions of a space-fractional heat equation to functions with polynomial growth at infinity. This can be seen as the parabolic counterpart of the theory put forth in [DSV19] for the fractional elliptic case.

We recall that, given $s \in(0,1)$, the fractional Laplacian of a function $u : \mathbb{R}^n \to\mathbb{R}$ is defined as

where p.v. denotes the principal value, and $c_{n,s} > 0$ is a normalizing constant depending only on $n$ and $s$. We also use the notation $B_\varepsilon^c(x) := \mathbb{R}^n \setminus B_\varepsilon(x)$.

For a more detailed study of the fractional Laplacian and more general non-local operators, we refer to [Kas09; RS14; AV19].

We stress that the definition in (1) is well-posed if:

• $u$ is sufficiently regular near $x$, e.g., $u$ is $C^\alpha$-regular at $x$, for some $\alpha> 2s$;

• $u$ satisfies the growth condition at infinity

A defining feature of non-local operators is that the behavior of a function at infinity may deeply affect the value of the operator. This makes the growth condition in (2) a genuine source of difficulties in the regularity theory of non-local equations and free boundary problems, for instance when considering rescalings or blow-up sequences.

In this spirit, several recent developments in fractional elliptic regularity theory have benefited from the notion of harmonic functions “up to a polynomial” (see, e.g., [AR20; DFV22; Dip+22; FR22; AT24; RTW25; GK25; RW26; Dip+ar]). Moreover, the regularity results established in [DSV19] introduced several necessarily sharp refinements for handling the polynomial case, making them particularly well suited to delicate regularity questions, even in settings where polynomial growth is not explicitly assumed (see, e.g., [CFW18; Cir+23]).

The aim of this work is therefore to introduce an alternative notion of (viscosity) solution for non-local parabolic equations, in order to make sense of the fractional heat operator under a milder growth assumption.

Remark 1.2. We stress that, in the classical setting, the lack of any growth assumption allows one to consider solutions with polynomial growth (and, in fact, any growth) at infinity. For instance, the function $u(t,x) := 2nt + |x|^2$ satisfies

By analogy, one would like to say that the function $v(t,x) := c_{n,s}t-|x|^{2s}$, for a suitable constant $c_{n,s}>0$ depending only on $n$ and $s$, satisfies

Let $u : (-\infty,0] \times\mathbb{R}^n \to\mathbb{R}$ be a function with polynomial growth at infinity in space. The main idea is to consider a sequence of smooth cut-off functions $\{\eta_R\}_R$ such that the fractional Laplacian of $\eta_Ru$ is well-defined. Then, we obtain that the space-fractional heat operator of $u$, that is,

diverges as $R \to+\infty$, and it can be expressed as a given function plus a (non-uniquely determined) polynomial remainder. We will make a precise sense of this in Definition 2.3 below.

To conclude this introductory part, we present a possible application of our notion of solution for non-local parabolic equations.

The problem addressed in this paper stems from a question by Felix Otto: if a smooth function solves a homogeneous parabolic equation modulo a polynomial in space-time and has polynomial growth, must it itself be a space-time polynomial? This question, as well as our notion of solution, was motivated by problems arising in singular stochastic partial differential equations (SPDEs) and Hairer’s regularity structures¹, where polynomials play a distinguished role.

Indeed, in the framework of “subcritical” semilinear SPDEs, Hairer’s theory proposes a generalized notion of Taylor expansion that splits functions into a “rough” term and a regular polynomial remainder. Due to the local subcriticality, non-linear fluctuations remain controllable at a small scale. This implies that solutions of the semilinear equation resemble those of the corresponding linear problems, and can be constructed via a suitable approximation and limit procedure from a regularized problem.

We refer to [FH14; Hai14; FGH19; OW19; OST23] for both introductory overviews and detailed discussions on regularity structures and related topics.

Building on the formalism of solutions “up to a polynomial”, we answer Felix Otto’s question with our main result, that is a Liouville-type theorem which establishes that any function that is $s$-caloric, up to a polynomial, on all of time-space and has polynomial growth must be a polynomial both in space and time (see Theorem 2.10 below).

This Liouville-type result also relies on a sharp version, valid “up to polynomials”, of the Schauder estimates for solutions to a space-fractional heat equation (see Theorem 2.5 below).

A useful structural result is also that solutions modulo $k$ are stable under a uniform control of the mass and locally uniform convergence (see the forthcoming Theorem 2.7).

The extension of the elliptic case, introduced in [DSV19], to the parabolic setting requires significant additional arguments. Indeed, besides the technical transition from elliptic to parabolic regularity theory, a bespoke analysis is needed to account for the time dependence of the spatial polynomial representation of the solutions, in particular to distinguish between general time-dependent coefficients in the spatial polynomials and genuinely time-dependent polynomial coefficients.

Notation and main results

For our purposes, let us recall that if $\alpha:= (\alpha_1,\ldots,\alpha_n) \in\mathbb{N}^n$ is a multi-index, for every $x \in\mathbb{R}^n$, we write

Moreover, we say that a multi-index $\alpha\in\mathbb{N}^n$ is of degree at most $k$, and we simply write $|\alpha| \le k$, if

Then, for a general vector space $W$ (finite-dimensional or not), let us define

and

We also adopt the conventions

and

In what follows, we will focus on

and, for any interval $I\subseteq\mathbb{R}$ and for any $\sigma>0$,

Namely, $\bigl(C^\sigma(I)\bigr)_k^{[x]}$ is the family of polynomials in $x$ with degree at most $k$ and whose coefficients are $C^\sigma(I)$-functions of $t$, while $\mathbb{R}_k^{[t,x]}$ is the smallest space in which the coefficients are monomials in $t$, with the additional constraint that the degree of the total polynomial in $x$ and $t$ has degree at most $k$.

Furthermore, given $\gamma>0$, we conform to the notation $C^\gamma=C^{[\gamma],\{\gamma\}}$ for the space of all $C^{[\gamma]}$ functions with $\{\gamma\}$-Hölder continuous highest-order derivatives. Here, as customary, $[\gamma]$ and $\{\gamma\}$ denote the integer part and the fractional part of $\gamma$, respectively.

Throughout the rest of this paper, we consider a fixed $\sigma\in(0,1)$.

Since we work within the framework of viscosity solutions for parabolic equations (see Definition 2.2 below), we recall the definition of the standard parabolic cylinder of radius $r>0$ and height $r^{2s}$, namely

extending also this notation to $Q_\infty:= (-\infty,0]\times\mathbb{R}^n$.

Besides, for later convenience, let us introduce the closed, restricted cylinder

Moreover, let us consider, for any $k\in\mathbb{N}$, the measure $\omega_k$ on $\mathbb{R}^n$, which is absolutely continuous with respect to the Lebesgue measure, and is defined as