Technical Summary: Spreading speeds for nonlocal Fisher–KPP equations with time-dependent asymmetric kernels

Problem Statement

This paper investigates the Cauchy problem for nonautonomous Fisher–KPP equations featuring nonlocal diffusion. The governing equation is:
$$\begin{cases} \partial_t u(t, x) = \int_{\mathbb{R}} K(t, x -y)[u(t, y) -u(t, x)]dy + u(t, x)f(t, u(t, x)), & t > 0, x \in \mathbb{R}, \\ u(0, x) = u_0(x), & x \in \mathbb{R}, \end{cases}$$
where $u(t, x)$ represents population density, $u_0$ is a non-negative, compactly supported initial condition, and the reaction term $f$ is of Fisher–KPP type.

The core challenge addressed is the determination of spreading speeds (both rightward $c^*_+$ and leftward $c^*_-$) under two specific complexities not simultaneously handled in many prior works:

1. Time-dependence: The dispersal kernel $K(t, y)$ and the reaction rate depend on time, modeling heterogeneous environments (e.g., seasonal changes).
2. Asymmetry and Non-locality: The kernel $K(t, y)$ is not required to be symmetric in space, and its support is not required to contain a neighborhood of the origin. This generalizes standard local diffusion (Brownian motion) and symmetric nonlocal models.

Methodology

The authors employ a combination of comparison principles, refined maximum principles, and the construction of sub- and super-solutions. Key methodological components include:

1. Uniform Mean Value Framework

To handle time heterogeneity without assuming periodicity, the paper utilizes the concept of uniform mean value (Definition 1.3). A function $h$ has a uniform mean value $\langle h \rangle$ if the limit $\lim_{T \to \infty} \frac{1}{T} \int_0^T h(t+s) dt$ exists uniformly for $s \ge 0$. This framework encompasses periodic, almost periodic, and uniquely ergodic time dependencies.

2. Refined Maximum Principle

Standard strong maximum principles often fail for nonlocal equations where the kernel's support does not include the origin (i.e., no local diffusion). The authors prove a refined maximum principle (Lemma 2.4). This lemma establishes that if the kernel is positive on a set of positive measure in time and space, the solution becomes strictly positive on a half-line (or the whole line) after a finite time, provided the initial data is positive on a sufficiently long interval. This overcomes the lack of immediate smoothing effects typical of parabolic equations.

3. Linear Determinacy via Variational Formulas

The spreading speeds are derived using a linear determinacy principle. By analyzing the linearized equation near the unstable state $u=0$ with an ansatz of the form $e^{-\lambda(x - \int_0^t c(s)ds)}$, the authors define a function $H(\lambda)$ representing the time-averaged exponential moment:
$$H(\lambda) = \left\langle \int_{\mathbb{R}} K(\cdot, y)[e^{\lambda y} - 1]dy + r(\cdot) \right\rangle,$$
where $r(t) = f(t, 0)$. The spreading speeds are then defined as:
$$c^*_+ = \inf_{\lambda \in (0, i_+)} \frac{H(\lambda)}{\lambda}, \quad c^*_- = \inf_{\lambda \in (i_-, 0)} \frac{H(\lambda)}{-\lambda},$$
where $i_+$ and $i_-$ are the boundaries of the interval where the exponential moment is finite.

4. Sub- and Super-Solution Construction

• Upper Estimates: Proved using exponential super-solutions of the form $A \exp\{-\lambda x + \int_0^t m_\lambda(s) ds\}$, where $m_\lambda$ relates to the kernel and reaction term.
• Lower Estimates: Constructed using localized exponential sub-solutions with moving centers. The centers move with speeds determined by the uniform mean values of truncated kernel integrals. A key technical step involves handling the "endpoint case" where the infimum of the speed formula occurs at the boundary of the admissible $\lambda$ interval. This is addressed via an approximation method (truncating the kernel) to convert endpoint minima into interior minima.

Key Results

1. Existence and Formula for Spreading Speeds

Theorem 1.8 states that under assumptions of KPP-type reaction and exponentially bounded, time-dependent kernels with uniform mean values, the spreading speeds $c^*_+$ and $c^*_-$ exist and are given exactly by the linearly determined formulas above.

• The speeds are determined by the time-averaged properties of the kernel and reaction rate, rather than instantaneous values.
• The result holds even if the kernel is asymmetric, leading to potentially different speeds in opposite directions ($c^*_+ \neq c^*_-$).

2. Sign of the Sum of Speeds

The paper establishes a relationship between the sum of the spreading speeds and the net dispersal:
$$\text{sgn}(c^*_+ + c^*_-) = \text{sgn}\left( \int_{\mathbb{R}} \langle K(\cdot, y) \rangle dy \right).$$
If the time-averaged kernel has a net positive mass, at least one direction of propagation is positive. If the average kernel is zero, both speeds are zero.

3. Convergence to the Stable State

Inside the spreading region defined by $(-c^*_-, c^*_+)$, the solution $u(t, x)$ converges uniformly to the stable state $u=1$ as $t \to \infty$. Specifically, for any $\epsilon > 0$:
$$\lim_{t \to \infty} \sup_{(-c^*_ - + \epsilon)t \le x \le (c^*_+ - \epsilon)t} |u(t, x) - 1| = 0.$$
Outside this region (specifically $x \ge (c^*_+ + \epsilon)t$ or $x \le (-c^*_ - - \epsilon)t$), the solution converges uniformly to 0.

Significance and Claims

The paper claims to provide a rigorous extension of spreading speed theory to nonautonomous and asymmetric nonlocal Fisher–KPP equations.

• Generality of Time Dependence: Unlike previous works restricted to periodic media, this result applies to almost periodic and uniquely ergodic time dependencies via the uniform mean value assumption.
• Relaxation of Kernel Support: The proof does not require the kernel's support to contain a neighborhood of the origin. This is significant because many biological dispersal models involve long-distance jumps where local interaction is absent. The authors explicitly note that this necessitates the development of the refined maximum principle, as standard strong maximum principles are unavailable.
• Asymmetry: The framework naturally accommodates asymmetric kernels, allowing for distinct rightward and leftward spreading speeds, a feature often ignored in symmetric models.
• Linear Determinacy: The paper confirms that the linear determinacy principle holds in this complex setting, where the "linear selected speed" is determined by the time-average of the exponential moments rather than a fixed moment.

The authors position their work as filling a gap in the literature regarding propagation in time-heterogeneous, nonlocal, and asymmetric environments, providing a unified approach that covers periodic, almost periodic, and uniquely ergodic cases without requiring the compactness of the semiflow typically found in local diffusion equations.