Heat equations driven by mixed local-nonlocal operators with exponential nonlinearity

Technical Summary: Heat Equations Driven by Mixed Local-Nonlocal Operators with Exponential Nonlinearity

Problem Statement
This paper investigates the Cauchy problem for a semilinear heat equation driven by a mixed local-nonlocal operator $L := -\Delta + (-\Delta)^s$ with $s \in (0, 1)$, featuring an exponential nonlinearity $f(u)$. The equation is given by:
$$\begin{cases} \partial_t u(x, t) + Lu(x, t) = f(u(x, t)), & (x, t) \in \mathbb{R}^d \times (0, \infty), \\ u(0, x) = u_0(x), & x \in \mathbb{R}^d, \end{cases}$$
where the nonlinearity $f$ exhibits exponential growth at infinity (specifically $f(u) \sim e^{|u|^p}$ as $|u| \to \infty$ with $p > 1$) and satisfies $f(0)=0$. The operator $L$ combines the classical Laplacian (modeling local diffusion) and the fractional Laplacian (modeling nonlocal, jump-type diffusion). The study focuses on establishing local and global well-posedness for initial data $u_0$ that may be unbounded, necessitating a functional framework beyond standard Lebesgue spaces.

Methodology
The authors employ a functional analytic approach centered on Orlicz spaces, specifically the exponential Orlicz space $\exp L^p(\mathbb{R}^d)$ and its subspace $\exp L^p_0(\mathbb{R}^d)$. This choice is motivated by the failure of Lebesgue spaces to accommodate the critical scaling and growth conditions associated with exponential nonlinearities, analogous to the role of Sobolev spaces in critical polynomial cases.

Key methodological components include:

1. Functional Framework: The analysis is conducted in $\exp L^p_0(\mathbb{R}^d)$, defined as the closure of $C^\infty_0(\mathbb{R}^d)$ in $\exp L^p(\mathbb{R}^d)$. This space is crucial because the heat semigroup $e^{-tL}$ is not generally bounded on the full space $\exp L^p(\mathbb{R}^d)$, but is bounded on $\exp L^p_0(\mathbb{R}^d)$.
2. Semigroup Estimates: The authors derive refined mapping properties and decay estimates for the heat semigroup $e^{-tL}$ associated with the mixed operator. These estimates utilize heat kernel bounds and interpolation arguments in Orlicz spaces, capturing the interplay between the local ($-\Delta$) and nonlocal ($(-\Delta)^s$) components.
3. Fixed-Point Arguments:
   • Local Well-posedness: Proven via a decomposition technique where the initial data $u_0$ is split into a smooth part $v_0 \in C^\infty_0$ and a small perturbation $w_0$. The smooth part is handled via classical semigroup arguments, while the perturbation is solved using the Banach fixed-point theorem in a suitable Banach space adapted to $\exp L^p$.
   • Global Existence: Established for small initial data using a contraction mapping principle in a complete metric space equipped with time-weighted norms. The analysis distinguishes cases based on the relationship between the fractional order $s$ and the critical threshold $d(p-1)/(2p)$.

Key Contributions and Results

1. Local Well-posedness (Theorem 1.3):
   The paper establishes the existence and uniqueness of a mild solution $u \in C([0, T], \exp L^p_0(\mathbb{R}^d))$ for any initial data $u_0 \in \exp L^p_0(\mathbb{R}^d)$. The nonlinearity is assumed to satisfy a Lipschitz-type condition with exponential growth: $|f(u) - f(v)| \leq C|u-v|(e^{\lambda|u|^p} + e^{\lambda|v|^p})$. The solution is shown to be locally bounded in $L^\infty(\mathbb{R}^d)$ for $t > 0$.

2. Global Existence for Small Data (Theorem 1.4):
   Under the assumption that the nonlinearity behaves like $|u|^m$ near the origin (with $m \geq p$ and satisfying specific growth constraints related to the Fujita exponent), the authors prove the existence of global weak-mild solutions for sufficiently small initial data in $\exp L^p(\mathbb{R}^d)$. The proof requires distinguishing three regimes based on the parameter $s$ relative to $d(p-1)/(2p)$:

   • $s < d(p-1)/(2p)$
   • $s = d(p-1)/(2p)$
   • $s > d(p-1)/(2p)$ (requiring $m > p$)

3. Large-Time Decay Estimates:
   The authors derive explicit decay rates for the solution in Lebesgue spaces $L^q$. The decay estimate (Equation 1.8) reveals a unique asymptotic transition:

   • Short-time regime ($t \to 0$): The solution mimics the decay behavior of the local heat equation, governed by an effective power $\tilde{m}$ related to the local smoothing effects.
   • Long-time regime ($t \to \infty$): The estimate transitions to recover the purely nonlocal asymptotics characteristic of the fractional Laplacian $(-\Delta)^s$.

Significance and Claims
The paper claims to bridge the gap between local and nonlocal diffusion theories in the context of exponential nonlinearities. Specifically:

• It identifies $\exp L^p_0(\mathbb{R}^d)$ as the appropriate functional setting for the mixed operator with exponential growth, addressing the density issues present in the full Orlicz space.
• It demonstrates that the presence of the local diffusion term does not alter the fundamental global existence behavior dictated by the fractional component (the critical Fujita exponent remains $p_c = 1 + 2s/d$), but it significantly influences the short-time dynamics.
• The derived decay estimates highlight a continuous interpolation between singular local dynamics and heavy-tailed fractional dissipation, showing that nonlocal effects are not instantaneous but emerge over time.

The work extends previous results on polynomial nonlinearities in mixed operators (e.g., [4, 10, 28]) and exponential nonlinearities in pure local or pure nonlocal settings (e.g., [16, 23, 24, 30, 31, 34]) to the mixed local-nonlocal case.