Blow-up of solutions to semilinear parabolic equations driven by mixed local-nonlocal operators with large initial data

Technical Summary: Blow-up of Solutions to Semilinear Parabolic Equations Driven by Mixed Local-Nonlocal Operators with Large Initial Data

Problem Statement
The paper investigates the finite-time blow-up of nonnegative solutions to the Cauchy problem for semilinear parabolic equations driven by a mixed local–nonlocal operator $L$. The equation is given by:
$$\begin{cases} \partial_t u + Lu = f(u) & \text{in } \mathbb{R}^N \times (0, +\infty), \\ u(\cdot, 0) = u_0 & \text{in } \mathbb{R}^N, \end{cases}$$
where the operator is defined as $L = -a\Delta + b(-\Delta)^s$ with parameters $a \ge 0$ and $b > 0$. The reaction term $f$ satisfies structural hypotheses including local Lipschitz continuity, convexity, superlinear growth at infinity, and the Osgood condition ($\int_1^\infty \frac{dz}{f(z)} < \infty$). The prototype case is $f(u) = u^p$ with $p > 1$.

While the literature extensively covers the purely local case ($a=1, b=0$) and the purely nonlocal case ($a=0, b=1$), and previous work (specifically [8]) established blow-up for small $p$ ($p \le 1 + 2s/N$) with arbitrary nontrivial data, this study focuses on the regime where the initial datum $u_0$ is sufficiently large. The primary objective is to prove that for any $p > 1$, if the initial data is large enough, the solution blows up in finite time, even in the presence of the mixed operator.

Methodology
The authors adapt the Kaplan method (originally developed for the heat equation) to the framework of mixed operators. The core of the methodology involves three main steps:

1. Construction of a Kaplan Function: The authors define a "Kaplan function" $\kappa$ as a positive, smooth, integrable function satisfying the subsolution inequality $-L\kappa + \lambda\kappa \ge 0$ pointwise in $\mathbb{R}^N$ for some $\lambda > 0$.

   • Unlike the purely local case where the Kaplan function is a Gaussian ($e^{-k|x|^2}$), the authors explicitly construct a Kaplan function suitable for the mixed operator. The constructed function takes the form:
     $$\kappa(x) = C \varepsilon^{N/2} (1 + \varepsilon|x|^2)^{-\beta}$$
     where $C, \varepsilon, \beta$ are positive constants.
   • The construction requires careful handling of the nonlocal part $(-\Delta)^s$. The authors utilize properties of the hypergeometric function and scaling arguments to estimate the action of $L$ on $\kappa$, ensuring the inequality holds for sufficiently large $\lambda$.

2. Derivation of a Differential Inequality: For a solution $u(t)$, the authors define a functional $\Phi(t) = \int_{\mathbb{R}^N} \kappa(x) u(x,t) \, dx$. By multiplying the PDE by $\kappa$ and integrating, they derive the ordinary differential inequality:
   $$\Phi'(t) + \lambda \Phi(t) \ge f(\Phi(t))$$
   This derivation relies on integration by parts for the local part and specific integral identities (involving the bilinear form $B(f,g)$) for the nonlocal part. Crucially, they employ cut-off functions and limit arguments ($R \to \infty$) to handle the nonlocal tail terms, relying on the integrability properties of the constructed $\kappa$.

3. Blow-up Analysis: The authors show that if the initial datum satisfies $\Phi(0) > s_f(\lambda)$ (where $s_f(\lambda)$ is a threshold derived from the reaction term), the solution to the differential inequality blows up in finite time. This implies the non-existence of global classical solutions for the original PDE.

Key Results

• General Blow-up Criterion (Theorem 3.4): The paper establishes a general condition for finite-time blow-up. If a Kaplan function $\kappa$ exists satisfying specific regularity and decay bounds, and the initial data $u_0$ satisfies $\int_{\mathbb{R}^N} \kappa u_0 \, dx > s_f(\lambda)$, then no global classical solution exists.
• Explicit Construction (Theorem 3.7): The authors explicitly construct the Kaplan function $\kappa_\varepsilon(x) \propto (1 + \varepsilon|x|^2)^{-\beta}$ for the mixed operator $L = -a\Delta + b(-\Delta)^s$. They prove that for $\beta > N/2$, this function satisfies the required subsolution inequality for a specific range of $\lambda$ dependent on the scaling parameter $\varepsilon$.
• Main Theorem (Theorem 3.9): Combining the general criterion with the explicit construction, the authors prove that for the mixed operator, if the initial datum is sufficiently large (quantified by inequality 3.32), the solution blows up in finite time.
• Recovery of Known Results (Theorem 3.10): As a corollary, the authors apply their main result to the case $f(u) = u^p$ with $1 < p < 1 + 2s/N$. They demonstrate that for this range of $p$, blow-up occurs for any nontrivial initial datum (not just large ones), thereby recovering the result previously established in [8] using different methods.

Significance and Claims
The authors state that their result is new even in the special case of the fractional Laplacian ($a=0, b=1$). While blow-up for large initial data is well-known for the classical heat equation, extending this to the fractional case and the mixed operator framework required overcoming significant technical hurdles related to the nonlocal nature of the operator.

Specifically, the paper highlights:

• The necessity of a different form for the Kaplan function in the mixed/nonlocal setting compared to the classical local setting.
• The successful adaptation of the Kaplan method to operators involving fractional Laplacians, which is non-trivial due to the lack of explicit local formulas for the operator's action on test functions.
• The provision of an explicit, verifiable condition on the initial data that guarantees blow-up for any $p > 1$ in the mixed operator context.

The work contributes to the qualitative theory of mixed local-nonlocal parabolic equations, a field of growing interest due to applications in stochastic dynamics (combining diffusion and jump processes) and optimal foraging models. The paper does not propose new applications but rather solidifies the theoretical foundation for understanding the long-time behavior of solutions in this specific operator setting.