On the Nonexistence of Global Solutions for Nonlocal Parabolic Equations with Forcing Terms

TL;DR

This work analyzes a nonlinear diffusion equation with a nonlocal nonlinearity $\|u(t)\|_q^\alpha|u|^p$ and a time-space forcing term $t^{\varrho}\mathbf w(x)$ on $\mathbb{R}^N$. Using the heat semigroup and Duhamel formulation, it establishes local well-posedness and unconditional uniqueness in Lebesgue spaces under suitable integrability conditions, and derives a nonexistence result for global weak solutions in a Fujita-type regime when $\int \mathbf w>0$ and $p$ is below a critical threshold. It also proves global existence for small initial data and forcing under a critical exponent $p_c(\varrho,\alpha,q)$, highlighting how the interplay between the nonlocal nonlinearity and the forcing term governs long-time behavior. The results extend classical Fujita-type theory to nonlocal nonlinear parabolic problems with time-space forcing and quantify the precise smallness regimes needed for global-in-time solutions. These findings have potential implications for diffusion phenomena in heterogeneous media and related applied models.

Abstract

The purpose of this work is to analyze the well-posedness and blow-up behavior of solutions to the nonlocal semilinear parabolic equation with a forcing term: \[ \partial_t u - Δu = \|u(t)\|_{q}^α|u|^p + t^{\varrho} \mathbf{w}(x) \quad \text{in} \quad \mathbb{R}^N \times (0, \infty), \] where $N \geq 1$, $p, q \geq 1$, $α\geq 0$, $\varrho > -1$, and $\mathbf{w}(x)$ is a suitably given continuous function. The novelty of this work, compared to previous studies, lies in considering a nonlocal nonlinearity $\|u(t)\|_{q}^α|u|^p$ and a forcing term $t^{\varrho} \mathbf{w}(x)$ that depend on both time and space variables. This combination introduces new challenges in understanding the interplay between the nonlocal structure of the equation and the spatio-temporal forcing term. Under appropriate assumptions, we establish the global existence of solutions for small initial data in Lebesgue spaces when the exponent $p$ exceeds a critical value. In contrast, we show that the global existence cannot hold for $p$ below this critical value, provided the additional condition $\int_{\mathbb{R}^N} \mathbf{w}(x) \, dx > 0$ is satisfied. The main challenge in this analysis lies in managing the complex interaction between the nonlocal nonlinearity and the forcing term, which significantly influences the behavior of solutions.

Theorems & Definitions (24)

• Definition 1.1
• Theorem 1.1
• Remark 1.1
• Theorem 1.2
• Remark 1.2
• Theorem 1.3
• Remark 1.3
• Theorem 1.4
• Lemma 2.1
• Lemma 2.2