Global and nonglobal solutions for a mixed local-nonlocal heat equation

TL;DR

This paper analyzes global and nonglobal existence for a parabolic equation with a mixed local-nonlocal operator $\mathcal{L}=-\Delta+(-\Delta)^s$ ($0<s<1$) by studying the nonlinear problem $u_t+\mathcal{L}u= h(t) f(u)$ in $\mathbb{R}^N$. The authors employ a global supersolution of the form $(1+\gamma) e^{-t\mathcal{L}}u_0$ and derive lower-bound kernel estimates to obtain a sharp dichotomy between global existence for small data and finite-time blow-up, recovering the Fujita exponent $1+\frac{2s}{N}$. They formulate an Osgood-type criterion and apply it to Fujita-type nonlinearities, obtaining precise thresholds for $f(u)=u^p$, $f(u)=(1+u)[\ln(1+u)]^p$, and related logarithmic nonlinearities, depending on the growth of $h(t)$ (e.g., $h(t)\sim t^\rho$). These results extend the understanding of mixed local-nonlocal diffusion models and provide explicit global-nonglobal criteria with clear implications for blow-up and global solvability in nonlinear diffusion problems.

Abstract

In this work, we establish optimal conditions concerning the global and nonglobal existence of solutions of a semilinear parabolic equations governed by a mixed local-nonlocal operator. Furthermore, our findings recover the Fujita exponent recently derived by Biagi, Punzo and Vecchi, as well as by Del Pezzo and Ferreira.

Theorems & Definitions (17)

• Remark 1
• Definition 1
• Proposition 1
• Definition 2
• Lemma 2
• Lemma 3: Lower bound estimate for $\mathfrak{p}_t$
• proof
• Lemma 4
• proof
• Proposition 5