Nonexistence of global solutions to the Grushin heat equation with nonlocal and local nonlinearities
Ahmad Z. Fino, Arlúcio Viana
TL;DR
This paper analyzes the nonexistence of global-in-time nonnegative solutions for a Grushin-type heat equation with mixed nonlinear memory and local reaction terms. It develops a test-function framework adapted to the degenerate Grushin operator and memory effects to derive sharp Fujita-type exponents across several regimes, including pure local, memory-only, and memory-plus-source dynamics. The results extend classical blow-up criteria to hypoelliptic, degenerate diffusion operators and provide precise thresholds ($p_c$, $p_0$, etc.) that separate global existence from finite-time blow-up. The findings deepen understanding of diffusion with memory and nonlocal reactions in subelliptic geometries, with potential implications for related degenerate parabolic problems. The methods deliver a unified approach to identifying critical exponents in complex, nonlocal reaction-diffusion systems on Grushin-type spaces.
Abstract
In this paper, we investigate the nonexistence of global solutions to the Grushin-type heat equation with nonlinear reaction terms, including cases involving memory effects: $$ \left\{\begin{array}{ll} \displaystyle {u_{t}-Δ_{\mathcal{G}} u = k_1 \int_0^t(t-s)^{-γ}\abs{u}^{p_1-1}u(s)\,\mathrm{d}s} + k_2|u|^{p_2-1}u, & (z,t)\in {\mathbb{R}}^{N+k}\times (0,\infty), \displaystyle{u(z,0)= u_0(z),\qquad\qquad}&\displaystyle{z\in {\mathbb{R}}^{N+k},} \end{array} \right. $$ where $Δ_{\mathcal{G}}$ denotes the Grushin operator, $u_0 \in L^1_{\mathrm{loc}}(\mathbb{R}^{N+k})$, $γ\in[0,1)$, $k_1,k_2 \geq 0$, and $p_1,p_2>1$. We establish sharp nonexistence results for global-in-time positive solutions, thereby completing the picture of global existence versus blow-up and allow us to identify the corresponding Fujita-type critical exponents in certain parameter regimes. The analysis relies on the test function method, adapted to handle both the degeneracy of the Grushin operator and the influence of the memory term.