Finite time blow-up for a nonlinear parabolic equation with smooth coefficients
Oscar Jarrin, Gaston Vergara-Hermosilla
TL;DR
The paper addresses whether smooth solutions to the nonlinear parabolic equation $\partial_t u = \Delta u + |\nabla u|^2 \mathfrak{b}$ can blow up in finite time. It develops a virial-type framework, introducing the gradient field $\mathrm{v} = \nabla u$ and a weighted functional $I(t) = \int_{[-1,1]^n} \mathrm{v}(t,x) \cdot (\mathfrak{b}(x) \mathrm{w}(x))\,dx$, to obtain a differential inequality of the form $I'(t) \ge c_1 I(t)^2 - c_2$. Under structured assumptions on the coefficient, specifically $\mathfrak{b}(x) = \prod_i \mathfrak{b}_i(x_i)$ with $\mathfrak{b}_i(0)=0$, $\mathfrak{b}_i \ge 0$, $\mathfrak{b}_i \in \mathcal{S}$, and for initial data $u_0 \in H^s$ with $s>n/2+3$ satisfying certain positivity conditions on $I(0)$, the functional $I(t)$ blows up in finite time, which implies $\|u(t)\|_{H^s}$ blows up as well. The analysis combines a carefully crafted virial argument with local well-posedness results, providing a concrete positive answer to the finite-time blow-up question in the smooth-coefficient regime and extending prior work that relied on distributional or fractional models. This contributes a rigorous blow-up mechanism for nonlinear gradient-driven parabolic equations and clarifies the impact of smooth coefficient structures on solution longevity.
Abstract
In this article, we consider an n-dimensional parabolic partial differential equation with a smooth coefficient term in the nonlinear gradient term. This equation was first introduced and analyzed in [E. Issoglio, On a non-linear transport-diffusion equation with distributional coefficients, Journal of Differential Equations, Volume 267, Issue 10 (2019)], where one of the main open questions is the possible finite-time blow-up of solutions. Here, leveraging a virial-type estimate, we provide a positive answer to this question within the framework of smooth solutions.