Properties of Solutions to the Full Fractional Heat Operator Equation

TL;DR

This work analyzes the fully nonlocal master equation $$(\partial_t-\Delta)^s u = a(x) f(u)$$ with $s\in(0,1)$ and an indefinite weight $a(x)$. The authors adapt the moving planes method to the nonlocal parabolic operator to prove that bounded, uniformly continuous positive solutions are strictly increasing in the first spatial direction and, under symmetry of $a$, inherit corresponding symmetries. They establish a sharp dichotomy for existence: when $a(x)$ grows, nonexistence ensues; for growth forms $a(x)\sim|x_1|^{\alpha}$ and $f(u)\sim u^{r}$, a critical exponent $r^{*}=\min\{1+\frac{4s^{2}}{n}, \frac{n+2+2s-2\alpha}{n+2-2s}\}$ separates subcritical existence $(1<r<r^{*})$ from nonexistence at or above the threshold. In the subcritical regime, solutions exist globally and gain regularity via the fractional heat semigroup, while in the critical/supercritical regime, suitable initial data can blow up, with a comprehensive blow-up analysis provided. These results extend local operator theory to the fully fractional setting and introduce techniques potentially applicable to broader nonlocal problems.

Abstract

In this paper, we consider the following indefinite fully fractional heat equation involving the master operator . Under certain assumptions of the indefinite nonlinearity and its weight, we prove that there is no positive bounded solution, which is based on the monotonicity of the solution along the first direction that is proved by employing the method of moving planes. Besides, if the weight satisfy other conditions, we come to different conclusions according to the behavior of the nonlinearity at infinity. To overcome the difficulties caused by the operator, we lead in some mathematics tools that, as we believe, will be useful in studying problems involving other fractional operators or nonlinearities.