Master equations with indefinite nonlinearities

TL;DR

This paper investigates a fractional heat equation with an indefinite nonlinearity, proving the nonexistence of positive bounded solutions using novel methods involving the moving planes technique and sub-solution construction.

Contribution

It introduces new approaches to analyze fully fractional master equations, establishing nonexistence results and providing tools applicable to other fractional PDEs.

Findings

No positive bounded solutions exist under mild conditions.

Solutions are strictly increasing along the x_1 direction.

New methods developed for fractional elliptic and parabolic problems.

Abstract

In this paper, we consider the following indefinite fully fractional heat equation involving the master operator \begin{equation} (\partial_t -\Delta)^{s} u(x,t) = x_1u^p(x,t)\ \ \mbox{in}\ \R^n\times\R , \end{equation} where $s\in(0,1)$, and $-\infty < p < \infty$. Under mild conditions, we prove that there is no positive bounded solutions. To this end, we first show that the solutions are strictly increasing along $x_1$ direction by employing the direct method of moving planes. Then by constructing an unbounded sub-solution, we derive the nonexistence of bounded solutions. To circumvent the difficulties caused by the fully fractional master operator, we introduced some new ideas and novel approaches that, as we believe, will become useful tool in studying a variety of other fractional elliptic and parabolic problems.