Master equations with indefinite nonlinearities

TL;DR

We study the fully fractional master equation $(\partial_t-\Delta)^s u(x,t)=x_1 u^p(x,t)$ in $\mathbb{R}^n\times\mathbb{R}$ with $s\in(0,1)$ and prove nonexistence of positive bounded solutions under mild conditions. The authors develop a direct moving planes argument tailored to the nonlocal operator to obtain strict monotonicity in the $x_1$-direction, then construct unbounded subsolutions to derive a contradiction. To handle the lack of time-space separability, they introduce a novel cutoff-based auxiliary-function method with singular-integral estimates, avoiding limiting arguments. The results cover $p>1$ (nonexistence), $p<0$ (nonexistence), and, for $0<p<1$, nonexistence under the established monotonicity, yielding tools applicable to broader fractional elliptic and parabolic problems with indefinite nonlinearities.

Abstract

In this paper, we consider the following indefinite fully fractional heat equation involving the master operator \begin{equation} (\partial_t -Δ)^{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.

Theorems & Definitions (13)

• Theorem 1.1
• Theorem 1.2
• Remark 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 2.1
• proof : Proof
• Theorem 2.2
• proof : Proof
• proof : Proof of Theorem \ref{['thm1']}