Liouville theorem for fully fractional master equations and its applications

Abstract

In this paper, we study the fully fractional master equation \begin{equation}\label{pdeq1} (\partial_t-\Delta)^s u(x,t) =f(x,t,u(x,t)),\,\,(x, t)\in \mathbb{R}^n\times \mathbb{R}. \end{equation} First we prove a Liouville type theorem for the homogeneous equation \begin{equation}\label{pdeq0} (\partial_t-\Delta)^s u(x,t) = 0,\,\,(x, t)\in \mathbb{R}^n\times \mathbb{R}, \end{equation} where $0<s<1$. When $u$ belongs to the slowly increasing function space $$\mathcal{L}^{2s,s}(\mathbb{R}^n\times\mathbb{R})=\left\{u(x,t) \in L^1_{\rm loc} (\mathbb{R}^n\times\mathbb{R}) \mid \int_{-\infty}^{+\infty} \int_{\mathbb{R}^n} \frac{|u(x,t)|}{1+|x|^{n+2+2s}+|t|^{\frac{n}{2}+1+s}}\operatorname{d}\!x\operatorname{d}\!t<\infty\right\} $$ and satisfies an additional asymptotic assumption $$\liminf_{|x|\rightarrow\infty}\frac{u(x,t)}{|x|^\gamma}\geq 0 \; ( \mbox{or} \; \leq 0) \,\,\mbox{for some} \;0\leq\gamma\leq 1, $$ in the case $\frac{1}{2}<s < 1$, we prove that all solutions of (\ref{pdeq0}) must be constant. This result includes the previous Liouville theorems on harmonic functions \cite{ABR} and on $s$-harmonic functions \cite{CDL} as special cases. Then we establish the equivalence between nonhomogeneous pseudo-differential equations (\ref{pdeq1}) and the corresponding integral equations. We believe that these integral equations will become very useful tools in further analysing qualitative properties of solutions, such as regularity, monotonicity, and symmetry. In the process of deriving the Liouville type theorem, through very delicate calculations, we obtain an optimal estimate on the decay rate of $(\partial_t-\Delta)_{\rm right}^s \varphi(x,t)$. This sharp estimate will become a key ingredient and an important tool in investigating master equations.