Liouville type theorems for dual nonlocal evolution equations involving Marchaud derivatives
Yahong Guo, Lingwei Ma, Zhenqiu Zhang
TL;DR
This work studies Liouville-type properties for the dual nonlocal evolution equation $\partial_t^\alpha u + (-\Delta)^s u = 0$ in $\mathbb{R}^n\times\mathbb{R}$ with $0<\alpha<1$, $0<s<1$. It develops a tempered-distribution framework via the space $\mathcal{L}_{2s,\alpha}(\mathbb{R}^n\times\mathbb{R})$ and proves a sharp decay estimate for the dual operator on Schwartz functions, enabling a Fourier-analytic rigidity argument. Under a mild asymptotic growth condition (for $\tfrac{1}{2}<s<1$) the distributional solutions are shown to be constant, generalizing classical Liouville results for harmonic and $s$-harmonic functions. The paper also derives an integral representation for nonhomogeneous problems and establishes the equivalence between the pseudo-differential formulation and the corresponding integral equation, with potential applications to a broad class of nonlocal parabolic problems.
Abstract
In this paper, we establish a Liouville type theorem for the homogeneous dual fractional parabolic equation \begin{equation} \partial^α_t u(x,t)+(-Δ)^s u(x,t) = 0\ \ \mbox{in}\ \ \mathbb{R}^n\times\mathbb{R} . \end{equation} where $0<α,s<1$. Under an asymptotic assumption $$\liminf_{|x|\rightarrow\infty}\frac{u(x,t)}{|x|^γ}\geq 0 \; ( \mbox{or} \; \leq 0) \,\,\mbox{for some} \;0\leqγ\leq 1, $$ in the case $\frac{1}{2}<s < 1$, we prove that all solutions in the sense of distributions of above equation must be constant by employing a method of Fourier analysis. Our result includes the previous Liouville theorems on harmonic functions \cite{ABR} and on $s$-harmonic functions \cite{CDL} as special cases and it is still novel even restricted to one-sided Marchaud fractional equations, and our methods can be applied to a variety of dual nonlocal parabolic problems. In the process of deriving our main result, through very delicate calculations, we obtain an optimal estimate on the decay rate of $\left[D_{\rm right}^α+(-Δ)^s\right] \varphi(x,t)$ for functions in Schwartz space. This sharp estimate plays a crucial role in defining the solution in the sense of distributions and will become a useful tool in the analysis of this family of equations.