We settle the open question concerning the Harnack inequality for globally positive solutions to non-local in time diffusion equations by constructing a counter-example for dimensions $$d\ge \beta $$ d ≥ β , where $$\beta \in (0,2]$$ β ∈ ( 0 , 2 ] is the order of the equation with respect to the spatial variable. The equation can be non-local both in time and in space but for the counter-example it is important that the equation has a fractional time derivative. In this case, the fundamental solution is singular at the origin for all times $$t>0$$ t > 0 in dimensions $$d\ge \beta $$ d ≥ β . This underlines the markedly different behavior of time-fractional diffusion compared to the purely space-fractional case, where a local Harnack inequality is known. The key observation is that the memory strongly affects the estimates. In particular, if the initial data $$u_0 \in L^q_{loc}$$ u 0 ∈ L loc q for q larger than the critical value $$\tfrac{d}{\beta }$$ d β of the elliptic operator $$(-\Delta )^{\beta /2}$$ ( - Δ ) β / 2 , a non-local version of the Harnack inequality is still valid as we show. We also observe the critical dimension phenomenon already known from other contexts: the diffusion behavior is substantially different in higher dimensions than $$d=1$$ d = 1 provided $$\beta >1$$ β > 1 , since we prove that the local Harnack inequality holds if $$d<\beta $$ d < β .

中文翻译：

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关于非局部扩散方程的抛物线 Harnack 不等式

如果初始数据 $$u_0 \in L^q_{loc}$$ u 0 ∈ L loc q for q 大于椭圆算子 $$\tfrac{d}{\beta }$$ d β $(-\Delta )^{\beta /2}$$ ( - Δ ) β / 2 ，Harnack 不等式的非局部版本仍然有效，正如我们所展示的。我们还观察到从其他上下文中已知的临界维度现象：如果 $$\beta >1$$ β > 1 ，则扩散行为在更高维度上与 $$d=1$$ d = 1 大不相同，因为我们证明如果 $$d<\beta $$ d < β，则局部 Harnack 不等式成立。