Fractional diffusion equations (FDEs) were shown to provide very competitive descriptions of challenging phenomena of anomalously diffusive transport or long-range interactions. However, FDEs introduce mathematical issues that are not common in the context of integer-order diffusion equations. For instance, the homogeneous Dirichlet boundary-value problems of linear elliptic FDEs with smooth data in one space dimension may generate solutions with singularities that do not seem physically relevant, which are in sharp contrast to their integer-order analogues do. We prove the wellposedness of the Dirichlet boundary-value problem of one dimensional variable-order linear space-fractional diffusion equations (sFDEs). We further prove that their solutions have the similar regularities as their integer-order analogues if the order has an integer limit at the boundary or have the same singularity near the boundary as their constant-order sFDE analogues if the order has a non-integer limit at the boundary. In particular, we prove that constant-order sFDEs with a variable-order modification indeed generate solutions with significantly improved regularities.

结果表明，分数扩散方程（FDE）可以非常有竞争力地描述异常扩散传输或远距离相互作用的挑战性现象。但是，FDE引入了在整数阶扩散方程中不常见的数学问题。例如，在一个空间维度上具有平滑数据的线性椭圆形FDE的齐次Dirichlet边值问题可能会产生奇异性的解决方案，这些奇异性在物理上似乎并不相关，这与它们的整数阶类似物形成鲜明对比。我们证明了一维变阶线性空间分数维扩散方程（sFDEs）的Dirichlet边值问题的适定性。我们进一步证明，如果阶数在边界处具有整数限制，或者边界附近具有与常数阶sFDE类似物相同的奇异性（如果阶数具有非整数极限），则它们的解具有与整数阶类似物相似的规则性。在边界。特别是，我们证明了具有可变顺序修改的恒定顺序sFDE确实生成了具有显着改善的规则性的解。