AI 中文总结
研究无限圆柱域上分数阶拉普拉斯算子线性和半线性方程大解，通过研究有限圆柱域上边界爆破解的性质，进而在无限圆柱上构造大解，揭示非局部算子下不同边界爆破现象及解的存在性等。
AI 中文摘要
我们研究了在某些但并非所有方向上无界的区域上涉及分数阶拉普拉斯算子的线性和半线性方程的大解。在区域边界上爆破的解通常称为大解。对于局部算子，这种行为仅在存在低阶非线性项时出现。然而，对于非局部算子，此类解的存在更为微妙。与局部情况不同，即使在线性方程的情况下，算子的非局部性质也会导致不同的边界爆破现象。在本文中，我们研究了有限圆柱域上分数阶线性和半线性方程边界爆破解的存在性和定性行为，并利用这些性质在无限圆柱上构造大解。
英文摘要
We investigate large solutions of linear and semi-linear equations involving the fractional Laplacian on domains that are becoming unbounded in some, but not all, directions. Solutions that blow up on the boundary of a domain are commonly called large solutions. For local operators, such behaviour arises only in the presence of lower-order nonlinear terms. However, for nonlocal operators, the existence of such solutions is more subtle. In contrast to the local case, the nonlocal nature of the operator gives rise to different boundary blow-up phenomena even in the case of linear equations. In this article, we study the existence and qualitative behaviour of boundary blow-up solutions to fractional linear and semi-linear equations on finite cylindrical domains and employ these properties to construct large solutions on infinite cylinders.
