Solutions of differential equations that blow up on the boundary of a domain are called large solutions or boundary blow-up solutions.

We investigate large solutions for the fractional Laplacian on a sequence of cylinders $ B_{1}^{n-1}(0)\\times [-\\ell,\\ell]$ as $\\ell\\to +\\infty$ and obtain large solutions on infinite cylinder $\\Omega_{\\infty}:=B_{1}^{n-1}(0)\\times\\mathbb{R}$ to the following problem
\\[
\\begin{cases}
(-\\Delta)^{s} u = f(x,u) & \\text{in } \\Omega_{\\infty},\\\\
u = g & \\text{in } \\mathbb{R}^{n} \\setminus \\Omega_{\\infty},\\\\
\\delta_{\\Omega_{\\infty}}^{s}(x)\\,u(x) \\xrightarrow[x\\to \\theta]{x\\in \\Omega_{\\infty}} h(\\theta) & \\text{for } \\theta \\in \\partial\\Omega_{\\infty},
\\end{cases}
\\]
under suitable assumptions on $\\big(f, g,h)$. The investigation is carried out for both linear and semi-linear equations. Further, for a linear equation, we identify three distinct types of large solutions on an infinite cylinder. These are solutions arising from the data $(f,0,0)$, $(0,g,0)$, and $(0,0,h)$.