Boundary-blow-up problems in a fractal domain
Boundary blow up problems for the problem \(\Delta u(x)= f(u(x))\) are considered in a bounded domain \(\Omega\subset \mathbb{R}^n\), \(n\geq 2\), which satisfies a uniform interior and exterior cone condition, and with \(f(t)= e^t\) or \(f(t)= t^p\) where \(p\in (1, +\infty)\). The author gives uniform a priori lower and upper bounds for the growth of the solutions and their gradients. A maximum principle is used to show the existence and the uniqueness of a solution. As an application, the problem in a two-dimensional domain whose boundary is the von Koch snowflake curve are presented.
scientific article; zbMATH DE number 889005