Singular perturbations and dead core behavior
Let u(x) be a smooth function satisfying u''(x)\(\geq 0\) on [0,1] and consider the Dirichlet problem \[ (*)\quad \epsilon y''=f(y-u(x)),\quad x\in (0,1),\quad y(0)\geq u(0),y(1)\geq u(1); \] here \(\epsilon\) is a small positive parameter, \(f(0)=0\), and f is continuous and increasing. It is shown that there exists a unique solution \(y_{\epsilon}\) of (*), and \(y_{\epsilon}(x)\to u(x)\) uniformly on compact subsets of (0,1). If additionally \(u''=0\), necessary and sufficient conditions are given for \(y_{\epsilon}(x)\) to coincide with u(x) for x in some compact subset \([p_{\epsilon},q_{\epsilon}]\subset [0,1]\), and estimates on \(p_{\epsilon}\) and \(1-q_{\epsilon}\) are given. Generalizations to equations of more general form \(\epsilon y''=f(x,y)\) are given, as are generalizations to less smooth u.
scientific article; zbMATH DE number 4004685
scientific article