The Blow-Up Time for Solutions of Nonlinear Heat Equations with Small Diffusion

Consider a nonlinear heat equation $u_t - \varepsilon \Delta u = f(u)$ in a cylinder $\{ x \in \Omega ,t > 0\} $, with , u Vanishing on the lateral boundary and $u = \phi _\varepsilon (x)$ initially $(\phi _\varepsilon \geqq 0)$. Denote by $T_\varepsilon $ the blow-up time for the solution. Asymptotic estimates are obtained for $T_\varepsilon $ as $\varepsilon \to 0$.