STABLE BLOW-UP PATTERNS

For the semilinear heat equation ut = Δu + eu in a convex domain Ω ⊂ Rn, given any b ε{lunate} Ω we show the existence of solutions which blow up in finite time exactly at b and whose final profile has the form u(T, x) ≈ -2 ln |x - b| + ln |ln |x - b|| + ln 8, T being the blow-up time. Using a suitable set of rescaled coordinates, this asymptotic behavior is proved to be stable with respect to small perturbations of the initial conditions. © 1992.