Blow-up behavior of one-dimensional semilinear parabolic equations

Abstract

Consider the Cauchy problem u(t) - u(xx) - F(u) = 0; x is-an-element-of R, t>0 u(x, 0) = u0(x); x is-an-element-of R where u0 (x) is continuous, nonnegative and bounded, and F(u) = u(p) with p > 1, or F(u) = e(u). Assume that u blows up at x = 0 and t = T > 0. In this paper we shall describe the various possible asymptotic behaviours of u(x, t) as (x, t) --> (0, T). Moreover, we shall show that if u0(x) has a single maximum at x = 0 and is symmetric, u0(x) = u0(- x) for x > 0, there holds 1)If F(u) = u(p) with p > 1, then lim u(xi((T - t)\log (T - t)\)1/2, t) t up T x(T - t)1/(p - 1) = (p - 1) - (1/(p - 1)) [1 + (p - 1)xi2/4p] - 1/(p - 1)) uniformly on compact sets \xi\ less-than-or-equal-to R with R > 0, 2) If F(u) = e(u), then lim (u(xi((T - t)\log (T - t)\)1/2, t) + log(T - t)) = - log [1 + xi2/4] t up T uniformly on compact sets \xi\ less-than-or-equal-to R with R>0.