Blow-up behaviour of one-dimensional semilinear parabolic equations

Consider the Cauchy problem \begin{cases} u_{t}−u_{xx}−\mathrm{F}(u) = 0;\:\:&x \in ℝ,\:t > 0 \\ u(x,0) = u_{0}(x);\:\:&x \in ℝ \end{cases} where u_0(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 u_0(x) has a single maximum at x = 0 and is symmetric, u_0(x) = u_0(−x) for x > 0 , there holds 1) If \mathrm{F}(u)=u^p with p > 1 , then \begin{align*} &\lim\limits_{t\uparrow \mathrm{T}} u\left(\xi((\mathrm{T}−t)|\log(\mathrm{T}−t)|)^{1/2},t\right) (\mathrm{T}−t)^{1/ (p−1)} \\ &\qquad = (p−1)^{−(1/(p−1))}\left[1 + \frac{(p−1)\xi^{2}}{4p}\right]^{−(1/(p−1))} \end{align*} uniformly on compact sets |ξ| ≦ R with R > 0 , 2) If F(u) = e^u , then \lim \limits_{t\uparrow \mathrm{T}}\left(u(\xi((\mathrm{T}−t)|\log (\mathrm{T}−t)|)^{1/ 2},t) + \log (\mathrm{T}−t)\right) = −\log\left[1 + \frac{\xi^2}{4}\right] uniformly on compact sets |ξ| ≦ R with R > 0 . Résumé On considère le problème de Cauchy \begin{cases} u_{t}−u_{xx}−\mathrm{F}(u) = 0;\:\:&x \in ℝ,\:t > 0 \\ u(x,0) = u_{0}(x);\:&x \in ℝ \end{cases} où u_0(x) est une fonction continue, non négative et bornée, et F(u) = u^p avec p > 1 ou F(u) = e^u . Nous supposons que u explose au point x = 0 en temps T > 0 . Dans ce travail, nous obtenons tous les comportements asymptotiques possibles de la solution u(x, t) quand (x, t) → (0, T) .