Blow-up behaviour of one-dimensional semilinear parabolic equations

Consider the Cauchy problem {cases} uₜ−uₓₓ−F(u) = 0;\:\:&x ∈ ℝ,\:t > 0 \\ u(x,0) = u₀(x);\:\:&x ∈ ℝ {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 F(u)=u^p with p > 1 , then {align*} &limt↑ T u(ξ((T−t)|log(T−t)|)1/2,t) (T−t)1/ (p−1) \\ & = (p−1)−(1/(p−1))[1 + {(p−1)ξ²}{4p}]−(1/(p−1)) {align*} uniformly on compact sets |ξ| ≦ R with R > 0 , 2) If F(u) = e^u , then lim t↑ T(u(ξ((T−t)|log (T−t)|)1/ 2,t) + log (T−t)) = −log[1 + ξ^2/4] uniformly on compact sets |ξ| ≦ R with R > 0 . Résumé On considère le problème de Cauchy {cases} uₜ−uₓₓ−F(u) = 0;\:\:&x ∈ ℝ,\:t > 0 \\ u(x,0) = u₀(x);\:&x ∈ ℝ {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) .