Differentiability of the Blow-up Curve for one Dimensional Nonlinear Wave Equations

Abstract

Consider the Cauchy problem

$$rac{{{artial ^2}u}}{{artial {t^2}}} - umimits_{i = 1}^N {rac{{{artial ^2}u}}{{artial x_i^2}}} = F(u)(x n {athbb{R}^N},t >0),$$

$$ueft( {x,0} ight) = feft( x ight)eft( {x n {athbb{R}^N}} ight),$$

$${u_t}eft( {x,0} ight) = geft( x ight)eft( {x n {athbb{R}^N}} ight)$$

with N ≦ 3. It is well known that if F(u) is superlinear, i.e.,

$$Feft( u ight)A{eft| u ight|^p}ifeft| u ight| o nfty eft( {A >0,p > 1} ight) $$

(0.1)

then solutions generally blow up in finite time; see [2]–[6].