CONVERGENCE OF A BLOW-UP CURVE FOR A SEMILINEAR WAVE EQUATION

We consider a blow-up phenomenon for partial derivative t(u)(2)u(epsilon) - epsilon(2)partial derivative(2)(x)u(epsilon) = F(partial derivative(t)u(epsilon)), The derivative of the solution partial derivative(t)u(epsilon) blows-up on a curve t = T-epsilon(X) if we impose some conditions on the initial values and the nonlinear term F. We call T-epsilon blow-up curve for partial derivative(2)(t)u(epsilon) - epsilon(2)partial derivative(2)(x)u(epsilon) = F(partial derivative(t)u(epsilon)). In the same way, we consider the blow-up curve t = (T) over tilde (x) for partial derivative(2)(t)u = F(partial derivative(t)u). The purpose of this paper is to show that, for each x, T-epsilon(x) converges to (T) over tilde (x) as epsilon -> 0.