The blow-up rate for a non-scaling invariant semilinear wave equations

Abstract

We consider the semilinear wave equation $${\partial }_t^{2}u-\Delta u=f(u),(x,t)\in {\mathbb{R}}^N\times [0,T),(1)$$ with $f(u)=|u{|}^{p-1}u{\log }^a(2+u^2)$, where $p>1$ and $a\in \mathbb{R}$. We show an upper bound for any blow-up solution of (1). Then, in the one space dimensional case, using this estimate and the logarithmic property, we prove that the exact blow-up rate of any singular solution of (1) is given by the ODE solution associated with $(1)$, namely $u^{\prime \prime }=|u{|}^{p-1}u{\log }^a(2+u^2)$ Unlike the pure power case ($g(u)=|u{|}^{p-1}u$) the difficulties here are due to the fact that equation (1) is not scale invariant.

37 pages