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

Abstract

We consider the semilinear wave equation(1)∂t2u−Δu=f(u),(x,t)∈RN×[0,T), with f(u)=|u|p−1uloga⁡(2+u2), where p>1 and a∈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″=|u|p−1uloga⁡(2+u2). Unlike the pure power case (g(u)=|u|p−1u) the difficulties here are due to the fact that equation (1) is not scale invariant.