Initial boundary value problem for semilinear hyperbolic equations and parabolic equations with critical initial data

Abstract

We study the initial boundary value problem of semilinear hyperbolic equations u t t − Δ u = f ( u ) u_{tt}-\Delta u=f(u) and semilinear parabolic equations u t − Δ u = f ( u ) u_{t}-\Delta u=f(u) with critical initial data E ( 0 ) = d E(0)=d (or J ( u 0 ) = d J(u_0)=d ), I ( u 0 ) > 0 I(u_0)>0 , and prove that there exist non-global solutions under classical conditions on f f .