Blowup of solutions for improved Boussinesq type equation

Abstract

The paper studies the existence and uniqueness of local solutions and the blowup of solutions to the initial boundary value problem for improved Boussinesq type equation utt−uxx−uxxtt=σ(u)xx. By a Galerkin approximation scheme combined with the continuation of solutions step by step and the Fourier transform method, it proves that under rather mild conditions on initial data, the above-mentioned problem admits a unique generalized solution u∈W2,∞([0,T];H2(0,1)) as long as σ∈C2(R). In particular, when σ(s)=asp, where a≠0 is a real number and p>1 is an integer, specially a<0 if p is an odd number, the solution blows up in finite time. Moreover, two examples of blowup are obtained numerically.