Cauchy problem for a class of nonlinear dispersive wave equations arising in elasto-plastic flow

Abstract

The paper studies the existence, both locally and globally in time, stability, decay estimates and blowup of solutions to the Cauchy problem for a class of nonlinear dispersive wave equations arising in elasto-plastic flow. Under the assumption that the nonlinear term of the equations is of polynomial growth order, say α, it proves that when α > 1 , the Cauchy problem admits a unique local solution, which is stable and can be continued to a global solution under rather mild conditions; when α ⩾ 5 and the initial data is small enough, the Cauchy problem admits a unique global solution and its norm in L 1 , p ( R ) decays at the rate ( 1 + t ) − ( p − 2 ) ( 2 p ) for 2 < p ⩽ 10 . And if the initial energy is negative, then under a suitable condition on the nonlinear term, the local solutions of the Cauchy problem blow up in finite time.