Nonlinear pseudodifferential equations on a half-line with large initial data

We study the initial-boundary value problem for nonlinear pseudodifferential equations, on a half-line, $$displaylines{ u_{t}+mathcal{lambda}| u| ^{sigma}u+mathcal{L} u=0,quad(x,t)in{mathbb{R}^{+}}imes{mathbb{R}^{+}},cr u(x,0)=u_{0}(x),quad xin{mathbb{R}}^{+}, }$$ where $lambda>0$ and pseudodifferential operator $mathcal{L}$ is defined by the inverse Laplace transform. The aim of this paper is to prove the global existence of solutions and to find the main term of the asymptotic representation in the case of the large initial data.