ASYMPTOTICS FOR NONLINEAR NONLOCAL EQUATIONS ON A HALF-LINE.

We study the initial-boundary value problem for a general class of nonlinear pseudo-differential equations on a half-line \[ \hspace*{5pc} \left\{\begin{array}{ll}u_{t}+\mathcal{N}(u,u_{x})+\mathcal{L}u=f,&\quad (x,t)\in{\mathbf{R}^{+}}\times{\mathbf{R}^{+}},\\[5pt] u(x,0)=u_{0}(x),&\quad x\in{\mathbf{R}}^{+},\\[5pt] \partial_{x}^{j-1}u(0,t)=h_{j}(t)&\quad\mbox{for}\ j=1,\ldots,M, \end{array} \right.\hspace*{5.3pc}(0.1) \label{W} \] where the number M depends on the order of the pseudo-differential operator $\mathcal{L}$ on a half-line. The nonlinear term $\mathcal{N}(u,u_{x})$ is such that $|\mathcal{N}(u,v)| \leq C|u|^{\rho}|v|^{\sigma}$ as u, v → 0, with ρ, σ > 0. Pseudo-differential operator $\mathcal{L}$ is defined by the inverse Laplace transform. The aim of this paper is to prove the global existence of solutions to the initial-boundary value problem (0.1) and to find the main term of the asymptotic representation of solutions taking into account the influence of inhomogeneous boundary data and a source on the asymptotic properties of solutions. [ABSTRACT FROM AUTHOR]