Neumann problem for a nonlinear nonlocal equation on a half-line

Abstract

Our goal is to study the global existence and large time asymptotic behavior of solutions to the Neumann initial-boundary value problem for the nonlinear nonlocal equation on a half-line

$$\{\begin{array}{l}{u}_t+\mathcal{N}(u,u_x)+ℒu=f,(t,x)\in {\mathbf{R}}^{+}\times {\mathbf{R}}^{+},\hfill \\ u(0,x)=u_0(x),x\in {\mathbf{R}}^{+},\hfill \\ {\partial }_{x}u(t,0)=h(t),t\in {\mathbf{R}}^{+},\hfill \end{array}$$

where the nonlinear term is $\mathcal{N}(u,u_x)=u^{\rho }{u}_x^{\sigma }$, with , and $ℒ$ is a pseudodifferential operator defined by the inverse Laplace transform

$$ℒu=\frac{1}{2\pi i}{\displaystyle {\int }_{-i\infty }^{i\infty }{e}^{px}}E{p}^{\frac{5}{2}}\left(\widehat{u}(t,p)-\left(\frac{u(t,0)}{p}+\frac{{\partial }_{x}u(t,0)}{p^2}\right)\right)dp$$

where $\widehat{u}(p)={\displaystyle {\int }_0^{\infty }{e}^{-px}u(x)dx}$. We prove that if the initial data $u_0\in {\mathbf{L}}^{1,a}\cap {\mathbf{L}}^{\infty }$ for $a\in [0,1)$, then there exists a unique solution

$$u\in \mathbf{C}([0,\infty );{\mathbf{L}}^{1,a})\cap \mathbf{C}((0,\infty );{\mathbf{L}}^{\infty }\cap {\mathbf{W}}_{\infty }^1),$$

for the inital-boundary value problem. We also obtain the large time asymptotic formulas for the solutions...