Asymptotic profile of solutions for the critical Sobolev type equation on a half-line

We study nonlinear Sobolev type equations on half-line \[ \{
\begin{array} [c]{c} \partial_{t}u+\mathbb{L}u=\lambda|u|^{\rho}u_{x}^{\sigma}, x\in\mathbf{R}^{+}, t>0,
u(0,x)=u_{0}(x), x\in\mathbf{R}^{+}, \end{array}
. \] with $\rho+\sigma=\frac52,\rho>0,\sigma>0,\lambda\in\mathbf{C}$. The linear operator $\mathbb{L}$ is defined as \[ \mathbb{L}=\mathcal{L}^{-1}K(p)\mathcal{L}. \] Here $\mathcal{L}^{-1}$ and $\mathcal{L}$ are Laplace transform and inverse Laplace transform with respect to space variable $x$ and
\begin{equation*} K(p)=p^{2}\sum_{j=0}^{m}a_{j}p^{2j}\left(\sum_{l=0}^{m+1}b_{l}p^{2l}\right) ^{-1}, \end{equation*}
$m>0$ is integer number.The aim of this paper is to prove the global existence of solutions to the initial-boundary value problem and to find the main term of the asymptotic representation of solutions in the critical convective case.