An Initial-Boundary Value Problem for a Modified Transitional Korteweg-de Vries Equation

We study the following modified transitional Korteweg-de Vries equation $u_t+f\left(t\right)u^p{u}_x+u_{xxx}=0$ , $\left(x,t\right)\in R^{+}\times R^{+}$ , ($p\ge 2$ is an even integer) with initial value $u\left(x,0\right)=g\left(x\right)\in H^4\left(R^{+}\right)$ and inhomogeneous boundary value $u\left(0,t\right)=Q\left(t\right)\in C^2\left(\left[0,\infty \right)\right)$ . Under the conditions either (i) $f\left(t\right)\le 0$ , $f^{\prime }\left(t\right)\ge 0$ or (ii) $f\left(t\right)\le ？\alpha$ where $\alpha >0$ , we prove the