For the following parabolic PDE system, $u(x,t)$ and $v(x,t)$ are functions of independent variables $x$ and $t$, $x\in[a,b]$.
\begin{equation} \begin{cases} \frac{\partial}{\partial t}u=\frac{\partial^{2}u}{\partial x^{2}}+u^{3}+uv^{2},&& a<x<b,t>0\\ \frac{\partial}{\partial t}v=\frac{\partial^{2}v}{\partial x^{2}}+\frac{2}{u}\frac{\partial u}{\partial x}\frac{\partial v}{\partial x}+2[\frac{1}{u}\frac{\partial^{2}u}{\partial x^{2}}-\frac{1}{u^{2}}(\frac{\partial u}{\partial x})^{2}+u^{2}]v, && a<x<b,t>0\\ u(x,0)=u_{0}(x)\in C^{\infty}(x),u_{0}(x)>0,v(x,0)=v_{0}(x)\in C^{\infty}(x), &&a\leq x\leq b\\ u(a,t)=u(b,t),v(a,t)=v(b,t), &&t\geq0 \end{cases} \end{equation}
Question 1: Does the system have local existence?
If the system have local existence:
Question 2: If $u(x,t)$ is bounded on an interval $[0,\epsilon)$, then does $\frac{\partial u}{\partial x}(x,t)$ bounded on $[0,\epsilon)$?
Question 3: If $u(x,t)$ and $v(x,t)$ are bounded on an interval $[0,\epsilon)$, then does $\frac{\partial u}{\partial x}(x,t)$ bounded on $[0,\epsilon)$?
Question 4: If $u(x,t)$ is bounded on an interval $[0,\epsilon)$, then does $v(x,t)$ bounded on $[0,\epsilon)$?