Uniform convergence of the method of lines in the case of the first boundary-value problem for a nonlinear second-order parabolic equation

Let $u(t,x)$ be a solution of the first initial–boundary-value problem for the nonlinear equation
$$ \dfrac{\partial u}{\partial t}=F\biggl(t,x,u,\dfrac{\partial u}{\partial x},\dfrac{\partial^2u}{\partial x^2}\biggr),\qquad 0<t\leqslant T,\quad 0<x<1 $$
with initial condition
$$ u(0,x)=\omega(x),\quad 0<x<1 $$
and boundary conditions $u(t,0)=u(t,1)=0$, $0<t\leqslant t$, such that $\biggl|\dfrac{\partial^4u}{\partial x^4}(t,x)\biggr|\leqslant C$. Assume that the function $F(t,x,u,p,r)$ is smooth and is such that
$$ \dfrac{1}{r-\overline{r}}\biggl[F(t,x,u,p,r)-F(t,x,u,p,\overline{r})\biggr]\geqslant\alpha>0 $$
in a small neighborhood of the solution under consideration. Then, the longitudinal scheme of the method of lines converges uniformly with order $h^2$ to the solution under consideration. One considers the case of less smooth solutions and of more general equations. One gives theorems which show explicit estimates for the step $h$, under which one can guarantee a nonlocal solvability of the Cauchy problem for systems of ordinary differential equations by the method of lines.