Assume that $\Omega\subset\mathbb{R}^n(n\geq3)$ is a bounded open set with smooth boundary, $\varphi\in H^1_0(\Omega)$, $F(t)$ is differentiable in $\mathbb{R}$ and $F'$ is bounded. Given a PDE $$ \begin{cases} u_t-\displaystyle{\sum}_{i,j=1}^na^{ij}(x)u_{x_ix_j}=F(u)&\text{a.e. }(x,t)\in\Omega\times(0,T],\\u|_{t=0}=\varphi&\text{in }L^2(\Omega) \end{cases} $$ where the matrix function $[a^{ij}(x)]_{n\times n}\in C^{\infty}(\bar{\Omega})$ is positive definite and symmetric. I want to know that if there exist a weak soluton $u\in L^{\infty}(0,T;H^1_0(\Omega))\cap L^2(0,T;H^2(\Omega))$ such that $u_t\in L^2(\Omega\times(0,T])$. If it exists, how to prove it? Thanks!

Start by showing that there exists a weak solution with lower regularity (standard $L^2$ energy), i-e $$ u\in C([0,T];L^2)\cap L^2(0,T;H^1_0) \qquad\mbox{with}\qquad u_t\in L^2(0,T;H^{-1}). $$ (This can be achieved opening any textbook about quasilinear parabolic equations.)

Then improve the regularity: since $F$ is Lipschitz it is easy to see that $$ f:=F(u)\in L^2(0,T;L^2) $$ hence $u$ is a weak solution of the frozen Initial-Boundary-Value problem $$ \left\{ \begin{array}{ll} u_t+Lu=f & \mbox{in }Q_T\\ u=0 & \mbox{on } [0,T]\times \partial\Omega\\ u|_{t=0}=\varphi & \mbox{in }\Omega \end{array} \right. $$ with $\varphi\in H^1_0$ and $f\in L^2(0,T;L^2)$, as well as $L$ a good, smooth, uniformly elliptic operator. Standard results (improved regularity) for this linear problem gives the desired regularity for $u$ (see e.g. Evans' book "PDEs" section 7.1.3).