# Existence of weak solutions of a parabolic PDE

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).