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 $$u_t-\sum_{i,j=1}^na^{ij}(x)u_{x_ix_j}=F(u),a.e.(x,t)\in\Omega\times(0,T],$$ $$u|_{t=0}=\varphi\text{ in }L^2(\Omega)$$ 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!

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!

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 $$u_t-\sum_{i,j=1}^na^{ij}(x)u_{x_ix_j}=F(u),a.e.(x,t)\in\Omega\times(0,T],$$ $$u|_{t=0}=\varphi\text{ in }L^2(\Omega)$$ where the matrix function $[a^{ij}(x)]_{n\times n}\in C^{\infty}(\bar{\Omega})$ is positive definite and symmetric. I want ro 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!

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 $$u_t-\sum_{i,j=1}^na^{ij}(x)u_{x_ix_j}=F(u),a.e.(x,t)\in\Omega\times(0,T],$$ $$u|_{t=0}=\varphi\text{ in }L^2(\Omega)$$ 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!