Consider the general semilinear elliptic second-order PDE $$ u_t-\mathcal L u=f\left(t,x,u,\nabla u\right) $$ where $\mathcal L$ is an elliptic linear operator (like minus the Laplace operator), $t \in [0,T],$ $x$ is in a bounded smooth domain $\Omega,$ and the boundary conditions are $$ u(T,x)=c, \quad \forall x\in\Omega $$ where $c$ is a constant. $f$ can be nonlinear in $u$ and $\nabla u$ with at most quadratic growth in both of them, and has no singularity in both of them and $x$. For example, $$ f\left(t,x,u,\nabla u\right) = x^\alpha u^2 + ∇u \cdot ∇u $$ is a possible form. Mind that this is a Cauchy problem with final data, a situation which fits with the "backward" heat operator $\partial_t+\Delta$.

Suppose we know from the original problem that the solution to this PDE is bounded by two smooth $\mathcal C^2(\Omega)$ functions: $$\underline u \leq u \leq \overline u, \quad \forall (t,x)\in[0,T]\times \Omega.$$ My question is: is there a result (with possible conditions on $f$) to provide an existence result for a (weak) solution to this PDE ? any reference would be greatly appreciated.

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