Conditions for the existence of a solution to a semilinear second-order PDE with a-priori bounds

Question

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.

Answer 1

After a lot of reading, I came across the (enlightening) paper:

On principally linear elliptic differential equations of the second order, Nagumo 1954.

Basically, the (classical) results there shows that for any bounded domain $\Omega$ for your space variables, if you have lower and upper solutions for a quasi-linear / semi-linear PDE, and that your nonlinear operator in the PDE has at most quadratic growth in the gradient, then you can have (in this order) bounds on the solution and on the gradient in your bounded domain. I recommend reading it for simple scalar PDEs.

That being said, I did not need to find upper and lower solutions since I already had nice smooth à-priori bounds for my PDE. But for some simple PDEs, finding upper and lower solutions can be easy by trying constants, or simple form functions.

I wanted to leave this here in case someone with à-priori knowledge on the solution of their quasi-linear or semi-linear PDE and less than quadratic growth in their nonlinearity wanted some simple applicable result to prove existence, without having to go through all the weak-solution theory, which I think could take a few months to master.