I am trying to prove global existence and uniqueness, or even better, find an analytical solution, or at least some form to get rid of one of the dimensions to the following semilinear PDE
$$
0=u_{t}-c_{0}+\left(c_{1}x-c_{2}x^{3}y\right)u_{x}+c_{3}x^{2}u_{xx}+c_{4}y^{2}u_{yy}+c_{5}y^{3}u{}^{2},\\
\textrm{with terminal condition: } \quad u(T,x,y)=c_6 \quad \forall (x,y) \in (U_x,U_y)
$$
where the only nonlinearity comes from the $u^2$ term, and where $c_i$ are positive constants for all $i \in \{0,\dots,6\},$ $t \in [0,T]$ is the time, and the variables $x$ and $y$ lie in some open sets.
If it can help, with an implicit numerical scheme in $y$, the numerical solution does not explode and appears smooth. This PDE is the Hamilton-Jacobi-Bellman equation of a stochastic control problem, and then simplified using some ansatzs. I was unable to further simplify the PDE by getting rid of one of the $x$ or $y$ dimensions. My intuition is that the solution has a form with an hyperbolic tangent, because of the $u^2$ term. Also, if you fix the $y^3$ term, then there is a closed-form solution with a $\tanh.$
I also wanted to know, if I succeed in showing local existence near $T$, can I use a non-explosion argument using bounds, to transform local into global existence ?
Thank you very much for any help or guidance regarding my problem.
I found a way to further analyze this PDE. By using the form $$u\left(t,x,y\right)=\sum_{n=0}^{+\infty}u_{n}(x,y)\left(T-t\right)^{n},\quad\quad u_{0}(x,y)=-c_{6} $$ and identifying, we can end up with an infinite system $$\begin{cases} u_{0}(x,y) & =-c_6\\ u_{1}(x,y) & =P(x^{0},y^{3})\\ 2u_{2}(y,S) & =P(x,y^{9})\\ ... & ....\\ mu_{m}(y,S) & =P(x^{m-1},y^{3^{m}})\\ ... & ... \end{cases}$$ where $P(x^i,y^j)$ stands for a known polynomial of order $i$ in $x$ and order $j$ in $y.$
For a time $t$ near $T,$ I believe I can show that $(T-t)^n$ goes to zero faster than the highest powers of $x$ and $y$ blow up, especially if I consider bounds on $x$ and $y.$
Now my question is, with a local existence near $T,$ can I use a-priori bounds of the solution to transform this into global existence ?
Thank you again for your help.