Question

Hi, I have a PDE of the type $$v_t+av_{xx}+bv_{yy}-\gamma v^2_y=0$$ with $a,b,\gamma>0$. The final condition is $$v(T,x,y)=g^n(y)h(x)$$ where $g^n$ is a sequence of smooth functions converging to a step function discontinuous at zero, and $h$ is a smooth function. The domain is $[0,T]\times \mathbb R^2$.
Existence should not be a problem, standard results guarantee that we have a $C^{122}$ solution for each $n$.
I am interested to prove that when $n\to\infty$ the quantity $v^n_y$ (I added a superscript to stress dependence on $n$) does not explode for $t < T$ (of course it will explode at $T$ due to the terminal condition). This is true when there is no $x$-variable, since in that case $v_y$ solves another PDE which is explicitly solvable (that's the burgers equation) and you obtain that $v_y$ is bounded by something proportional to $\frac{1}{\sqrt{T-t}}$. However in this slightly more general context I'm unable to prove it.
Does anybody have an idea? Maybe based on some apriori estimates for PDEs (that I do not really master perfectly).
Thank you in advance!