# Non linear PDE solution

Could someone advise if it is possible to solve the following PDE with Mathematica? I am quite a beginner in Mathematica so any input would be highly appreciated.

$$\displaystyle\frac{1}{2} \sigma^2\frac{\partial^2 u(x,y)}{\partial x^2}+\frac{1}{2} \sigma^2\frac{\partial^2 u(x,y)}{\partial y^2}+a\frac{\partial u(x,y)}{\partial x}\left(\frac{\frac{\partial u(x,y)}{\partial x}+\frac{\partial u(x,y)}{\partial y}}{3\frac{\partial u(x,y)}{\partial x}+\frac{\partial u(x,y)}{\partial y}}\right)^2-r u(x,y)=0.$$, where $$\sigma$$, a and r are come constants.

The domain for $$\sigma=0.85$$, $$r=0.05$$ and $$a=50$$ is specified as follows $$y \leq 0.52 + 0.46 x; x \leq 0.52 + 0.46 y; x\geq 0; y \geq 0; x \leq 0.97; y \leq 0.97$$

witht the boundary conditions are

1. $$u(x,y)=0$$ for $$x=0,y\leq0.46$$;
2. $$u(x,y)=249.5 \mathrm{e}^{-34.5784 x} (-1 + \mathrm{e}^{34.6 x})$$ for $$y=0, x\leq0.46$$;
3. $$\frac{\partial u(x,y)}{\partial x}=1$$ for $$0.46\leq x\leq 0.97$$, on $$x = 0.52 + 0.46 y$$;
4. $$\frac{\partial u(x,y)}{\partial y}=0$$ for $$0.46\leq y\leq 0.97$$, on $$y = 0.52 + 0.46 x$$.