Consider the function $u(\boldsymbol{x},t)$, where $u:\mathbb{R}^n \times \mathbb{R}_+ \rightarrow \mathbb{R}$ ($\mathbb{R}_+$ denotes nonnegative reals). My question is related to the PDE $\frac{du}{dt}=a \Delta u + b (\Delta u)^2$ s.t. $u(\boldsymbol{x},t=0)=f(\boldsymbol{x})$ for given $f$. The coefficients $a$ and $b$ are given real scalars and $\Delta$ is the Laplace operator with respect to $n$ components in $\boldsymbol{x}=(x_1,\dots,x_n)$.
When $b=0$, this becomes the standard heat equation whose solution is a known analytical form, i.e. the convolution of the heat kernel with $f$. I was wondering if an analytical solution also exists when $b \neq 0$, and if so, what form does it have?
Note: $(\Delta u)^2$ means squared Laplacian, e.g. $(\Delta (x^3+y^3))^2 = (6x+6y)^2$.
Thanks,
Golabi