Hello,


I am considering the following non-linear heat equation


$$
\left(\frac{\partial}{\partial t}-\nu\: \Delta \right) u(t,x) = F(t,x) \sigma(u(t,x)),\qquad (t,x)\in R\_+\times R^d
$$
where $F(t,x)$ is certain external heat source and $\sigma$ is a Lipschitz continuous function. Let's call $\nu$ the diffusion coefficient. When I calculate certain properties of the above equation, I find that when $d=1$, the results do not depend on the value of $\nu$, however when $d\ge 2$, the value $\nu$ becomes important. In particular, for the properties that I am studying, we need that $\nu > 1/(2\pi)$ !?.. Does anyone have ever had such critical dependence on $\nu$? Or does anyone know any phenomenon (like phase transitions) depending on $\nu$?


Thank you very much for any hints! 


Anand



Are there any physical phenomena of the heat transfer critically depending on diffusion coefficient? 
What you describe is very much expected from the statistical physics principle "there is no phase transition in one-dimensional systems with short-range interactions at $T>0$." See [Lower Critical Dimension](http://en.wikipedia.org/wiki/Critical_dimension#Lower_critical_dimension) in Wikipedia. Since you have a PDE your interactions are short-range. Since you have noise, this corresponds to $T>0$, i.e. non-zero temperature. The principle states that you should not observe a phase transition when you vary $\nu$ in 1 dimension, but you may in 2 or 3 dimensions.


