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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!


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Could you elaborate on what properties do you have in mind? Because a dilatation of a space variable $u(x,t)=v(x\nu^{-1/2},t)$ reduces the problem to the case $\nu=1$. So the property you are interested in have to be not invariant under linear transforms. – Andrew Aug 8 '11 at 18:40
Could you elaborate more on the exact problem you are studying? Furthermore, even in $d=1$ the solution depends on $\nu$ (Example: $F=0$, then we have the fundamental solution that is $\nu$-dependent). Not quite sure what you are looking for – Michael Kissner Aug 8 '11 at 18:40
@Michael Kissner, yes, the fundamental solution is dependent on $\nu$ for all dimensions, but without $F$, the solution doesn't depend on $ \nu$ on a critical manner. It is essentially change of the time scale. :-) – Anand Aug 8 '11 at 18:48
@Anand Then it would be nice to write the function $\sigma$. Say if it is а power function, there is a notion of a critical exponent. It depends on dimension. So the value of the exponent what is subcritical for $n=1$ could be critical for $n=2$. – Andrew Aug 8 '11 at 19:17
As far as I understand the first was the work of Fujita…. Many others followed. There are numerous works of Pokhozhaev Perhaps references to some recent results could be found there. – Andrew Aug 8 '11 at 19:46
up vote 4 down vote accepted

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 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.

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Thanks Paul Tupper for your hints. It is very helpful. I am still working on this problem. It takes time. :-) – Anand Aug 10 '11 at 8:23

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