2
$\begingroup$

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

$\endgroup$
9
  • 4
    $\begingroup$ 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. $\endgroup$
    – Andrew
    Aug 8, 2011 at 18:40
  • 2
    $\begingroup$ 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 $\endgroup$ Aug 8, 2011 at 18:40
  • $\begingroup$ @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. :-) $\endgroup$
    – Anand
    Aug 8, 2011 at 18:48
  • 1
    $\begingroup$ @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 $n=2$. $\endgroup$
    – Andrew
    Aug 8, 2011 at 19:17
  • 1
    $\begingroup$ As far as I understand the first was the work of Fujita zentralblatt-math.org/zmath/en/advanced/…. Many others followed. There are numerous works of Pokhozhaev mathnet.ru/php/person.phtml?option_lang=eng&personid=12566. Perhaps references to some recent results could be found there. $\endgroup$
    – Andrew
    Aug 8, 2011 at 19:46

1 Answer 1

4
$\begingroup$

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.

$\endgroup$
1
  • $\begingroup$ Thanks Paul Tupper for your hints. It is very helpful. I am still working on this problem. It takes time. :-) $\endgroup$
    – Anand
    Aug 10, 2011 at 8:23

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.