4
$\begingroup$

In a nutshell, if $u$ is a solution to $$ \partial_r^2 u(r)+ \frac{1}{r} \partial_r u(r) - u(r) ( 1- u(r)) = 0, \quad \text{for} \; r>r_0>0\\ \lim_{r \to \infty} u(r) = 0, \quad \text{and} \quad u(r_0) = u_0 > 0 . $$ I'd like to know the behavior of $u(r)$ as $r \to \infty$.

Motivation: $u$ is the steady-state concentration of a substance whose dynamics are determined by a reaction-diffusion equation obtained by adding a time derivative to the first line (the Fisher-KPP), and there is an influx of the substance at $r_0$. The setting is two-dimensional, but assumed to be radially symmetric; if you want to be more general replace $1/r$ by $(d-1)/r$.

Without the first derivative term the equation can be written explicitly with elliptic functions, having $u(r) \sim \exp(-r)$ as $r \to \infty$; but I have good reason to think $u(r) \sim \exp(-r)/\sqrt{r}$ in this case.

$\endgroup$
  • 2
    $\begingroup$ The book "Advanced mathematical methods for scientists and engineers" is a classic reference for treating problems like this. The method given by Alexandre is discussed at length. $\endgroup$ – Tom Dickens Nov 23 '13 at 3:37
5
$\begingroup$

I do not have a complete solution, just a comment. As $u(r)\to 0$, one may try to neglect the quadratic term. Then the equation becomes linear, and its solution tending to $0$ is called the modified Bessel function $K_0$. $K_0(x)=Y_0(ix)$, where $Y_0$ is the Weber function ("second" solution of the Bessel equation). It has asymptotic behavior $$K_0(x)=cr^{-1/2}e^{-r}(1+O(1/r)).$$ This is consistent with what you wrote. Now I think that rejection of the quadratic term can be justified in this case with some standard perturbative method.

Some evidence that such an approximation can be justified comes from $u^{\prime\prime}-u(1-u)=0$, which can be exactly solved using elliptic functions. The linearlization that I propose would give $u^{\prime\prime}-u=0$, whose solution tending to $0$ is $e^{-r}$ and this is confirmed by exact solution in elliptic functions.

I believe that a rigorous justification of this asymptotics can be obtained from the papers: J. Malmquist, Acta Math. 73 (1940), 87–129; 74 (1941), 1–64, 109–128; MR0003898 and M. Iwano, Intégration analytique d'un système d'équations différentielles non linéaires dans le voisinage d'un point singulier. I. Ann. Mat. Pura Appl. (4) 44 1957 261–292, MR0096838.

In English: W. Wasow, Asymptotic expansions for ordinary differential equations, John Wiley and sons, NY 1965, he has a brief chapter IX on non-linear equations, explaining the results of Iwano and Malmquist.

$\endgroup$
  • $\begingroup$ Hey, thanks! That looks just right. I don't read French, and the book @Tom_Dickens refers to seems like a good intro, but lacks theorems; but a bit of searching around will end me up at the right place. (still, if you know a good non-French reference, I'd appreciate it!) $\endgroup$ – petrelharp Nov 23 '13 at 19:40
  • $\begingroup$ Tom probably means the book of Bender and Orszag. It indeed contains no proofs. I know one of the authors, and he thinks that engineers and scientists do not really need mathematical proofs. I added an English reference with some proofs to my answer. $\endgroup$ – Alexandre Eremenko Nov 24 '13 at 0:12

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

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