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.

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    $\begingroup$ 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

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.

  • $\begingroup$ 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

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