Timeline for Existence and uniqueness of an Euler-type ODE with varying parameters
Current License: CC BY-SA 4.0
13 events
| when toggle format | what | by | license | comment | |
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| Aug 24, 2021 at 17:23 | vote | accept | Laithy | ||
| Aug 24, 2021 at 16:34 | comment | added | Willie Wong | I would suggest asking the corrected version as a new question, especially since my guess would be that you get existence of unique solution in this case. | |
| Aug 24, 2021 at 14:56 | comment | added | Laithy | @williewongThank you so much!! I now understand it very well. I actually made a mistake when I typed the equations. The right side with the nonlocal part should be decaying to 0. I fixed it now in the original post. Do you think we can prove existence/uniqueness for the corrected version? | |
| Aug 24, 2021 at 14:51 | comment | added | Willie Wong | To be clear: the problem is with the asymptotic limit. For any $p, f_0, f_1\in \mathbb{R}$, you can always solve (globally on $[1,\infty)$) the linear inhomogeneous ODE $$ (r^2 - 2ra) f'' + 2(r-a) f' - (4a + m(m+1)) f = -4a p$$ with initial data $f(1) = f_0$ and $f'(1) = f_1$. The claim is that when $p\neq 0$, no solution has $\lim_{r\to\infty} f(r) = 0$. I suspect the only possible values for $\lim_{r\to\infty} f(r)$ are $\{\pm \infty, \zeta p\}$. | |
| Aug 24, 2021 at 14:42 | comment | added | Willie Wong | Also: if you replace $-4a f(1)$ by $-4a p$ for any $p\in \mathbb{R}$, the same argument shows that $f$ is eventually monotonic. If $p > 0$, then the argument says no local max with $f(r) > \zeta p$ and no local min with $f(r) < \zeta p$. Such a condition rules out any solution with $r_1 < r_2 < r_3 < r_4$ such that $f(r_2) < \min(f(r_1), f(r_3))$ and $f(r_3) > \max(f(r_2), f(r_4))$. This leads to eventual monotonicity. Then the asymptotic argument can kick in and also rule out any solution that limits to $0$ at $\infty$. | |
| Aug 24, 2021 at 14:39 | history | edited | Willie Wong | CC BY-SA 4.0 |
added 338 characters in body
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| Aug 24, 2021 at 14:27 | comment | added | Willie Wong | @Laithy: I added more details on the proof of monotonicity. | |
| Aug 24, 2021 at 14:21 | history | edited | Willie Wong | CC BY-SA 4.0 |
added 474 characters in body
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| Aug 24, 2021 at 3:58 | comment | added | Iosif Pinelis | @Laithy : It is seen from my answer that, if we replace $-4af(1)$ with a nonzero constant, then there is no solution vanishing at $\infty$. | |
| Aug 24, 2021 at 1:40 | comment | added | Laithy | It surprises me that if we replace the non-local part $-4af(1)$ with say a constant, then there always exists a solution. So it seems that there is no positive $a$ in which for every C there is a solution. I didn't understand your argument for monotonicity. Say $f(1)>0$. So local max implies $f(r)\leq \zeta f(1)$. How does that imply monotonicity? | |
| Aug 23, 2021 at 23:47 | comment | added | Willie Wong | @IosifPinelis and here I am, thinking "I know the ODE is explicitly solvable, but I am too lazy to look up the solution." | |
| Aug 23, 2021 at 23:31 | comment | added | Iosif Pinelis | Very nice! I thought of such qualitative analysis, but was too lazy for that. | |
| Aug 23, 2021 at 22:36 | history | answered | Willie Wong | CC BY-SA 4.0 |