Timeline for Unconditional nonexistence for the heat equation with rapidly growing data?
Current License: CC BY-SA 3.0
14 events
| when toggle format | what | by | license | comment | |
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| S Jul 17, 2013 at 14:36 | history | suggested | SBF | CC BY-SA 3.0 |
fixed formulas
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| Jul 17, 2013 at 14:29 | review | Suggested edits | |||
| S Jul 17, 2013 at 14:36 | |||||
| Aug 7, 2011 at 2:20 | history | edited | George Lowther | CC BY-SA 3.0 |
fix equation; edited body
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| Aug 6, 2011 at 22:44 | comment | added | paul garrett | Very interesting riff, @George L.! | |
| Aug 6, 2011 at 21:41 | history | edited | George Lowther | CC BY-SA 3.0 |
typos
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| Aug 6, 2011 at 11:04 | comment | added | George Lowther | Thanks for the comments! Regarding the point in your first comment, I think it is possible to build on this method to show that uniqueness fails in the worst possible way. Consider $u\in C^\infty(\mathbb{R})$ and any solution $f\in C^\infty((0,\infty)\times\mathbb{R})$ to the heat equation. It is not assumed that $f(t,x)$ satisfies any particular initial condition, or even that the limit exists as $t\to0$. Then, there will exist a sequence $f_n$ of smooth solutions to the heat equation with initial condition $u$ so that, restricting to $t > 0$, $f_n$ tends to $f$ in the compact-open topology. | |
| Aug 6, 2011 at 3:21 | comment | added | Terry Tao | Ah, you patched (1) already :-). I just wanted to add that the Hahn-Banach theorem can be used to dualise (3), and it seems to say something like a (tempered distributional) solution to an inhomogenous heat equation with compactly supported forcing term and zero initial data cannot be compactly supported at any future time, which is a sort of "parabolic unique continuation" result. | |
| Aug 6, 2011 at 3:13 | vote | accept | Terry Tao | ||
| Aug 6, 2011 at 3:12 | comment | added | Terry Tao | Ah, a nice "pyramid scheme" to recursively build a solution - I like it! I think (1) can be patched by dividing the solution by $\sqrt{t}$ first (one can see from the fundamental solution that this keeps the solution continuous on the interval where the initial data vanishes). One may also have to cap t to be bounded for (1)-(3), but that doesn't really damage (4), as one just needs control on $P_t(\tilde u+v)(x)$ for $t \leq n$, say. Incidentally it seems this gives a new proof of nonuniqueness for the heat equation... | |
| Aug 6, 2011 at 2:59 | history | edited | George Lowther | CC BY-SA 3.0 |
edited body
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| Aug 6, 2011 at 2:56 | comment | added | George Lowther | ...fixed. I think its good now, other than probable small mistakes or typos. | |
| Aug 6, 2011 at 2:54 | history | edited | George Lowther | CC BY-SA 3.0 |
fix proof; added 3 characters in body; edited body; edited body
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| Aug 6, 2011 at 2:27 | comment | added | George Lowther | hmm, I think there's still some issues with (1) here. I missed out a factor of $1/\sqrt{4\pi t}$, but it should be fixable... | |
| Aug 6, 2011 at 2:03 | history | answered | George Lowther | CC BY-SA 3.0 |