Timeline for Bound for zero-crossings of heat equation
Current License: CC BY-SA 4.0
16 events
| when toggle format | what | by | license | comment | |
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| Oct 21, 2023 at 10:39 | comment | added | NancyBoy | Hi @Lorenzo Pompili, I posted my attempt here I you want to have a look! | |
| Oct 19, 2023 at 10:58 | comment | added | NancyBoy | I forgot to mention your pseudo @Lorenzo Pompili, sorry | |
| Oct 19, 2023 at 10:38 | comment | added | NancyBoy | is it ok for you if I show you what I've done so far (following your advice) and show you where I am stuck (in another answer to this post) ? | |
| Oct 15, 2023 at 9:21 | comment | added | Lorenzo Pompili | Let us continue this discussion in chat. | |
| Oct 15, 2023 at 9:01 | comment | added | NancyBoy | Last question: where do you need in your proof that $t\leq 1 $ ? | |
| Oct 15, 2023 at 8:43 | history | bounty ended | NancyBoy | ||
| Oct 15, 2023 at 8:32 | comment | added | NancyBoy | Thank you vrry much for all these details ! | |
| Oct 15, 2023 at 8:32 | vote | accept | NancyBoy | ||
| Oct 14, 2023 at 21:30 | comment | added | Lorenzo Pompili | * that $\partial_u$ should be $\partial_xu$ | |
| Oct 14, 2023 at 21:28 | comment | added | Lorenzo Pompili | I guess for small $t$ you really need to use that $f$ approaches $0$ smoothly as $x\to 0$, so instead of bounding $f(x)$ with $-1$ for negative $x$, it would be better to use $f(x)\geq x\cdot(1+\sup_{y\in[-1,0]} |f’(y)|)$, $x\leq 0$. Doing the same proof with this bound for $x\leq0$ would probably work. | |
| Oct 14, 2023 at 21:16 | comment | added | Lorenzo Pompili | Sure. For the first inequality, I use that $f(x)\geq -1$ for negative $x$ (because of the assumption on the $L^\infty$ norm), that $f(x)\geq\varepsilon$ on $[1,2]$, and $f(x)\geq 0$ on the remaining intervals. Concerning the second question, for the bound on $t\in[0,1]$ it is enough to show that $\partial_u(t,x_t)>0$ for all $t\in[0,1]$ (you start from $u(t,x_t)=0$, differentiate w.r.t. $t$, and obtain $\frac{d}{dt}x_t=\frac{u_{xx}(t,x_t)}{u_x(t,x_t)}$). It should follow by similar methods, but I don’t immediately see how. My bounds are too loose for small $t$ | |
| Oct 14, 2023 at 19:29 | comment | added | NancyBoy | Thank you very much @Lorenzo Pompili for your detailed answer! I have two questions: (i) : Could you provide details on how you get the very first inequality ? (ii) : Do you have an idea for getting a bound on $t\in[0,1]$ ? | |
| Oct 14, 2023 at 17:35 | history | edited | Lorenzo Pompili | CC BY-SA 4.0 |
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| Oct 14, 2023 at 16:21 | history | edited | Lorenzo Pompili | CC BY-SA 4.0 |
added 188 characters in body
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| Oct 14, 2023 at 16:04 | history | edited | Lorenzo Pompili | CC BY-SA 4.0 |
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| Oct 14, 2023 at 15:56 | history | answered | Lorenzo Pompili | CC BY-SA 4.0 |