Timeline for Integral estimate for the solution of the heat equation
Current License: CC BY-SA 4.0
17 events
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| Apr 7, 2020 at 8:09 | comment | added | Andrew | Solution of the Cauchy problem for the heat equation is not unique. And why for arbitrary $f$ the integral converges? | |
| Apr 5, 2020 at 9:48 | comment | added | Giorgio Metafune | @Nico Thanks, but that inequality seems to be weaker. For the $\Delta$ it gives $$\int_{R^N} (-\Delta u) u|u|^{p-2} \ge 0,$$ which is true by integration by parts. For $I-\Delta $ I am wondering about $$\int_{R^N} (-\Delta u)f|f|^{p-2}$$ with $f=u-\Delta u$ which is closer to the parabolic question. | |
| Apr 4, 2020 at 15:25 | comment | added | Nico | @GiorgioMetafune, it's Theorem 1 in this paper: doi.org/10.1090/S0002-9939-1993-1160303-9. | |
| Apr 4, 2020 at 13:57 | comment | added | Giorgio Metafune | @Nico It seems that the discussion never started. Where can I find the inequality you mentioned? Thanks | |
| Apr 3, 2020 at 22:27 | comment | added | Nico | Zyl, the "stationary version" of this estimate is called "Stroock-Varopoulos inequality". @Giorgio Metafune, keep me in the loop: I'm currently considering a problem where (a more general version of) this kind of result would be very helpful. | |
| Mar 31, 2020 at 19:49 | comment | added | Thomas Kojar | @Zyl what happened when you substituted the inhomogeneous heat equation (you get phi(f(x)) (T* f(x)-u(x,T))? You can also use the exactt solution i.e. heat kernel mollified by f. | |
| Mar 25, 2020 at 9:28 | comment | added | Giorgio Metafune | I have some ideas how to tackle the problem, but I should work out the details, first. If you like, we could keep in contact via mail and then post the solution, if works. I do not know how to write to you, my address is [email protected]. | |
| Mar 24, 2020 at 16:02 | comment | added | user140746 | @GiorgioMetafune I don't know. As for specific $\phi$, I'm mostly interested in power-type i.e. $\phi(s) = |s|^{m-1}s$. | |
| Mar 24, 2020 at 15:32 | comment | added | Giorgio Metafune | Do you know what happens for the elliptic counterpart, that is for $u-\Delta u=f$? Or for specific $\phi?$. Thank you | |
| Mar 24, 2020 at 10:19 | comment | added | Giorgio Metafune | Thank you, I see now. | |
| Mar 24, 2020 at 10:10 | comment | added | user140746 | @GiorgioMetafune I don't know why it should be true, but I wish it to be the case. There was a typo and I meant $-\Delta$ in the integral. | |
| Mar 24, 2020 at 10:09 | history | edited | user140746 | CC BY-SA 4.0 |
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| Mar 24, 2020 at 9:31 | comment | added | Giorgio Metafune | When $\phi(t)=t$ the converse inequality is true. | |
| Mar 24, 2020 at 9:10 | comment | added | Giorgio Metafune | Why this should be true? | |
| Mar 24, 2020 at 2:10 | review | Close votes | |||
| Apr 2, 2020 at 0:16 | |||||
| Mar 24, 2020 at 1:23 | history | edited | Alexandre Eremenko | CC BY-SA 4.0 |
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| Mar 23, 2020 at 23:33 | history | asked | user140746 | CC BY-SA 4.0 |