Timeline for Parabolic smoothing for semilinear PDE
Current License: CC BY-SA 4.0
18 events
| when toggle format | what | by | license | comment | |
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| S Oct 15 at 19:03 | history | bounty ended | CommunityBot | ||
| S Oct 15 at 19:03 | history | notice removed | CommunityBot | ||
| Oct 10 at 4:01 | comment | added | Student | @WillieWong, I just realized that this might not be true in general because finite Strichartz norm and global existence imply dissipation of $\dot{H}^1$ norm to zero which is not true in the presence of bubbling at infinity. | |
| Oct 9 at 16:10 | history | edited | Student | CC BY-SA 4.0 |
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| Oct 7 at 21:10 | comment | added | Student | @WillieWong, I added Theorem 1 in their paper here sites.math.rutgers.edu/~brezis/PUBlications/147-J-fulltext.pdf | |
| Oct 7 at 21:09 | history | edited | Student | CC BY-SA 4.0 |
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| Oct 7 at 20:21 | comment | added | Willie Wong | Can you include a statement of the Brezis-Cazenave theorem in your question statement? I don't have access to it at the moment. | |
| Oct 7 at 18:07 | history | edited | Student | CC BY-SA 4.0 |
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| S Oct 7 at 17:36 | history | bounty started | Student | ||
| S Oct 7 at 17:36 | history | notice added | Student | Authoritative reference needed | |
| Oct 6 at 17:26 | history | edited | Student | CC BY-SA 4.0 |
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| Oct 6 at 2:00 | history | edited | LSpice | CC BY-SA 4.0 |
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| Oct 5 at 23:34 | history | edited | Student | CC BY-SA 4.0 |
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| Oct 4 at 21:50 | comment | added | Student | Thank you for pointing that out! It seems to only work when $T < +\infty.$ In that case one can bound $\int_{t_0}^{T} \int_{\mathbb{R}^n} |u|^{10} \leq C \log(T/t_0)$ since $u$ decays like $t^{-1/(p-1)}=t^{-1/4}$ and so in particular as $t_0 \to T$ this goes to zero. Do you think there is a way to make this work also when $T = + \infty$? My impression was that the heat flow would make the solution more regular at later times $t_0$ which could then give us more decay at later times. | |
| Oct 4 at 18:32 | comment | added | Willie Wong | Doesn't Brezis-Cazenave already give that $\|u(t)\|_{L^6}$ is uniformly bounded? Then since $L^{10}$ is between $L^{6}$ and $L^\infty$, just interpolate to get uniform boundedness of $\|u(t)\|_{L^{10}}$ when $t\in [t_0, T)$, from which what you want follow. | |
| Oct 4 at 17:04 | comment | added | Student | @WillieWong Thank you for the comment! In my application, I would be happy to have that estimate hold when $t>t_0>0$ uniformly. In particular I would like $\int_t^T \int_{\mathbb{R}^n} |u|^{2p} \to 0$ as $t\to T.$ | |
| Oct 4 at 14:20 | comment | added | Willie Wong | Roughly speaking you gain two space and one time derivative when you solve the heat equation. For details, you can look at Krylov's 2008 book on elliptic and parabolic PDEs in Lebesgue and Sobolev spaces. // For fixed $t$ the claimed inequality is certainly true; but I suspect you actually want the bound to be uniform as $t\searrow 0$? | |
| Oct 4 at 3:28 | history | asked | Student | CC BY-SA 4.0 |