Timeline for Existence and uniqueness for reaction-diffusion equations
Current License: CC BY-SA 4.0
19 events
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| S May 14, 2019 at 14:01 | history | bounty ended | CommunityBot | ||
| S May 14, 2019 at 14:01 | history | notice removed | CommunityBot | ||
| May 9, 2019 at 11:09 | history | edited | Oleg | CC BY-SA 4.0 |
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| S May 6, 2019 at 12:18 | history | bounty started | Oleg | ||
| S May 6, 2019 at 12:18 | history | notice added | Oleg | Draw attention | |
| May 4, 2019 at 22:01 | history | edited | Oleg | CC BY-SA 4.0 |
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| May 3, 2019 at 15:34 | comment | added | Oleg | @WillieWong Thank you for the comments! You are totally right: if $u_0$ is continuous, then this case is described in Lunardi's boook and local existence follows e.g. by the Picard iterations. But what to do if $u_0\in L^2$? | |
| May 3, 2019 at 15:16 | comment | added | Willie Wong | Your linked question is somewhat different. In the linked question $f$ is continuous but not Holder continuous, and the issue is local existence. In your case since $f$ is Lipschitz in $u$, you should be able to get local existence for $C^0$ initial data just using Picard iteration. Your dissipativity condition should give you also a priori, time-dependent upper bounds on $|u|$ using just the maximum principle, if I am not mistaken. This should give you most things you need for what you want. | |
| May 3, 2019 at 15:00 | comment | added | Willie Wong | @MichaelRenardy: is it really true? If $f$ is independent of $z$, the putative inequality would read $0 \leq C - C|z_1 - z_2|^2$ for all $z_1, z_2$, which certainly is not always holding. The condition assumed seems to me to be requiring $f$ to be asymptotically in $z$ behaving like a strictly concave function. | |
| May 3, 2019 at 11:12 | history | edited | Oleg | CC BY-SA 4.0 |
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| May 3, 2019 at 11:01 | history | edited | Oleg | CC BY-SA 4.0 |
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| May 3, 2019 at 10:55 | comment | added | Oleg | @MichaelRenardy Thanks for your comment, but I also assumed that $f$ is continuous in $t$ in $x$. In this case if $f$ does not depend on $u$, then it is bounded on $[0,T]\times \mathbb{T}^d$ and there is no blow-up. | |
| May 3, 2019 at 10:07 | history | edited | Oleg | CC BY-SA 4.0 |
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| May 3, 2019 at 9:43 | history | edited | Oleg | CC BY-SA 4.0 |
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| May 3, 2019 at 9:42 | comment | added | Oleg | @user35593 Yes, $u_0$ should ideally be in $L_2$, but existence for continuous $u_0$ would also be fine for me. | |
| May 3, 2019 at 6:24 | comment | added | user35593 | Do you have some initial conditions as well? | |
| May 2, 2019 at 20:56 | history | edited | Oleg | CC BY-SA 4.0 |
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| May 2, 2019 at 20:37 | history | edited | Oleg |
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| May 2, 2019 at 13:54 | history | asked | Oleg | CC BY-SA 4.0 |