Timeline for localization of the $L^p$ variation for heat equation
Current License: CC BY-SA 3.0
4 events
| when toggle format | what | by | license | comment | |
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| Oct 21, 2013 at 13:36 | comment | added | leo monsaingeon | I am starting to believe that my statement $\lim\limits_{t\searrow 0}\frac{1}{t}|u(t)|_{L^p(E^c)}=0$ is too much to ask for, since this really means differentiability at time $t=0$ in $L^p(E^c)$ with zero derivative. Instead, can we get an upper bound on how fast $|u(t)|_{L^p(E^c)}$ grows for small $t>0$? Maybe some estimate $|u(t)|_{L^p(E^c)}\leq Ct^{\alpha}$ with $\alpha<1$? but this may be again too much to ask for. | |
| Oct 17, 2013 at 16:45 | history | edited | leo monsaingeon | CC BY-SA 3.0 |
added 2 characters in body
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| Oct 17, 2013 at 11:30 | comment | added | leo monsaingeon | I just realized that my question can be summarized as follows: For initial datum $0\leq u_0\in L^1\cap L^p(R^d)$ with initial support $supp(u_0)=E$ of finite measure, is it true that the restriction $t\mapsto u(t)|_{E^c}$ of $u(t,x)$ to the complement of the initial support $E^c$ is $\mathcal{C}^1$ from $[0,\infty)$ to $L^p(E^c)$ and that the time derivative at time $t=0$ is zero? | |
| Oct 17, 2013 at 10:59 | history | asked | leo monsaingeon | CC BY-SA 3.0 |