Timeline for Counterexamples in PDE
Current License: CC BY-SA 3.0
4 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 24, 2011 at 2:45 | comment | added | Rbega | I mean that there is a solution $u$ to the heat equation on $R^n$ so that $u|_{t=0}=0$ but $u\neq 0$ for $t>0$. However, this can't happen if for $t>0$ $u=(e^{|x|})$ (by using the backwards heat kernel) and so one does have uniqueness under some spatial growth assumptions. | |
| Jun 24, 2011 at 1:57 | comment | added | timur | @RBega Checking the proof shows that u is bounded. About the parabolicity, I don't know, De Giorgi himself calls it "of parabolic type" in the paper. Do you mean there is non-uniqueness for the heat equation bounded in time for each spacial point but unbounded in space? | |
| Jun 24, 2011 at 1:37 | comment | added | Rbega | In what sense is that equation parabolic? Shouldn't there be half as many time derivatives as spatial derivatives? Also what growth assumptions are you imposing. There us non-uniqueness for the Cauchy problem for the standard heat equation after all if you don't impose any growth restrictions. | |
| Jun 24, 2011 at 1:14 | history | answered | timur | CC BY-SA 3.0 |