Timeline for Decay of Solutions to the Heat equation
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 9, 2015 at 18:29 | vote | accept | Matthias Ludewig | ||
| Jul 23, 2014 at 15:35 | comment | added | Andrew | Namely dividing integral $\int_M \Gamma(x,y,t)u_0(x)\,dx$ into two: over $B_R$ and $M\backslash B_R$. Taking $x$ with $d(x)$ large enough the first part will be small due to the factor $e^{-c_1d^2(x,y)/t}$ and the second part can be made small taking $R$ s.t. $|u_0|\le \varepsilon$ on $M\backslash B_R$. | |
| Jul 23, 2014 at 15:34 | comment | added | Andrew | Denote $d(x,y)$ the distance function, $d(x)=d(x,0)$ and $B_R=\{x\in M|d(x)<R\}$. If an estimate for $\Gamma$ like $$ |\Gamma(x,y,t)|\le Ct^{-n/2}e^{-c_1d^2(x,y)/t+c_2t} $$ holds then it is straightforward to obtain the required property. For $\mathbb R^n$ such an estimate is known and for $M$ can be done (I think) in the same way. | |
| Jul 23, 2014 at 12:41 | comment | added | Matthias Ludewig | A function $u$ is in the space $C_0(\mathbb{R}^n)$ if for each $\varepsilon >0$, there exists a compact set $K$ such that $|u|< \varepsilon$ outside of $K$. | |
| Jul 23, 2014 at 11:50 | comment | added | Andrew | What exactly is meant under vanishing of solutions at infinity? | |
| Jul 23, 2014 at 10:59 | comment | added | Matthias Ludewig | What about the vanishing condition at infinity? You only showed smoothess, right? | |
| Jul 23, 2014 at 9:32 | history | edited | Andrew | CC BY-SA 3.0 |
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| Jul 23, 2014 at 8:23 | history | answered | Andrew | CC BY-SA 3.0 |