Timeline for Reference request: continuity of the derivatives of the (fundamental) solution to a parabolic equation
Current License: CC BY-SA 4.0
6 events
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| Mar 3, 2022 at 19:47 | comment | added | GJC20 | Here you may assume that $b,\sigma$ are as nice as possible, and $\alpha$ should belong to the class of non-increasing Holder continuous functions taking values in $[0,1]$. The motivation to study this stability is to show the uniqueness at mathoverflow.net/questions/417113/… | |
| Mar 3, 2022 at 19:46 | comment | added | Connor Mooney | Yes, that is the book. Regarding your other question, I don't know. However, if $u_t + b(t)u_x + \sigma^2/2(1+\alpha(t))^2u_{xx} = 0$ then after the change of variable $u(t,x) = v(f(t),x-h(t))$ with $h' = b$ and $f' = 1/2(1+\alpha)^2$ we see that $v_t + \sigma^2(f^{-1}(t), x+h(f^{-1}(t))v_{xx} = 0$, which simplifies the structure of the equations being considered and reduces the problem to understanding small perturbations of a single Lipschitz coefficient. | |
| Mar 3, 2022 at 19:46 | comment | added | GJC20 | Thank you so much for your quick reply. While with this transformation, is $v$ still a fundamental solution? Here $v(f^{-1}(s),x)=\delta_{y+h\circ f^{-1}(s)}(x)$. Do you think this matters for the estimation? I do appreciate if you could write an answer (even an outline) to my other question | |
| Mar 3, 2022 at 18:44 | vote | accept | GJC20 | ||
| Mar 3, 2022 at 18:44 | comment | added | GJC20 | Thank you very kindly for the answer Connor. Do you mean this book worldscientific.com/worldscibooks/10.1142/3302 ? Btw, this post follows from my question at mathoverflow.net/questions/415175/…. Could you please take a look and let me know whether the dependency of fundamental solutions on the coefficients have been studied. Many thanks | |
| Mar 3, 2022 at 15:54 | history | answered | Connor Mooney | CC BY-SA 4.0 |