Timeline for Heat conduction type equation in 4D
Current License: CC BY-SA 4.0
6 events
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| Dec 6, 2022 at 16:12 | comment | added | Fetchinson0234 | You are absolutely right, I don't question that $u$ as given is always a solution for any $\xi$. Similarly, $\phi = e^{\xi x_4} / x^2$ is always a Green's function for any $\xi$. But the decay properties are set up for $\phi$, not $u$. So the puzzle I'd like to understand is this: why is it that apparently we have no problem with constructing a $u$-solution for imaginary $\xi$ such that the corresponding Green's function is decaying, but we cannot construct a $u$-solution for real $\xi$ such that the corresponding Green's function is decaying. Is this really the case? Kind of strange if true. | |
| Dec 6, 2022 at 16:05 | comment | added | Carlo Beenakker | I presume you don't question that $u$ in the answer solves the heat equation with all boundary conditions; so if there is a second solution the heat equation would not have a unique solution? (not impossible, but I always thought the heat equation uniquely determined the flow) | |
| Dec 6, 2022 at 16:01 | comment | added | Fetchinson0234 | Respectfully, the down vote came from me, because the crux of the problem is not addressed I believe. The issue is this: on general grounds one can obtain the Green's function from $u$. For purely imaginary $\xi$ we know what the appropriate (decaying) Green's function is and we can construct $u$. For real $\xi$ we also know what the appropriate (decaying) Green's function is but we cannot construct $u$. The decay property requirements are for the Green's function, not $u$. | |
| Dec 6, 2022 at 15:55 | comment | added | Carlo Beenakker | I am bit at a loss to understand the down vote: the OP asks for the solution of the 4+1 dimensional heat equation with a complex parameter $\xi$, for a given initial condition and subject to the requirement that the solution decays at infinity in each coordinate; the solution given in the answer satisfies those requirements; I presume the solution is unique, so what other answer could be possible? Am I missing something? | |
| Dec 6, 2022 at 9:30 | history | edited | Carlo Beenakker | CC BY-SA 4.0 |
added 23 characters in body
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| Dec 6, 2022 at 9:17 | history | answered | Carlo Beenakker | CC BY-SA 4.0 |