Consider the following heat equation with a perturbation given by a second order mixed derivative: $$u_t - u_{tx} - u_{xx} = 0$$ Does this equation have a name? How can one prove a wellposedness result for it?
3
-
1$\begingroup$ I am not aware of a name for it. How you would prove wellposedness depends on what boundary conditions you want to add to it. If you consider it on the whole real line, Fourier transform in x will give you easy results. $\endgroup$– Michael RenardyCommented Sep 23, 2020 at 1:34
-
1$\begingroup$ Doing the change of variables $s = 2t - x$, $y = x$, in the $s, y$ coordinates your equation reads $2 u_s + u_{ss} - u_{yy} = 0$. If you take $s$ to be the "time variable" this is just a damped wave equation. $\endgroup$– Willie WongCommented Sep 23, 2020 at 5:19
-
$\begingroup$ the equation with $u_{tx}$ replaced by $u_{txx}$ appears in many physical problems (search for "heat conduction involving two temperatures"); in what context does your equation arise? It is not invariant for $x\mapsto -x$, which seems strange. $\endgroup$– Carlo BeenakkerCommented Sep 23, 2020 at 8:14
Add a comment
|