two dimensional heat equation
Please I really need some help for this exercise, I can't solve it for any ways...
I need to prove the maximum principle for the two dimensional heat equation with zero boundary data. Really I need help for this...
The book (Asmar) state without prove the maximum principle for one dimension and is this:
1 Answer
$\begingroup$Anytime you see a finite domain, standard heat equation, think separation of variables. Try to use $u(x,y,t) = T(t)X(x)Y(y),$ and you'll get 3 ODE's to solve, something along the lines of \begin{align} T'(t) + c^2(\lambda_x + \lambda_y)T(t) &= 0\\ X'' - \lambda_x X &=0\\ Y'' - \lambda_y Y &=0. \end{align} This should be the motivated approach by seeing the form of the solution, namely 3 functions, each of a single variable, multiplied together. Try it and see how it goes.
A little intuition behind why $m \leq u(x,t) \leq M$ is as follows,
The solution form is exponentially decaying in time, and the trig functions are bounded, so for $t$ growing, the solution relaxes (Show this rigorously by bounding the size of your solution). When $t$ is zero, the solution satisfies the boundary conditions, and those are bounded as well. It is not hard to translate this into a mathematical statement.
