Question

Let u(x; t) satisfy the PDE
Ut = k Uxx; 0 < x < L; t > 0;
u(0; t) = u(L; t) = 0; t > 0;
u(x; 0) = u0(x); 0 <= x <= L;
where k is a positive constant. Show that
integral from 0 to L of u2(x; t) dx <= integral from 0 to L of Uo^2(x) dx for all t >= 0.
What can be said about u(x; t) if u0(x) = 0 on [0; L]?

Answer 1

Hint: Multiply the PDE by u and integrate (by parts) with respect
to x and use the boundary conditions. Then show that the function
f(t) := integral from 0 to L of u2(x; t) dx is non-increasing given that d/dt u2 = 2ut u
{ interchanging integration and dierentiation appropriately