Consider the equation
ut = uxx, 0  x  1, t> 0
with boundary conditions u(0, t) = u(1, t) = 0 and initial condition u(x, 0) = 4x(1 x).
•

Consider the equation
ut = uxx, 0  x  1, t> 0
with boundary conditions u(0, t) = u(1, t) = 0 and initial condition u(x, 0) = 4x(1 x).
• Show that 0  u  1 for all t > 0.
• Show that u(x, t) = u(1 x, t) for all t > 0.
• Show that R 1
0 u2(x, t)dx is a strictly decreasing function of t, unless u(x, t) = 0 for all
0  x  1.