Detrimine the diffusion equation ut=kuxx

Question

Solve the following problem:

Suppose that u(x,t) satisfies the diffusion equation  u_t=ku_xx

for 00, and the Robin boundary conditions  u_x(0,t)-a₀u(0,t)=0 and u_x (L,t)+ a_Lu(L,t)=0

where k,L a₀, and a_L are all positive constants.

Show that ∫^L₀ [u(x,t)]² dx is a decereasing function of t.

Provide step by step calculations.