Consider the problem $u_t$ = $u_{xx}$ − $au$ 0< x <1, t>0, a=constant≠0 $u(x,0) = f(x)$ 0 < x < 1 $u(0,t) = 0$, $u(1, t)$ = 0 t > 0. (a) If a > 0, what are the possible steady state solutions? (b) Solve the time-dependent problem via separation of variables method when a > 0. What happens to the solution as $t → ∞$? (c) If a < 0, what are the possible steady state solutions?
In class, we have only done problems with $u_t$ = $u_{xx}$ so I am not sure how to do it with the $-au$ attached.