MHB Dissipative Phenomena: Diffusion Equation with Source

[simo1]

i was given the question below, and have submitted my working but i cannot move forward from where i stopped on the photo

Consider the diffusion equation (with source)
u_t (x; t) - Du_xx (x; t) + ku(x; t) = 0; 0 < x < ℓ; t > 0; under conditions of zero-flux at the boundary points x = 0, x = ℓ and the initial
condition
u(x; 0) = a(x); a(x) given:
The following problem is concerned with applications of the energy identity
(u_t (
; t); φ(
; t)) + D(ux (
; t); φx (
; t)) + k(u(
; t); φ(
; t)) = 0:

The average value of a function f over the interval [0; ℓ] is dened as
Av(f) =(1 divide ℓ)∫ f(x)dx By letting φ = c

show that under the assumption A = 0
∥u(
; t)∥(less than or equal)  ∥a∥ exp{-(Dc + k)t}View attachment 2181:
Draw a conclusion about the long term behaviour of u(
; t).

 

[MarkFL]

I would suggest that if you are going to attach images of your work rather than typeset it with $\LaTeX$, using more lighting and higher resolution pics. It would be a real struggle (for me at least) to try to read your images.