Help for reaction-diffusion question, R-D-D with reaction function

[jac7]

Hi I would really appreciate it if someone could give me some hints on how to start this problem that I have set because I'm really stuck on it!

This is the problem:

"Consider R-D-D with reaction function f: $$\Re$$$$\rightarrow$$$$\Re$$ given by

f(u)=($$\alpha$$-u)^{3}u $$\forall$$ $$\textsl{u}$$ $$\in$$ $$\Re$$,

with $$\alpha$$ > 0 constant. The initial data u_{0} : [0,$$\alpha$$] $$\rightarrow$$ $$\Re$$ is such that

u_{0}(x) $$\in$$ [0,$$\alpha$$] $$\forall$$ x $$\in$$ [0,$$\alpha$$].

Establish that any solution u : $$\overline{D}_{T}$$ of R-D-D satisfies,

0$$\leq$$ u(x,t) $$\leq$$ $$\alpha$$ $$\forall$$ (x,t) $$\in$$ $$\overline{D}_{T}$$.
and deduce that R-D-D has a global solution on $$\overline{D}_{T}$$ for any T>0."

Please help, I'm so stuck!
Thank you

(I don't know why the alpha is so raised?!)

 

[gtoguy97]

Hints: 1. Start by proving that the solution is bounded by 0 and alpha by analyzing the reaction function. 2. Then, use the maximum principle to prove that the solution remains within the bounds for all time. 3. Finally, deduce that R-D-D has a global solution on $\overline{D}_{T}$ for any $T>0$.