Heat problem with an internal source of heat for which the maximum principle doesn't hold. The problem is the following and honestly I don't know how to solve it...
$$u_{t}=u_{tt}+2(t+1)+x(1-x) , 0<x<1, t>0$$ $$u(0,t)=0, u(1,t)=0$$ $$u(x,0)=x(1-x), 0<x<1$$
$u(x,t)=(t+1)x(1-x)$ is a solution and there is a question: what are the maximum and minimum values of the initial and boundary data? And show that for some values of $t>0$, the temperature distribution exceeds M ( the maximum value) at certain points in the bar.
If you could help me please, thanks for your time and help everyone.