Convergence of solution of non linear problem using heat equation I am considering the following non linear problem:
$u_{t}(t,x)-u_{xx}(t,x)+(u_{x}(t,x))^2=f(t,x)$ for $t>0, x \in (0,1)$
$u(0,x)=u_{0}(x)$ for $x \in [0,1]$
$u(t,0)=u(t,1)=0$ for $t>0$
where f is $C^{\infty}$ and $u_0$ is continuous, and they are both $\geq 0$ everywhere.
So far I've proved that $u \geq 0$ everywhere (the hint is using the equation solved by $v(t,x)=e^{-u(t,x)}$ - I found that $v$ is a sub solution to the heat equation so I applied maximum principle).
Now I'm supposed to show that $\int_{[0,1]}(e^{-u(t,x)}-1)^2dx$ goes to $0$ as t goes to infinity. I'm completely stucked at this. Can some give me a hint? Thank you in advance.
I was suggested to use Poincaré inequality but I'm not sure how to use it, since the hypothesis $\int_{[0,1]}(e^{-u(t,x)}-1)dx=0$ doesn't seem verified.
 A: Let $w=e^{-u}-1$, then $w$ solves the problem:
$$
\begin{cases}
w_t-w_{xx}=f & t>0,x\in(0,1)\\
w(0,x)=e^{-u_0(x)}-1 & x\in[0,1]\\
w(t,0)=0=w(t,1) & t>0
\end{cases}
$$
with $f(t,x)=-ce^{-\alpha t-u(t,x)}$. Fix now $E(t)=\int_0^1w(t,x)^2dx$ we obtain that:
$$
\int_0^1w\cdot w_tdx-\int_0^1 w\cdot w_{xx}dx=\int_0^1w\cdot f dx
$$
Integrating by part the second term and then applying Poincare inequality we get:
$$
\dfrac{1}{2}\partial_t\int_0^1w(t,x)^2+\int_0^1w(t,x)^2\leq\int_0^1w(t,x)\cdot f(t,x)dx
$$
Then fixing $H(t)=\int_0^1w(t,x)\cdot f(t,x)dx$ you get
$$
\dfrac{E'(t)}{2}+E(t)\leq H(t)
$$
Now if you multiply by $e^{2t}$ you recognise the derivative of $\frac{1}{2}E(t)e^{2t}$ on the LHS. Then integrating and applying TFC you get:
$$
E(t)\leq e^{-2t}\bigg(E(0)+2\int_0^tH(s)e^{2s}ds\bigg)
$$
Finally you can estimate $H(s)$ using the initial hypothesis of $f$:
$$
H(s)=ce^{-\alpha t}\int_0^1(1-e^{-u(t,x)})e^{-u(t,x)}dx\leq\dfrac{ce^{-\alpha t}}{4}
$$
And substituting in the previous equation and computing the integral you get the thesis letting $t\to\infty$ on both sides.
Edit: Recall that Poincarè works because $w$ is zero on the boundary.
