Let $u(t)$ be the solution of the following SPDE
$\frac{\partial u}{\partial t}(t,x)=\Delta u(t,x) +f(t,x,u(t,x))+g(u(t,x)) \frac{dW(t)}{dt}$
with initial condition $u_0(x)$ and Dirichlet boundary condition, and defined on domain $[0,1]\times [0,T].$ Here $g$ is bounded and satisfy Lipschitz conditions in $u$ variable, $W(t)$ is Brownian motion, $f$ has linear growth and satisfy Lipschitz condition in u.
Question: I want to apply Ito formula on $||u(t)||_{L^2([0,1])}^{2p}.$ Could someone please help?
I have tried in this way:
Let $D=[0,1]$, and $F(u(t))=||u(t)||_{L^2([0,1])}^{2p}= (\int_D |u(t,x)|^2 dx)^p.$
Now, I am trying to use the following Ito formula
$F(u(t))-F(u(0))=\int_0^t F'(u(s))du(s)+1/2\int_0^t F''(u(s))g^2(u(s))ds$.
Now,
$F'(u(t))=p(\int_D |u(t,x)|^2 dx)^{p-1} (2\int_D|u(t)|du(t) dx)=2p||u(t)||^{2p-2}(u(t),du(t)).$
$F''(u(t))=4p(p-1)||u(t)||^{2p-4}(u(t),du(t))^2+2p||u(t)||^{2p-2}$ ~~~is it correct?
Substituting in the formula , we have
$F(u(t))-F(u(0))=2p\int_0^t||u(t)||^{2p-2}(u(t),du(t)) du(s)+\frac{1}{2}\int_0^t (p(p-1)||u(t)||^{2p-4}(u(t),du(t))^2)+p||u(t)||^{2p-2})g^2(u(s))ds$.
Is it correct?