Ytukyg

I want to use Lax-Milgram theorem on the following problem
$$
begin{cases}
displaystyle frac{partial T}{partial t}left(x,tright)=alpha frac{partial^2 T}{partial x^2}left(x,tright) \
displaystyle Tleft(0,tright)=25\
displaystyle Tleft(L,tright)=60
end{cases}
$$
I need to find the weak formulation of this, so with $v$ a function that satisfies my conditions I write
$$
int_{0}^{L}frac{partial T}{partial t}left(x,tright)vleft(xright)text{d}x=int_{0}^{L}alpha frac{partial^2 T}{partial x^2}left(x,tright)vleft(xright)text{d}x
$$
Then it becomes
$$
int_{0}^{L}frac{partial T}{partial t}left(x,tright)vleft(xright)text{d}x=left[alphafrac{partial T}{partial x}left(x,tright) frac{partial v}{partial x}left(x,tright)right]^{L}_{0}-int_{0}^{L}alpha frac{partial T}{partial x}left(x,tright) frac{partial v}{partial x}left(x,tright)text{d}x
$$
But I think that i can't write it as $aleft(T,vright)=ell left(vright)$ because here I still have $frac{partial T}{partial t}left(x,tright)$ that blocks me.
How can I apply the Lax-Milgram theorem here ?