Good day!
Let $V = H^1(\Omega)$, $\Omega \subset \mathbb R^3$.
Consider the linear parabolic equation $y' + Ay = f$ where $f \in L^q(0,T;V')$, $y \in W = \{y \in L^p(0,T;V) \colon dy/dt \in L^q(0,T;V')$. $1/p + 1/q = 1$
$ A\colon W \to L^q(0,T;V') $ - linear operator
I can't find the theorem of existence of the solution of this equation. Usually it is written about the space $W_0 = \{y \in L^2(0,T;V) \colon dy/dt \in L^2(0,T;V')$.
Could you help me, please?
Thank you.