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.



Existence of the solution of a linear parabolic pde
It seems that $A=-\Delta$ is one of the operators satisfying your condition.
For this operator and any fixed $1<p<\infty$, the solution of $y'+Ay=f\in L^p(0,T;V')$ is in $W=\{y\in L^p(0,T;V):dy/dt\in L^p(0,T;V')$}.
Existence of solution of this type is called theory of "maximal $L^p$ regularity", which can be found in the paper 
"Maximal $L\_p$-regularity for Parabolic Equations, Fourier Multiplier Theorems and $H^\infty$-functional Calculus", written by Peer C. Kunstmann and Lutz Weis.


