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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.

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  • $\begingroup$ There should also be some assumption made on $A$ to guarantee that your equation is parabolic. Can you include that please? $\endgroup$
    – Willie Wong
    Commented May 9, 2014 at 8:16
  • $\begingroup$ @WillieWong It is weakened coercivity property of $A$. For example, for $L^2(0,T;V)$ space it is $(Ay,y) + \lambda\|y\|^2_H \geq \alpha\|y\|^2_V, \alpha > 0$, $H = L^2(\Omega)$ $\endgroup$
    – jokersobak
    Commented May 9, 2014 at 8:45
  • $\begingroup$ Hm, can you edit that into your question and, also in what sense are you considering the pairing $(Ay,y)$? By your definition that should be a scalar, but $\|y\|^2_H$ and $\|y\|^2_V$ are both time dependent objets. $\endgroup$
    – Willie Wong
    Commented May 9, 2014 at 9:10
  • $\begingroup$ @WillieWong This holds for any $t$. $V \subset H = H' \subset V'$ and $(Ay,y)$ means the value of functional $A(t)y$ on the element $y$. $\endgroup$
    – jokersobak
    Commented May 9, 2014 at 9:19
  • $\begingroup$ @jokersobak Take a look at the book by Zeidler on monotone operators. He does a nonlinear equation in this setting but yours obviously applies and will be simpler. $\endgroup$
    – student
    Commented May 9, 2014 at 20:28

1 Answer 1

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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.

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