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I am looking for a hint to a reference: I am dealing with a system of parabolic equations of the form

$$ \partial_t u_i=\sum_{j=1}^m a_{ij}(x,t)\Delta u_j,\quad i=1,\ldots,m $$ set on a open and bounded domain $\Omega \subset \mathbb{R}^d$ (smooth boundary), with initial data $$ u_i \in H^1(\Omega),\; i=1,\ldots,m $$ and homogeneous Neumann boundary conditions $\partial_n u_i=0$ on $\partial \Omega$ for $i=1,\ldots,m$.

I assume the system to be parabolic in the sense of Petrovskii and the coefficients have the regularity $$a_{ij} \in L^2((0,T);H^2(\Omega))\cap H^1((0,T);L^2(\Omega)).$$ (In particular, I am interested in spatial dimension $d\le 3$, so that by embedding, the coefficients are Hoelder-continuous)

I am looking for a theorem that states existence of unique solutions, again in $L^2((0,T);H^2(\Omega))\cap H^1((0,T);L^2(\Omega))$, with the corresponding a priori estimate.

I am aware of (some of) the works of Amann, the book by Eidelmann (both assume two much regularity on the coefficients) and also checked the book of Ladyzhenskaya but I cannot find a theorem which fits to my specific setting.

I guess that the work

"On boundary value problems for linear parabolic systems of differential equations of general form" by V. A. Solonnikov (Trudy Mat. Inst. Steklov., 1965, Volume 83, Pages 3–163) might be a good reference but cannot find the english translation anywhere.

So I would be very glad for a hint to a proper reference of the english version of the book above.

Many thanks!

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  • $\begingroup$ If I remember correctly V. A. Solonnikov considered uniformly parabolic systems, senior coefficients are bounded. $\endgroup$
    – Andrew
    Sep 8, 2018 at 5:41

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