I consider 1D system of six components with the following structure:

Finding W \in H^1_0([0,1],R^6) such that

a(W',V') + b_1(W',V) + b_2(W,V') + c(W,V) + d(V) = 0. for all V in H_0^1([0,1];R^6)

where

a, b_1, b_2, and c are bilinear forms and d is linear wr.t.H^1([0,1];R^6).

I have a full coupelng of the components, i.e. a,b_1 ,b_2 and c have a representation of 6x6 matrix fields and

a is elliptic.

Does someone know literature for existence, uniqueness, regularity for such a problem?