Convergence of Linear Systems of PDE

Assume we have a sequence of C1C^1 functions gk=(gk,1,…,gk,n)g^k = (g^{k,1},\dots,g^{k,n}) where each gk,ig^{k,i} are real valued functions on Rn\mathbb{R}^n whose coordinates I will denote as (x1,…,xn)(x_1,\dots,x_n). Given any function f=(f1,…,fn)f=(f^{1},\dots,f^{n}), I denote its partial derivative with respect to xix_i as fi=(f1i,…,fni)f_i=(f^{1}_i,\dots,f^{n}_i). Now consider the following system of linear PDE
n∑i=1Ai∘fki=gk,
\sum_{i=1}^n A_i \circ f^k_i = g^k,

where AiA_i are matrices with constant coefficients but not necessarily invertible or diagonalizable. Lets assume now that for each given gkg^k we can find a solution fkf^k on some compact U⊂RnU \subset \mathbb{R}^n. Lets also assume that gkg^k converges uniformly to some continuous function gg. Then do the solutions fkf^k converge to some function ff as well? In simple cases when it is an ODE or two dimensional linear system of PDE with nice matrices, the solutions fkf^k can be expressed as some integrals of gkg^k. And when gkg^k converge uniformly then the integrals converge. I am not sure how to do this in higher dimensions or whether if this is true.

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