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I'm working with the linear equations of poroelasticity (describing flow and deformation processes in porous media). The particular equations i'm working with (in 1D) are of the form

$$-\left(\lambda + 2\mu\right)\frac{d^2u}{dx} + \alpha\frac{dp}{dx} = f$$ $$\frac{d}{dt} \left(c_0p + \alpha \frac{du}{dx}\right) -\frac{1}{\mu_f} \frac{d}{dx}\left(k(x)\frac{dp}{dx}\right)=h$$

I'm familiar with PDE Theory as it applies to elliptic, parabolic, and hyperbolic equations (of a single dependent variable), but I don't know if this theory extends to include systems of linear PDE's as well. Is there an extension of PDE theory to linear systems of PDE's? In particular, how do I classify a system as opposed to a single equation?

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