Suppose $(\mu_t)_{t\geq 0}$ is a sequence of absolutely continuous measures on $\mathbb{R}^n$, which satisfy the distributional PDE $\partial_t \mu_t = A \mu_t$, where $A: \mathbb{R}^n\to \mathbb{R}^n$ is a linear function. Note that this implies that for any $\varphi\in C_c^\infty(\mathbb{R}^n)$, $$\frac{d}{dt}\int \varphi d\mu_t = \int A\varphi d\mu_t.$$
Is it possible from this to write down a closed form expression for $\mu_t$, perhaps in terms of exponentials? I know similar things can be said for Euclidean ODEs. Is there anything we can say if we assume $A$ is constant?