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Reference request: uniqueness for a certain PDE systems

I'm working on a system of the following form: $$(1) \,\,\,\,\,\ \begin{cases} u_{tt} + L_1u + L_2v= f, \\ \nabla u - \nabla v - \nabla v_t=0 \end{cases} $$ where $u(x,t)$ and $v(x,t)$ belong to sui …
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