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My question regards uniqueness of the pair $(u(t, x), v(t))$ which satisfy the following one dimensional time dependent heat equation with a(n) (also time varying) Robin boundary condition at the origin. Assume we are given the initial conditions of $v(0) \geq 0$ and $u(0, x) \geq 0.$ Do these give a unique solution to: \begin{align*} &u_t(t, x) = \frac{1}{2}u_{xx}(t, x) + v(t)u_x(t, x), \text{ when } x > 0\\ &u_x(t, 0) = 2v(t)u(t, 0)\\ &\frac{d}{dt}v(t) = \frac{1}{2}u(t, 0). \end{align*} Say these equations hold as functions, not just as distributions. Although if someone can answer the weak-uniqueness that'd also be useful.

PDE's are not my research area, so any references or additional comments would be greatly appreciated.

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