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We develop adaptive controllers for parabolic partial differential equations (PDEs) controlled from a boundary and containing unknown destabilizing parameters affecting the interior of the domain. These are the first adaptive controllers for unstable PDEs without relative degree limitations, open-loop stability assumptions, or domain-wide actuation. It is the first necessary step towards developing adaptive controllers for physical systems such as fluid, thermal, and chemical dynamics, where actuation can be only applied non-intrusively, the dynamics are unstable, and the parameters, such as the Reynolds, Rayleigh, Prandtl, or Peclet numbers are unknown because they vary with operating conditions. Our method builds upon our explicitly parametrized control formulae to avoid solving Riccati or Bezout equations at each time step. Most of the designs we present are state feedback but we present two benchmark designs with output feedback which have infinite relative degree.