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I am trying to work out the correct definition of a weak solution of $$ u_{xx}=u_t $$ on the domain $(0,1)\times[0,T]$ subject to say periodic boundary conditions $u(0,t)=u(1,t)$ and $u_x(0,t)=u_x(1,t)$. So far I have only come across definitions for the Dirichlet, Neumann or Robin boundary conditions (where after integration by parts we can replace the boundary terms by the boundary conditions). I think the difficulty arise since I only want $u(\cdot,t)\in H^1((0,1))$ yet the definition to also treat non-periodic test functions. Does anyone have any ideas/references to what the standard definitions in the literature are?

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