Suppose we consider the heat equation $$\partial_t u = \Delta u, x \in \text{int}D^2, t > 0$$ where $D^2$ is the closed unit disc in $\mathbb{R}^2$, subject to Neumann type boundary conditions $$\partial_\eta u(x, t) = A(t), x \in \partial M, t > 0$$ and the initial condition $$u(x, 0) = u_0(x), x \in D^2$$ Do we necessarily have to have $$\partial_\eta u(x, 0) = \lim_{t \to 0} A(t) ?$$ Is the answer same for the wave equation $$\partial^2_t u = \Delta u$$ with the same initial and boundary conditions?
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$\begingroup$ The choice of $u_0$ has nothing whatsoever to do with the bc (say $A\equiv 0$, and take any $u_0$ that does not satisfy Neumann bc's). Maybe I'm not understanding the question correctly? $\endgroup$– Christian RemlingJul 6, 2014 at 1:14
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$\begingroup$ If the initial condition does not satisfy the boundary condition, your solution will not satisfy the boundary condition for $t=0$, but will satisfy it at other times, if the conditions are appropriate. $\endgroup$– Alexandre EremenkoJul 6, 2014 at 5:49
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