[Math] Compatibility of initial and boundary conditions

Question

Suppose we consider the heat equation $$\partial_t u = \Delta u, x \in \operatorname{int}D^2, t > 0$$
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?

Answer 1

No, this is frequently not the case. The initial and boundary conditions may naturally disagree at the edge. For example, suppose we take a cold circular plate (temperature $0$) and start heating its edges at time $t=0$. Then the initial condition is $u(x,0)=0$ while the boundary condition is $A_\eta(x,t)=A>0$. The initial and boundary conditions do not agree along the edge of space-time cylinder, $\partial D^2\times \{0\}$. This will cause the normal derivative of $u$ to be discontinuous at the edge. But it will be a perfectly smooth solution in the open cylinder $\operatorname{int} D^2\times (0,\infty)$, will satisfy the initial condition on $\operatorname{int} D^2\times \{0\}$, and will satisfy the boundary condition on $\partial D^2\times (0,\infty)$.

Similarly for the wave equation.