Consider the parabolic equation $$ u_t = u_{xx}, $$ $$ u(0,t)=u(1,t)=0, $$ $$ u(x,0) = 1. $$
Initial and boundary conditions are inconsistent but a weak solution exists. It belongs to $C([0,T];L^2(0,1)]\cap L^2(0,T,H^1_0(0,1))$.
What this weak solution looks like? Does this solution satisfy the boundary conditions? (Computations has demonstrated that the solution decreases from $1$ to $0$ and it does not equal to $0$ on the boundary.)
The usual Fourier series method shows that a solution is given by $$u(x,t) = \sum_{n=1,3,5,\dots} \frac{2}{\pi n} e^{-n^2 \pi^2 t} \sin(n \pi x).$$ You can check that this satisfies the boundary and initial conditions in the following senses: it extends continuously to $([0,1] \times (0,\infty)) \cup ((0,1) \times \{0\})$, and the extension equals 0 on $\{0,1\} \times (0,\infty)$ and equals 1 on $(0,1) \times \{0\}$. It does not extend continuously to $\{0,1\} \times \{0\}$.
It's instructive to sum a bunch of terms of this series and graph the resulting approximation.