3. Let $Omega$ be a bounded open region with smooth boundary, and let $U_T = {2 imes (0,T]}$ and $T^c = U_T ackslash U_T$ for $T > 0$. For $u = u(x, t) = C^2(U_T)$, consider the problem $egin{cases} -k Delta u + u = f ext{ in } U_T, \ u = 0 ext{ on } x imes {t=0}, \ frac{partial u}{partial mathbf{n}} + au = g ext{ on } partial Omega imes [0, T], end{cases}$ where $f(x, t) in C(overline{U_T})$, $p(x) = C(overline{Omega})$ and $g(x, t) in C(partial Omega imes [0,T])$ are given functions and where $k$ and $a$ are positive constants. Prove or disprove (give a counterexample) that this problem has a unique solution.