The Problem
Let $\Omega\subseteq\mathbb{R}^d$ be open and bounded with $C^2$-boundary. Let $T>0$ and $u\in C^{1,2}\left((0,T)\times\Omega\right)$ be a solution of the heat equation $$\partial_t u-\Delta u=0\qquad\text{in }(0,T)\times\Omega$$ subject to a Dirichlet boundary condition $$u=0\qquad\text{on }(0,T)\times\partial\Omega$$ and a Neumann boundary condition $$\frac{\partial u}{\partial n}=0\qquad\text{on }(0,T)\times\partial\Omega.$$ Intuitively, I would say that the trivial solution $u=0$ is the only one, but I'm having trouble to show this.
My Question: Are there non-trivial solutions to the equation described above?
Remarks and Observations
I was working on my master's thesis, when I stumbled across the above problem. There, I had an additional assumption that $u\leq 0$ in $(0,T)\times\Omega$. In this special case, there is indeed only the trivial solution:
The strong maximum principle implies that any non-trivial solution must satisfy $u<0$ everywhere in $(0,T)\times\Omega$ and hence we can make use of a boundary point lemma (e.g. Lieberman: Second Order Parabolic Differential Equations, Lemma 2.6) to find $\frac{\partial u}{\partial n}>0$ on $(0,T)\times\partial\Omega$.
Also, in the general case, but assuming $T=\infty$, mass conservation and standard energy estimates imply that $$\int_\Omega u\ \mathrm{d}x=0.$$ However, this certainly doesn't help for small $T$ and even if we knew this, I don't see where we could go from there.
Finally, in the case the answer to my question turns out to by yes, does this also hold for more general (linear) second-order parabolic pde?