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I am still working on this,however, I don't know how to use energy method to prove the uniqueness.Any hint or suggestion from you would be appreciated.

Let U $\subset R^{n}$ be open, bounded, with smooth boundary $\partial U$, and $ T \gt 0$. Use energy methods to prove that the initial boundary-value problem

$u_{t}-\Delta u =f$ in $U_{T}$

$\frac{\partial u}{\partial \nu}=0$ on $\partial U \times [0,T]$

u=g on $U \times \{t=0\}$

where f=f(x,t),g=g(x) are given smooth functions, has at most one solution $u \in C^{2}_{1}$ $(\overline U_{T})$

Yang
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The basis of the energy method is that the energy integral $E(t)=\int_U u(x,t)^2\,dx$ does not increase in $t$ for solutions of the homogeneous PDE $u_t-\Delta u=0$. (Proof: $E'(t)=-2\int_U |\nabla_x u|^2\,dx\le 0$.) Apply this to the difference of two solutions of your initial value problem. Since $E(0)=0$, it follows that $E(t)=0$ for all $t$, hence the difference of solutions is identically zero for all times.

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