# Uniqueness of the solution for a semilinear parabolic PDE

Let $$T>0$$. Consider the following semilinear parabolic equation $$\partial_tu(x,t)-\Delta u(x,t)=f(u(x,t)) \quad \forall (x,t) \in (0,1) \times (0,T),$$ with homogeneous Neumann boundary conditions and given initial conditions in $$L^2(0,1)$$.

Assume that $$f:\mathbb{R}\longrightarrow\mathbb{R}$$ is continuously differentiable. Prove that the solution to the stated PDE is unique.

Attempt:

Let $$u_1,u_2$$ be two solutions. Then, using Green formula, we obtain $$\begin{split} \dfrac{1}{2} \dfrac{d}{dt}\int_{0}^1 \vert u_1(x,t)-u_2(x,t)\vert^2 dx&=\int_{0}^1 f(u_1(x,t))-f(u_2(x,t)) (u_1(x,t)-u_2(x,t))dx\\ &\leq C \int_{0}^1 \vert u_1(x,t)-u_2(x,t)\vert^2 dx \end{split}$$ where we have used the mean value theorem and such that $$C:=\underset{s \in [u_1(x,t),u_2(x,t)]}{\sup} f'(s)$$
which exists since $$f$$ is continuously differentiable.

By applying Gronwall inequality, we obtain $$u_1=u_2.$$

Is this proof correct? I can't spot the mistake.

$$\dfrac{1}{2} \dfrac{d}{dt} \int_0^1 \vert u_1(x,t)-u_2(x,t)\vert^2 dx\leq \int_{0}^1C(x,t) \vert u_1(x,t)-u_2(x,t)\vert^2dx$$
The mistake is assuming that the constant $$C$$ is uniform.