Given the initial boundary value problem \begin{align*} &u_t +uu_x = Du_{xx}, \quad 0<x<1, D>0 \\ &u(0,t) = u(1,t) = 0, \\ &u(x,0) = u_{0}(x), \quad 0<x<1 \end{align*} Assume there exists two solutions $u$ and $v$ to the the IBVP and define $w = u-v$. By defining 'Energy' $$E(t) = \frac{1}{2} \int_{0}^{1} w(x,t)^2 \ dx$$ show uniqueness of solutions to the IBVP.
Attempt: I started by showing $w$ has to satisfy $$w_t +\left( \frac{aw}{2} \right) _x = Dw_{xx}$$ with $a= u+v$ and initial condition $w(x,0) = 0$ and $w(0,t) = w(1,t) = 0$. Then, $$\frac{dE}{dt} = \int_{0}^{1} ww_t \ dx = \int_{0}^{1} Dww_{xx} \ dx - \frac{1}{2} \int_{0}^{1} (aw)_{x}w \ dx$$ Then by Integration by Parts got to: $$\frac{dE}{dt} = -D \int_{0}^{1} (w_{x})^2 \ dx - \frac{1}{4} \int_{0}^{1} a_{x}w^2 \ dx \leq - \frac{1}{4} \int_{0}^{1} a_{x}w^2 \ dx $$
I am trying to bound the last integral above by $-cE(t)$ and them employ Gronwall's Lemma to get $E(t) \leq \underbrace{E(0)}_{=0}e^{-ct} \Rightarrow E(t) = 0 \Rightarrow w=0$