Consider the following nonlinear BVP:
\begin{align} \frac{\partial u}{\partial t} - k \frac{\partial^2 u}{\partial x^2} + \lambda e^{\alpha u} &= f \\ u(0,t) &= T_0(t) \\ u(L,t) &= T_L(t) \\ u(x,0) &= u_0(x) \end{align}
in some region $(0,L)\times (0,T)$, where $k$, $\lambda$, $\alpha$, $L$ and $T$ are all positive. I am interested in proving that if solutions exist, they are unique. The nonlinear term makes the usual energy method difficult. The best I can come up with is if $u$ and $v$ are solutions to the original BVP, and if $w = u - v$, then $w$ satisfies:
$$\frac{\partial w}{\partial t} - k \frac{\partial^2 w}{\partial x^2} = \lambda (e^{\alpha v} - e^{\alpha u}).$$ But from there I cannot make any progress towards showing the usual argument that $\int w^2 \ dx$ must be decreasing or constant in time, and thus $w = 0$.
I have spent some time looking into the J.C. Evans textbook, as well as the probably overkill Handbook of Nonlinear Partial Differential Equations, without noticing anything useful at first glance. How can I make some progress here?