Does stability imply uniqueness for PDE/ODE

Question

If given some PDE, which has two solutions $u_1$ and $u_2$, where $u_1$ corresponds to initial condition $Φ_1$, $u_2$ corresponds to initial condition $Φ_2$, and $|u_1-u_2|$ depends on $|Φ_1-Φ_2|$ continuously, then, given an initial condition (for which the PDE has a solution), this PDE has a unique solution right?

Answer 1

If you have a stability estimate of the form $|u_1 - u_2| \le f(|\Phi_1 - \Phi_2|)$ with $f(0)=0$, then $\Phi_1 = \Phi_2$ implies $|u_1 - u_2| \le 0$ so $u_1 = u_2,$ i.e. the solution is unique.