I am wondering under what conditions do we need boundary/initial conditions to claim the uniqueness of the solution?
For example, let us consider the Laplace equation in 1D: $$ u''(x) = 0. $$ Based on this, we know that the solution should be the form of $u(x) = ax+b$. Based on this, to claim the uniqueness, two boundary conditions are required (to determine $a$ and $b$). Everything is clear in this case.
Now let us consider a 1D heat equation: $$ u_t = u_{xx}. $$ By just counting the number of derivatives in the equation, it seems that we need three extra conditions - one for the time-derivative, and two for the spatial derivatives.
Based on only this relation, a standard approach (that I know of) is to construct a fundamental solution that naturally requires an initial condition. Let $\Gamma(x,t)$ be a fundamental solution with $\Gamma(x,0) = f(x)$ (Cauchy problem). Ok, it seems that this is understandable, however, it is unclear to me that why not a boundary condition rather than initial. For example, why we don't consider a fundamental solution that satisfies $u_t=u_{xx}$ and $u(a,t) = \phi(t)$ for some $a$ in the spatial domain?
It seems that the right problem we should consider (for the uniqueness) is a heat equation with three extra conditions: for example, boundary/initial problem - $$ u_t = u_{xx} \text{ on } (-1,1)\times (0,T] $$ with $$ u(x,0) = f(x) \text{ on } (-1,1)\times \{0\}, \quad u(\pm1,t) = \phi_{\pm}(t) \text{ on } (0,T]. $$ The above now have three conditions that corresponds to the number of derivatives shown in the equation. For me, this seems the right problem setup.
Now I am confused. We know that there is a fundamental solution $\Gamma(x,t)$ that satisfies the equation of $u_t = u_{xx}$ and the initial condition $u(x,0)=f(x)$. If the above boundary/initial heat equation is well-posed (i.e., the solution exists), the well-poseness suggests that there are infinitely many fundamental solutions and we're looking for a particular fundamental solution that satisfies the boundary condition. Am I right? If this is not the case, I am not sure what is the point of learning the fundamental solution, as it becomes useless in solving the general problem...
Unlike the 1D Laplace equation, even the 1D heat equation, it is unclear for me how many boundary/initial conditions (in general the number of extra equations) are needed to claim the uniqueness of the solution to the PDE (assuming the existence).
Any answers/suggestions/comments will be very appreciated.