Your question, as I understand it, is:
In deterministic models, does there exist a solution (to the equations of motion or the differential equations governing the system e.g. the heat flow equations), and is it unique?
And I would say that this is the very definition of deterministic.
In mathematics, computer science and physics, a deterministic system
is a system in which no randomness is involved in the development of
future states of the system. A deterministic model will thus always
produce the same output from a given starting condition or initial
If your question instead is, "under what conditions are the solutions of a mathematical model of a physical system unique?", then this is more interesting mathematically, but probably physcically irrelevant. For example, take the heat flow equations. As long as the initial starting conditions of temperature - as a function of space and time - $T(x,y,z,t),$ are smooth in a differentiable sense, then the time evolution of the system will surely be unique.
However, if the initial starting conditions of temperature, $T(x,y,z,t),$ are discontinuous, then strange stuff might happen. You could even take it to the extreme and make it everywhere discontinuous. Then the time evolution of the system is probably not deterministic because the rules of the game, i.e. what should happen are probably not well-defined (although you could come up with your own rules for what should happen). This corresponds to the situation where every atom has a temperature which is not close to any of it's surrounding atoms - and for this it is probably very difficult, if not computationally impossible to even approximate the time evolution of the system due to the chaotic nature of "everywhere discontinuous".
So basically the differentiable equations in physics require a nice smooth (or at least, continuous or differentiable?) function in order to determine the time evolution of the system. Once this continuity/differentiability breaks down, you can no longer use the differential equations to predict the time evolution: you need to look to other laws (like laws of particle physics) to see what's going on.
Conclusion: Some systems will not be deterministic when the starting conditions are discontinuous, but will be deterministic when the starting system is continuous; other systems will not be deterministic when the starting conditions are continuous, but will be deterministic when the starting system is differentiable; other systems will not be deterministic when the starting conditions are differentiable, but will be deterministic when the starting system is twice differentiable, and so on. Which one of these is the case for a given system depends on what kind of equations govern the physical system and you would have to look at each system on a case-by-case basis...