Consider the $n \times n$ system $$u_t + A(u)u_x = F(u).$$
If the eigenvalues of $A$ are all real and distinct the system is called strictly hyperbolic.
What is the relationship between this definition and the existence of a strictly convex entropy? Are they equivalent?
They are not. First of all, the existence of a convex entropy is not meaningful for a system given in this quasi-linear form. The reason is that you might make a change $v=\phi(u)$ of unknown, but the convexity is not preserved by composition by the diffeomorphism $\phi$.
In addition, if $n\ge3$, a generic quasi-linear system does not admit conservation laws $\partial_t\eta(u)+\partial_q(u)=g(u)$, because the compatibility condition $\nabla\eta A=\nabla q$ is over-determined.
Now, if you give yourself a system of balance laws $u_t+f(u)_x=F(u)$, whose principal part is in conservation form, then the notion of convex entropy becomes meaningful, because you authorize only linear change of variables. Once again, a generic system with $n\ge3$ does not admit an entropy. So the system can be hyperbolic without having this convex entropy.
Of course, systems coming from thermodynamics are not generic. They were characterized by Godunov as those for which there are two functions $E(w),M(w)$ with $E$ strictly convex, such that $u=\nabla E(w)$ and $f(u)=\nabla M(w)$. Then the system is symmetric hyperbolic, in the sense that $$\nabla^2Ew_t+\nabla^2Mw_x=F(u)$$ where the matrices are symmetric (Hessians), and the first one is positive definite. Such systems do have a convex entropy, namely the Legendre transform $E^*(u)$. Conversely, a system of balance laws equipped with a strictly convex entropy can be written that way.
No, entropy convexity and hyperbolicity are not equivalent conditions. A necessary and sufficient condition for the system of differential equations to possess a strictly convex entropy is that the system is symmetrizable and hence hyperbolic. The symmetrizability condition is stronger than the condition of hyperbolicity, a system may have real eigenvalues and be therefore hyperbolic without being symmetrizable, and therefore without having a strictly convex entropy.
