Existence and uniqueness of entropy solutions for a scalar conservation law

Consider the conservation law

$ $ (\ast) \qquad u_t + \partial_x(u^\alpha) = 0$ $ where $ \alpha > 0$ .

For what values of $ \alpha$ is it known that there exists a (unique) entropy solution for the initial value problem associated to $ (\ast)$ ?

In particular, I’m interested in knowing what happens in the case $ \alpha \in (1,3]$ .