# 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 $$\alpha \in (1,3]$$.