On the Cauchy problem for scalar conservation laws on the Bohr compactification of ${\mathbb{R}}^n$

We study the Cauchy problem for a multidimensional scalar conservation law on the Bohr compactification of ${\mathbb{R}}^n$. The existence and uniqueness of entropy solutions are established in the general case of merely continuous flux vector. We propose also the necessary and sufficient condition for the decay of entropy solutions as time $t\to +\mathrm{\infty }$.