Nonlocal solvability od the Cauchy problem for the dissipative equation of the dislocation density with a quadratic non-linearity

A non-linear first-order partial differential equation describing a dislocation density changing under a coercion of the diffusion creep is considered. The global estimates of the solution and its derivatives up to third order with respect to spatial variables are derived and the conditions of the non-local solvability of the Cauchy problem are determined with using the method of an additional argument.