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On the blow-up of a solution of a non-local system of equations of hydrodynamic type

We study Oskolkov's non-linear non-local system of equations with a source and the applicability of an operator method. We prove the existence of a weak solution of the initial-boundary value problem for this system and investigate uniqueness conditions for a weak solution. Sufficient conditions for the blow-up of a solution of the problem are found and upper and lower bounds are obtained for the time of blowing up of a solution. We also consider the smoothness problem for a weak solution.