System of inhomogeneous integral equations of convolution type with power nonlinearity

We consider a system of inhomogeneous integral equations of convolution type with power nonlinearity arising in the description of the processes of fluid infiltration from a cylindrical reservoir into an isotropic homogeneous porous medium, propagation of shock waves in pipes filled with gas, cooling of bodies under radiation, following the Stefan-Boltzmann law, etc. Keeping in mind the indicated and other applications, nonnegative solutions of this system, continuous on the positive semiaxis, are of interest. Two-sided a priori estimates for the solution of the system are obtained, on the basis of which a complete metric space is constructed, and the unique solvability of this system in this space is proved by the method of weight metrics (an analogue of A. Belitsky's method). It is shown that the solution can be found by successive approximations of the Picard type; an estimate of the rate of their convergence is obtained. It is established that this solution is unique in the entire class of continuous positive functions. In the case of the corresponding homogeneous systems of integral equations of convolution type with power nonlinearity, the question of the existence of nontrivial solutions is also studied.