Mathematics & Physics 2016, 9(3), 364–373 УДК 517.9 On the Solvability of a System of Two Multidimensional Loaded Parabolic Equations with the Cauchy Data Galina V.Romanenko∗ Igor V.Frolenkov† Institute of Mathematics and Computer Science Siberian Federal University Svobodny, 79, Krasnoyarsk, 660041 Russia Received 17.01.2016, received in revised form 03.04.2016, accepted 15.06.2016 We study a multidimensional system of two loaded parabolic equations of a special kind with the Cauchy data. <...> Sufficient conditions for the existence of a solution in the class of smooth bounded functions are obtained. <...> The splitting method at differential level (the method of weak approximation) is used in the proof. <...> Keywords: inverse problem, direct problem, loaded equation, parabolic equation, weak approximation method, system of partial differential equations, Cauchy problem. <...> Introduction A study of inverse problems for systems of parabolic equations can be a time-consuming process, therefore, in [1] there was proposed an algorithm to the study of one-dimensional direct problems for systems of parabolic equations, to which inverse problems for loaded systems of a special kind can be reduced. <...> The article presents a generalization of the algorithm proposed in [1] to the multidimensional case. <...> In the present article we propose and investigate the following model: a system of two multidimensional loaded parabolic equations connected by the lower terms with the Cauchy data. <...> The obtained result can be used as a sufficient condition for existence of a solution to auxiliary direct problems. <...> To prove existence of a solution we use the weak approximation method, which is the splitting method at the differential level [2, 3]. <...> The Cauchy problem for a loaded Burgerstype system has been investigated in [7]. ∗galina.romanencko@yandex.ru †igor@frolenkov.ru ⃝ Siberian Federal University. <...> All rights reserved c – 364 – Galina V.Romanenko, Igor V.Frolenkov On the solvability of the system of two multidimensional . . . 1. <...> Statement of the problem In the space En of variables x1, . . . ,xn choose ri different points αi variable xi (i = 1,n). <...> If t∗ depends on the constants bounding the initial data and t∗ T then we say that the functions u(t,x), v(t,x) are a solution (1), (2) ‘in the small’. <...> The main result Suppose that the following conditions hold: Condition 1 <...>