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The existence of a solution to the problem modeling a steady state heat distribution in an inhomogeneous plane with a crack is proved. The singular terms of an asymptotic expansion of the heat flux near the ends of the crack are obtained in closed form.Definition. The solution to problem (1)-(3) is a function in that satisfies Eq. (1) in the domain , obeys boundary conditions (2) and (3) in the sense of principal values for from the interval (-1; 1), and is such that the functions , , and are bounded near the crack .The interval models a crack. Conditions (2) and (3) describe the jumps in the temper ature and the normal heat flux, respectively, across the crack. The necessity of setting these boundary con ditions for the given problem was discussed in [6].

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