Analogue of Tricomi problem for characteristically loaded hyperbolic-parabolic equation with variable coefficients

Abstract

In the work we study an analogue of Tricomi problems for characteristically loaded hyperbolic-parabolic equations with variable coefficients. We prove the unique solvability of the studied problem. The uniqueness of the solutions is proved by means of the maximum principle, while the existence is established by the method of integral equations.

Cauchy problem for a high‐order loaded integro‐differential equation

“…Initial and boundary value problems for loaded second‐ and third‐order equations of hyperbolic, parabolic, elliptic, and mixed types were investigated in the works of previous research 20–34 . We would like to note that problems for the loaded equations of high‐order differential equations have not been studied yet.…”

“…Initial and boundary value problems for loaded second-and third-order equations of hyperbolic, parabolic, elliptic, and mixed types were investigated in the works of previous research. [20][21][22][23][24][25][26][27][28][29][30][31][32][33][34] We would like to note that problems for the loaded equations of high-order differential equations have not been studied yet. The construction of a theory of unique solvability in the study of the correctness of problem formulation for the generalized multidimensional differential equation with loaded term is required both for the internal completeness of the theory of loaded differential equation and for numerous applications.…”

On an analogue of the Tricomi problem for a «pointwise» loaded hyperbolic-parabolic equation

“…Аналог задачи Трикоми для характеристически нагруженного уравнения (1) и для модельного уравнения смешанного гиперболо-параболического типа с производной при нагрузке исследован в [14] и [15]. В [16] исследован аналог задачи Трикоми для характеристически нагруженного уравнения гиперболо-параболического типа общего вида с переменными коэффициентами.…”

Boundary-Value Problem for a Loaded Mixed-Type Equation with a Characteristic Line of Type Change

“…Thus, if d i .x; y/; e i .x; y/ D 0; then Lu is a parabolic-hyperbolic operator, and we arrive at the equation considered in [2,3]. For a D 0 and b D 0; we get the equation investigated in [12]. If a D 0 and b D 0 or b D 0 and c D 0 and, in addition, ˛i D 0; then we get the equations studied in [13,14].…”

“…the operation of summation is carried out over the repeated index i D 1; 2; : : : ; n. It follows from relation (30) for (11) and (12), the properties of the integrodifferential equations, and [3], the functions K 2 .t; ⇠/ and g 2 .⇠/ are continuously differentiable for 0  t; ⇠  1: Moreover, for ⇠ D 0; Eq. (31) has a unique solution in the class of continuously differentiable functions.…”