# The Dirichlet problem for the higher order composite type equations with discontinuous coefficients

### Abstract

We study the Dirichlet problem for the composite type differential equations

$$D_t\big[(-1)^pD^{2p+1}_tu-h(x)u_{xx}\big]+a(x)u_{xx}+c(x,t)u=f(x,t)$$

in the domain $Q=\{(x,t)\,:\,x\in(-1,0)\cup(0,1),\,t\in(0,T),\,0<T<+\infty\}$, where

$p \geq 1$ is an integer, $D^k_t=\frac{\partial^k}{\partial t^k},$ and $D_t=\frac{\partial}{\partial t}$. The feature of such equations is that the coefficients $h(x)$ and $a(x)$ can have a discontinuity of the first kind when passing through the point $x = 0$. In addition to the usual Dirichlet boundary conditions, the problem under study also specifies the conjugation conditions on the line $x = 0$. Existence and uniqueness theorems are proved for regular solutions (those having all generalized Sobolev derivatives).

### References

[1]

Oleinik O. A., “Boundary value problems for linear equations of elliptic and parabolic type with discontinuous coefficients,” Izv. AN USSR. Ser. Math, 25, 3–20 (1961).

[2]

Il’in V. A., “On the solvability of the Dirichlet and Neumann problems for a linear elliptic operator with discontinuous coefficients,” Sov. Math., Dokl., 2, 228–231 (1961).

[3]

Il’in V. A., “The Fourier method for a hyperbolic equation with discontinuous coefficients,” Sov. Math., Dokl., 3, 12–16 (1962).

[4]

Ladyzhenskaya O. A., Solonnikov V. A., and Uraltseva N. N., Linear and Quasilinear Parabolic Equations [in Russian], Nauka, Moscow (1967).

[5]

Ilyin V. A. and Shishmarev I. A., “Potential method for the Dirichlet and Neumann problems in the case of equations with discontinuous coefficients,” Sib. Math. J., 2, No. 1, 46–58 (1961).

[6]

Bitsadze A. V., Equations of Mixed Type [in Russian], Izdat. Akad. Nauk SSSR, Moscow (1959).

[7]

Ladyzhenskaya O. A. and Stupyalis L., “About mixed type equations [in Russian],” Vestn. Leningrad. Gos. Univ., No. 18, 38–46 (1967).

[8]

Smirnov M. M., Mixed Type Equations [in Russian], Nauka, Moscow (1970).

[9]

Ladyzhenskaya O. A. and Stupyalis L., “Boundary value problems for mixed type equations,” Tr. MIAN USSR, 116, 101–136 (1971).

[10]

Stupyalis L., “Boundary value problems for for elliptic-hyperbolic equations,” Tr. MIAN USSR, 125, 211–229 (1973).

[11]

Juraev T. D., Boundary Value Problems for Equations of Mixed and Mixed-Composite Types [in Russian], Fan, Tashkent (1986).

[12]

Moiseev E. I., Mixed Type Equations with a Spectral Parameter [in Russian], Izdat. Moskov. Univ., Moscow (1988).

[13]

Rogozhnikov A. M., “Study of a mixed problem describing the oscillations of a rod consisting of several segments with arbitrary lengths,” Dokl. Math., 85, No. 3, 399–402 (2012).

[14]

Kuleshov A. A., “Mixed problems for the equation of longitudinal vibrations of a heterogeneous rod and for the equation of transverse vibrations of a heterogeneous string consisting of two segments with different densities and elasticities,” Dokl. Math., 85, No. 1, 98–101 (2012).

[15]

Rogozhnikov A. M., “Study of the mixed problem, describing the process of shaft vibrations, consisting of several sections with arbitrary lengths,” Dokl. Math., 444, No. 5, 488–491 (2012).

[16]

Smirnov I. N., “Oscillations described by the telegraph equation for a system consisting of several sections of different density and elasticity [in Russian],” Differ. Equ., 49, No. 5, 643–648 (2012).

[17]

Potapova S. V., “Boundary value problems for pseudoparabolic equations with a variable time direction. TWMS,” J. Inequalities Pure Appl. Math., 3, No. 1, 73 (2012).

[18]

Kozhanov A. I. and Potapova S. V., “The Dirichlet problem for a class of composite equations type with a discontinuous coefficient of the highest derivative [in Russian],” Dalnevost. Math. J., 14, No. 1, 48–65 (2014).

[19]

Kozhanov A. I. and Sharin E. F., “A conjugate problem for some higher order nonclassical equations, II [in Russian],” Mat. Zamet. SVFU, 21, No. 1, 18–28 (2014).

[20]

Kozhanov A. I. and Potapova S. V., “Conjugate problem for a third order equation with multiple characteristics and a positive function at the higher order derivative,” J. Math. Sci., New York, 215, No. 4, 510–516 (2016).

[21]

Kozhanov A. I. and Potapova S. V., “Transmission problem for odd-order differential equations with two time variables and a varying direction of evolution,” Dokl. Math., 95, No. 3, 267–269 (2017).

[22]

Kozhanov A. I., “A conjugation problem for a class of composite-type equations of variable direction [in Russian],” in: Nonclassical Equations of Mathematical Physics, pp. 96–109, Izdat. Sobolev Inst. Mat., Novosibirsk (2002).

[23]

Shubin V. V., “Boundary problems for third-order equations with discontinuous coefficients [in Russian],” Vestn. Novosib. Gos. Univ., 12, No. 1, 126–138 (2012).

[24]

Kozhanov A. I. and Sharin E. F., “A conjugation problem for some nonclassical differential equations of higher order [in Russian],” Ukr. Mat. Vestn., 11, No. 2, 181–202 (2014).

[25]

Antipin V. I., “The solvability of the boundary value problem for a third-order equation with changing time direction [in Russian],” Mat. Zametki YaGU, 18, No. 1, 8–15 (2011).

[26]

Pyatkov S. G., Popov S. V., and Antipin V. I., “On solvability of boundary value problem for kinetic operator-differential equations,” Integral Equ. Oper. Theory, 80, No. 4, 557–580 (2014).

[27]

Demidenko G. V., Equations and Systems Which Are Not Solved With Respect To Higher Derivative [in Russian], Nauch. Kniga, Novosibirsk (1998).

[28]

Kozhanov A. I., Composite Type Equations and Inverse Problems, VSP, Utrecht (1999).

[29]

Sviridyuk G. A., Linear Sobolev Type Equations and Degenerate Semigroup of Operators, VSP, Utrecht (2003).

[30]

Hayashi N., Kaikina E. I., Naumkin P. I., and Shishmarev I. A., Asymptotic for Dissipative Nonlinear Equations, Springer (2006).

[31]

Sveshnikov A. G., Al’shin A. B., Korpusov M. O., and Pletner Yu. D., Linear and Nonlinear Equations of Sobolev Type [in Russian], Fizmatlit, Moscow (2007).

[32]

Korpusov M. O., Blow-up in Non-Classical Non-Local Equations [in Russian], Librokom, Moscow (2011).

[33]

Kozhanov A. I., “Pseudo-hyperbolic and hyperbolic equations with growing lower terms [in Russian],” Vestn. Chelyab. Gos. Univ., 3, No. 5, 31–47 (1999).

[34]

Potapova S. V., “Boundary value problems for pseudohyperbolic equations with a varying time direction [in Russian],” Mat. Zametki YaGU, 18, No. 1, 108–124 (2011).

[35]

Kozhanov A. I. and Lukina G. A., “Pseudoparabolic and pseudohyperbolic equations in non-cylindrical time domains [in Russian],” Mat. Zametki SVFU, 26, No. 3, 31–47 (2019).

[36]

Trenogin V. A., Functional Analysis [in Russian], Fizmatlit, Moscow (2007).

*Mathematical notes of NEFU*, 28(4), pp. 17-29. doi: https://doi.org/10.25587/SVFU.2021.56.53.002.

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