# Boundary value problems for third-order pseudoelliptic equations with degeneration

### Abstract

We study the solvability in Sobolev spaces of the Dirichlet problem and other elliptic problems for the differential equations

\begin{equation}

u_{tt}+\alpha(t)\frac{\partial}{\partial t}(\Delta u)+Bu=f(x,t)\tag{*}

\end{equation}

$x\in\Omega\subset\mathbb{R}^n,\,t\in(0,T),$ where $\Delta$ if the Laplace operator acting in the variables $x_1,\dots, x_n$ and $B$ is a second-order elliptic operator acting in the same variables $x_1,\dots, x_n$. A feature of the equations (∗) is that the sign of the function is not fixed in them. Existence and uniqueness theorems for regular solutions (having all generalized Sobolev’s derivatives in the equation) are proved for the problems under study.

### References

[1]

Larkin N. A., “Existence theorems for quasilinear pseudohyperbolic equations,” Sov. Math. Dokl., 26, 260–263 (1982).

[2]

Liu Y. and Li H., “H1-Galerkin mixed finite element methods for pseudo-hyperbolic equations,” Appl. Math. Comput., 212, No. 2, 446–457 (2009).

[3]

Khudaverdiev K. and Veliev A., A Study of the One-Dimensional Mixed Problem for a Class of Pseudohyperbolic Third-Order Equations with Nonlinear Right-Hand Side [in Russian], ¸Çaşioĝlu, Baku (2010).

[4]

Mesloub S., “On an initial boundary value problem for a class of odd higher order pseudohyperbolic integrodifferential equations,” J. Appl. Math., article ID 464205 (2014).

[5]

Zhao Z. and Li H., “A continuous Galerkin method for pseudo-hyperbolic equations with variable coefficients,” J. Math. Anal. Appl., 473, No. 2, 1053–1072 (2019).

[6]

Dublya Z. D., “Dirichlet’s problem for a class of third-order equations [in Russian],” Differ. Uravn., 13, No. 1, 50–55 (1977).

[7]

Kozhanov A. I., “Boundary value problems and properties of solutions for equations of the third order,” Differ. Equations, 25, No. 12, 2143–2153 (1989).

[8]

Kozhanov A. I., “Boundary value problems for some classes of higher-order equations that are unsolved with respect to the highest derivative,” Sib. Math. J., 35, No. 2, 324–340 (1994).

[9]

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

[10]

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

[11]

Sobolev S. L., Some Applications of Functional Analysis in Mathematical Physics, Amer. Math. Soc., Providence, RI (1991).

[12]

Ladyzhenskaya O. A. and Uraltseva N. N., Linear and Quasilinear Elliptic Equations, Acad. Press, New York; London (1968).

[13]

Triebel H. Interpolation Theory, Function Spaces, Differential Operators, VEB Deutcher Verl. Wissenschaften, Berlin (1978).

[14]

Trenogin V. A. Functional Analysis [in Russian], Nauka, Moscow (1980).

[15]

Ladyzhenskaya O. A., “On the integral convergence estimates for approximate solutions in functionals to linear elliptic operators [in Russian],” Vestn. LGU, Ser. Mat., Mekh., Astron., No. 2, 60–69 (1958).

[16]

Kozhanov A. I., Larkin N. A., and Yanenko N. N., A Mixed Problem for Certain Classes of Third Order Equation, preprint No. 5, Inst. Teor. Prikl. Mat. SO AN SSSR, Novosibirsk (1980).

*Mathematical notes of NEFU*, 27(3), pp. 16-26. doi: https://doi.org/10.25587/SVFU.2020.63.12.002.

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