Boundary value problems for third-order pseudoelliptic equations with degeneration
Abstract
We study the solvability of boundary value problems in cylindrical domains $Q =\Omega\times(0, T)$, $\Omega\subset R^n$, $0 < T < +\infty$, for differential equations
$$h(t)\frac{\partial^{2p+1}u}{\partial t^{2p+1}}+(-1)^{p+1}\Delta u + c(x,t)u=f(x,t),$$
where $p$ is a non-negative integer, $h(t)$ is continuous on the segment $[0, T]$ a function such that $\phi(t) > 0$ for $t\in (0, T)$, $\phi(0)\geq 0$, $\phi(T)\geq 0$, and $\Delta$ is the Laplace operator in spatial variables $x_1,\dots, x_n$. The main feature of the problems under study is that, despite the degeneration, the boundary manifolds are not exempt to the bearing boundary conditions. We proved the existence and uniqueness theorems of the regular solutions, those having all Sobolev generalized derivatives included in the equation. Moreover, we describe some possible enhancements and generalizations of the obtained results.
References
[1] Mikhailov V. P., “About the first boundary value problem for one class of hypoelliptic equations [in Russian],” Mat. Sb., 63, No. 2, 229–264 (1964).
[2] Dubinskii Yu. A., “About one abstract theorem and its applications to boundary value problems for nonclassical equations [in Russian],” Mat. Sb., 79, No. 1, 91–117 (1969).
[3] Dubinskii Yu. A., “About some differential-operator equations of arbitrary order [in Russian],” Mat. Sb., 90, No. 1, 3–22 (1973).
[4] Romanko V. K., “Boundary value problems for one class of differential equations [in Russian],” Differents. Uravn., 10, No. 1, 117–131 (1974).
[5] Romanko V. K., “Unique solvability of boundary value problems for some operator-differential equations [in Russian],” Differents. Uravn., 13, No. 2, 324–335 (1977).
[6] Fichera G., “On a unified theory of boundary-value problems for elliptic-parabolic equations of second order,” Boundary Problems in Differential Equations, Proc. Symp., pp. 97–120, Univ. Wisconsin Press, Madison, WI (1960).
[7] Oleinik O. A. and Radkevich E. V., Equations with Nonnegative Characteristic Form [in Russian], Izdat. MGU, Moscow (2010).
[8] Egorov I. E., “First boundary problem for a nonclassical equation [in Russian],” Mat. Zametki, 42, No. 3, 403–411 (1987).
[9] Egorov I. E. and Fedorov V. E., Non-classical Higher-Order Equations of Mathematical Physics [in Russian], Izdat. VTs SO RAN, Novosibirsk (1995).
[10] Egorov I. E., Fedorov V. E., Tikhonova I. M., and Efimova E. S., “The Galerkin Method for Nonclasical Equations of Mathematical Physics,” in: AIP Conf. Proc., 1907, 020011 (2017).
[11] Kozhanov A. I. and Matsievskaya E. E., “Degenerate Parabolic Equations with a Variable Direction of Evolution,” Sib. Electron. Math. Rep., 718–731 (2019).
[12] Sobolev S. L., Some Applications of Functional Analysis in Mathematical Physics, Amer. Math. Soc., Providence, RI (1991).
[13] Ladyzhenskaya O. A., Solonnikov V. A., and Uraltseva N. N., Linear and Quasilinear Parabolic Equations [in Russian], Nauka, Moscow (1967).
[14] Ladyzhenskaya O. A. and Uraltseva N. N., Linear and Quasilinear Elliptic Equations, Acad. Press, New York; London (1968).
[15] Triebel H., Interpolation Theory, Function Spaces, Differential Operators, VEB Deutcher Verl. Wiss., Berlin (1978).
This work is licensed under a Creative Commons Attribution 4.0 International License.