A priori estimates for derivative solutions of one-dimensional inhomogeneous wave equations with an integral load in the main part

O. L. Boziev

Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika, no. 89 (2024), pp. 5-16

Résumé

In this paper, the second initial-boundary value problem with homogeneous boundary conditions for a one-dimensional modified wave equation is considered. The modification consists in replacing the coefficient at the second spatial derivative with an integral load. In our case, it is a power function of the integral of the squared modulus of the derivative of the equation solution with respect to the spatial variable. Equations with such a load are associated with some practically important hyperbolic equations with a power nonlinearity in the main part. This makes it possible to use previously found solutions of loaded problems to start the process of successive approximation to solutions of nonlinear problems reduced to them. In this case, with respect to the original nonlinear equation, the loaded equation contains a weakened nonlinearity. Linearization of the loaded equation makes it possible to find its approximate solution. The article considers three cases of the integral load. It is the squared norm of the derivative of the solution with respect to $x$ in the space $L_2$ in natural, inverse to natural, and two integer negative powers. The corresponding a priori inequalities are established. Their right-hand side is used to pass to linearized equations. Examples of linearization by this method of wave equations with an integral load in the main part are given.

Détail

Keywords: hyperbolic equation, integral load, linearization.
Mots-clés : a priori estimation

Bibliographie

[1] Bernshtein S.N., “Ob odnom klasse funktsionalnykh uravnenii s chastnymi proizvodnymi”, Izvestiya AN SSSR. Ser. matematicheskaya, 4:1 (1940), 17–26

[2] Woinowsky-Krieger S., “The effect of axial forces on the vibrations of hinged bars”, J. Appl. Mech., 17 (1950), 35–36

[3] Dickey R.W., “Infinite systems of nonlinear oscillation equations related to the string”, Proc. Amer. Math. Soc., 23 (1969), 459–468

[4] Crippa H.R., “On local solutions of some mildly degenerate Hyperbolic equations”, Nonlinear Analysis: Theory, Methods Applications, 21:8 (1993), 565–574

[5] Frota C.L., Medeiros L.A., Vicente A., “Wave equation in domains with nonlocally reacting boundary”, Differential and Integral Equations, 17 (2011), 1001–1020

[6] Pokhozhaev S.I., “Ob odnom klasse kvazilineinykh giperbolicheskikh uravnenii”, Mate maticheskii sbornik, 96(138):1 (1975), 152–166

[7] Pokhozhaev S.I., “Ob odnom kvazilineinom giperbolicheskom uravnenii Kirkhgofa”, Differentsialnye uravneniya, 21:1 (1985), 101–108

[8] Nishihara K., “Exponential decay of solutions of some quasilinear hyperbolic equations with linear damping”, Nonlinear Analysis: Theory, Methods Applications, 8:6 (1984), 623–636

[9] Ngoc L.T.P., Long N.T., “Linear Approximation and Asymptotic Expansion of Solutions in Many Small Parameters for a Nonlinear Kirchhoff Wave Equation with Mixed Nonhomogeneous Conditions”, Acta Appl Math., 112 (2010), 137–169

[10] Ono K., “Global solvability for mildly degenerate Kirchhoff type dissipative wave equations in Bounded Domains”, J. Math. Tokushima Univ., 55 (2021), 11–18

[11] Boziev O.L., “Priblizhennoe reshenie nagruzhennogo giperbolicheskogo uravneniya s odnorodnymi kraevymi usloviyami”, Vestnik Yuzhnouralskogo gosudarstvennogo universiteta. Ser. Matematika, fizika, mekhanika, 8:2 (2016), 14–18

[12] Boziev O.L., “Reshenie nelineinogo giperbolicheskogo uravneniya priblizhenno-analiticheskim metodom”, Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mekhanika, 2018, no. 51, 5–14

[13] Boziev O.L., “O linearizatsii giperbolicheskikh uravnenii s integralnoi nagruzkoi v glavnoi chasti s pomoschyu apriornoi otsenki ikh reshenii”, Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mekhanika, 2022, no. 80, 16–25

[14] Filatov A.N., Sharova L.V., Integralnye neravenstva i teoriya nelineinykh kolebanii, Nauka, M., 1976, 151 pp.

[15] Boziev O.L., “Apriornye otsenki proizvodnykh reshenii odnomernykh neodnorodnykh uravnenii teploprovodnosti s integralnoi nagruzkoi v glavnoi chasti”, Vestnik Yuzhno-Uralskogo gosudarstvennogo universiteta. Ser. Matematika, fizika, mekhanika, 15:2 (2023), 5–13