Regularity properties of integral problems for wave equations and applications

Abstract

In this paper, the integral problem for linear and nonlinear wave equations are studied.The equation involves elliptic operator L and abstract operator A in Hilbert space H. Here, assuming enough smoothness on the initial data given in corresponding interpolation spaces and operators the existence, uniqueness, L^{p}-regularity properties to solutions are established. By choosing the space H and operators L, A, the regularity properties to solutions of different classes of wave equations in the field of physics are obtained.

Keywords

• Apstract differential equations
• Wave equations
• Operator theory
• $L^{p}$-regularity property of solutions
• Fourier multipliers

Kaynakça

1. [1] M. Arndt and M. Griebel, Derivation of higher order gradient continuum models from atomistic models for crystalline solids Multiscale Modeling Simul. (2005)4, 531-62.
2. [2] A.C. Eringen, Nonlocal Continuum Field Theories, New York, Springer (2002).
3. [3] Z. Huang, Formulations of nonlocal continuum mechanics based on a new de?nition of stress tensor Acta Mech. (2006)187, 11-27.
4. [4] C. Polizzotto, Nonlocal elasticity and related variational principles Int. J. Solids Struct. ( 2001) 38 7359-80.
5. [5] F. Linares, Global existence of small solutions for a generalized Boussinesq equation, J. Differential Equations 106 (1993), 257-293.
6. [6] Y. Liu, Instability and blow-up of solutions to a generalized Boussinesq equation, SIAM J. Math. Anal. 26 (1995), 1527- 1546.
7. [7] C.D. Sogge, Lectures on Nonlinear Wave Equations, International Press,Cambridge, MA, 1995.
8. [8] G.B. Whitham, Linear and Nonlinear Waves, Wiley-Interscience, New York, 1975.

9. [9] J.L. Bona, R.L. Sachs, Global existence of smooth solutions and stability of solitary waves for a generalized Boussinesq equation, Comm. Math. Phys. 118 (1988), 15-29.
10. [10] A. Constantin and L. Molinet, The initial value problem for a generalized Boussinesq equation, Diff.Integral Eqns. (2002)15, 1061-1072.
11. [11] G. Chen and S. Wang, Existence and nonexistence of global solutions for the generalized IMBq equation Nonlinear Anal. Theory Methods Appl. (1999)36, 961-980.
12. [12] G.-Q. Chen, X. Deng & W. Xiang, Shock Diffraction by Convex Cornered Wedges for the Nonlinear Wave System, Archive for Rational Mechanics and Analysis, 211 (2014), 61-112.
13. [13] N.J. Zabusky, Nonlinear Partial Differential Equations, Academic Press, New York, 1967.
14. [14] V.G. Makhankov, Dynamics of classical solutions (in non-integrable systems), Phys. Lett. C 35(1978), 1-128.
15. [15] M. Ghisi, M. Gobbino and A. Haraux, Optimal decay estimates for the general solution to a class of semil-linear dissipative hyperbolic eqiations, J. Eur. Math. Soc. (JEMS), 18 (2016),1961-1982.
16. [16] H.A. Levine, Instability and nonexistence of global solutions to nonlinear wave equations of the form Pu = −Au + F(u), Trans. Amer. Math. Soc 192 (1974) 1-21.
17. [17] A. Pazy, Semigroups of linear operators and applications to partial di?erential equations. Springer, Berlin, 1983.
18. [18] J-R. Luo and T-J. Xiao, Decay rates for second order evolution equations in Hilbert spaces with nonlinear time-dependent damping, 9(2)(2020) 359-373.
19. [19] C.A. Silling, Reformulation of elasticity theory for discontinuities and long-range forces J. Mech. Phys. Solids (2000)48 175-209.
20. [20] S. Wang, G. Chen, Small amplitude solutions of the generalized IMBq equation, J. Math. Anal. Appl. 274 (2002) 846-866.
21. [21] S. Wang and G. Chen, Cauchy problem of the generalized double dispersion equation Nonlinear Anal. Theory Methods Appl. (2006 )64 159-173.
22. [22] L.S. Pulkina, A non local problem with integral conditions for hyperbolic equations, Electron. J. Differ. Equ.(1999)45, 1-6.
23. [23] H.O. Fattorini, Second order linear differential equations in Banach spaces, in North Holland Mathematics Studies, V. 108, North-Holland, Amsterdam, 1985.
24. [24] W. Arendt, C. Batty, M. Hieber, F.Neubrander; Vector-valued Laplace Transforms and Cauchy Problems. Monographs in Mathematics. 96. Birkhauser, Basel, 2001.
25. [25] A. Ashyralyev, N. Aggez, Nonlocal boundary value hyperbolic problems involving integral conditions, Bound.Value Probl., 2014 V (2014):214.
26. [26] R. Agarwal, V.B. Shakhmurov, Integral type Cauchy problem for abstract wave equations and applications, Applicable Analysis, 2021, DOI: 10.1080/00036811.2021.2021184A.
27. [27] G. Dore, S. Yakubov, Semigroup estimates and non coercive boundary value problems, Semigroup Form, 60 (2000), 93-121.
28. [28] E. Karapinar, I. M. Erhan, A. Öztürk, Fixed point theorems on quasi-partial metric spaces, Mathematical and Computer Modelling 57 (9-10)(2013), 2442-2448.
29. [29] V.B. Shakhmurov, The Cauchy problem for generalized abstract Boussinesq equations, Dynamic systems and applications, 25, (2016),109-122.
30. [30] V.B. Shakhmurov, Separable convolution-elliptic operators with parameters, Form . Math. 2015, 27(6), 2637-2660.
31. [31] V.B. Shakhmurov, Separable di?erential operators and applications, Journal of Moscow Math. Soc., 16 (2)(2016), 299-321.
32. [32] V. Shakhmurov, R. Shahmurov, The quality and blow-up properties of integral type problems for wave equations and applications, Journal of Applied Mathematics and Physics, (10)4, (2022), 1217-1239.
33. [33] H. Triebel, Interpolation theory, Function spaces, Differential operators, North-Holland, Amsterdam, 1978.
34. [34] M. Girardi, L. Weis, Operator-valued Fourier multiplier theorems on Besov spaces, Math. Nachr., 251(2003), 34-51.
35. [35] T. Kato, G. Ponce, Commutator estimates and the Euler and Navier-Stokes equations, Comm. Pure Appl. Math. (1988)41, 891-907.