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Applicable Analysis
An International Journal
Volume 102, 2023 - Issue 7
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Research Article

Integral type Cauchy problem for abstract wave equations and applications

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Pages 2097-2122 | Received 22 Jan 2021, Accepted 08 Dec 2021, Published online: 30 Dec 2021

References

  • Makhankov VG. Dynamics of classical solutions (in non-integrable systems). Phys Lett C. 1978;35:1–128.
  • Whitham GB. Linear and nonlinear waves. New York: Wiley–Interscience; 1975.
  • Wang S, Chen G. The Cauchy problem for the generalized IMBq equation in Ws,p(Rn). J Math Anal Appl. 2002;266:38–54.
  • Zabusky NJ. Nonlinear partial differential equations. New York (NY): Academic Press; 1967.
  • Arendt W, Batty C, Hieber M, et al. Vector-valued Laplace transforms and Cauchy problems. Basel: Birkhauser; 2001. (Monographs in Mathematics; 96).
  • Bahouri H, Gérard P. High frequency approximation of solutions to critical nonlinear wave equations. Am J Math. 1999;121:131–175.
  • Da Prato G, Giusti E. A characterization on generators of abstract cosine functions. Boll Del Unione Mat. 1967;22:367–362.
  • Ginibre J, Velo G. Generalized Strichartz inequalities for the wave equation. J Funct Anal. 1995;133:50–68.
  • Grillakis M. Regularity for the wave equation with a critical nonlinearity. Commun Pure Appl Math. 1992;45:749–774.
  • Kapitanski L. Weak and yet weaker solutions of semilinear wave equations. Commun Partial Differ Equ. 1994;19:1629–1676.
  • Klainerman S. Global existence for nonlinear wave equations. Commun Pure Appl Math. 1980;33:43–101.
  • Klainerman S, Machedon M. On the regularity properties of a model problem related to wave maps. Duke Math J. 1997;87:553–589.
  • Keel M, Tao T. Local and global well-posedness of wave maps on R1+1 for rough data. IMRN. 1998;1998:1117–1156.
  • Lindblad H, Sogge CD. Restriction theorems and semilinear Klein-Gordon equations in (1+3) dimensions. Duke Math J. 1996;85(1):227–252.
  • Shakhmurov VB. Nonlocal fractional differential equations and applications. Complex Anal Oper Theory. 2020;14:5.
  • Shakhmurov VB. Hardy type unique continuation properties for abstract Schrödinger equations and applications. Electron J Qual Theory Differ Equ. 2019;97:1–27.
  • Shakhmurov VB, Musayev H. Regularity properties of degenerate convolution-elliptic equations. Bound Value Probl. 2016;2016:5.
  • Ragusa A, Shakhmurov VB. The Navier-Stokes type problem with high order elliptic operator and applications. J Math. 2020;8(12):2256.
  • Triebel H. Interpolation theory, function spaces, differential operators. Amsterdam: North-Holland; 1978.
  • Amann H. Linear and quasi-linear equations. Vol. Basel: Birkhauser; 1995.
  • Aliev AB, Kazimov AA. Existence, non-existence and asymptotic behavior of global solutions to the Cauchy problem for systems of semilinear hyperbolic equations with damping terms. Nonlinear Anal. 2012;75(1):91–102.
  • Ashyralyev A, Aggez N. Nonlocal boundary value hyperbolic problems involving integral conditions. Bound Value Probl. 2014;2014:214.
  • Denk R, Hieber M, Prüss J. R-boundedness, Fourier multipliers and problems of elliptic and parabolic type. Mem Am Math Soc. 2003;166:788.
  • Davies EB, Pang MM. The Cauchy problem and a generalization of the Hille-Yosida theorem. Proc London Math Soc. 1987;s3-55:181–208.
  • Guidotti P, Lambers J, Solna K. 1D analysis of wave propagation in inhomogeneous media. Numer Funct Anal Optim. 2007;27(1):25–55.
  • Fattorini HO. Second order linear differential equations in Banach spaces. Amsterdam: North-Holland; 1985. (North Holland Mathematics Studies; vol. 108).
  • Triggiani R. A hidden gain of regularity on the strongly damped abstract wave equation: implications on the nonlinear model (Chapter 8). In: Advances in theory, methods and applications in dynamics and control (A felicitation volume in honor of A. V. Balakrishnan). Cambridge; 2011. p. 97–106.
  • Favini A, Goldstein GR, Goldstein JA, et al. Degenerate second order differential operators generating analytic semigroups in Lp and W1,p. Math Nachr. 2002;238:78–102.
  • Goldstein JA. Semigroup of linear operators and applications. Oxford; 1985.
  • Gorbachuk VI, Gorbachuk ML. Boundary value problems for differential-operator equations. Kiev: Naukova Dumka; 1984.
  • Krein SG. Linear differential equations in Banach space. Providence (RI): American Mathematical Society; 1971.
  • Shakhmurov VB. Linear and nonlinear degenerate differential operators and applications. Nonlinear Anal. 2020;191:1116332019.
  • Lunardi A. Analytic semigroups and optimal regularity in parabolic problems. Basel: Birkhauser; 2003.
  • Nirenberg L. On elliptic partial differential equations. Ann Scuola Norm Sup Pisa. 1959;13:115–162.
  • Shakhmurov VB. Nonlocal problems for abstract differential equations and applications. Moscow Math J. 2020;20(1):1–26.
  • Shakhmurov VB. Nonlocal problems for Boussinesq equations. Nonlinear Anal TMA. 2016;142:134–151.
  • Shakhmurov VB. The Cauchy problem for nonlocal abstract Schrodinger equations and applications. Anal Math Phys. 2021;11:807.
  • Shakhmurov VB. Embedding and separable differential operators in Sobolev-Lions type spaces. Math Notes. 2008;84(5-6):842–858.
  • Shakhmurov VB, Shahmurov R. The Cauchy problem for Boussinesq equations with general elliptic part. J Anal Math Phys. 2019;142(2):1689–1709.
  • Shakhmurov VB, Shahmurov R. The improved abstract Boussinesq equations and application. Mediterr J Math. 2021;18:95.
  • Shakhmurov VB. Embedding theorems in Banach-valued B-spaces and maximal B-regular differential operator equations. J Inequal Appl. 2006;2006:1–22.
  • Shahmurov R. On strong solutions of a Robin problem modeling heat conduction in materials with corroded boundary. Nonlinear Anal Real World Appl. 2012;13(1):441–451.
  • Shahmurov R. Solution of the Dirichlet and Neumann problems for a modified Helmholtz equation in Besov spaces on an annuals. J Differ Equ. 2010;249(3):526–550.
  • Shakhmurov VB. Regularity properties of nonlinear abstract Schrö dinger equations and applications. Int J Math. 2020;31(13):2050105.
  • Keyantuo V, Warma M. The wave equation with Wentzell-Robin boundary conditions on Lp-spaces. J Differ Equ. 2006;229:680–697.
  • Piskarev S, Shaw S-Y. Multiplicative perturbations of semigroups and applications to step responses and cumulative outputs. J Funct Anal. 1995;128:315–340.
  • Girardy M, Weis L. Operator-valued multiplier theorems on Besov spaces. Math Nachr. 2003;251:34–51.
  • Lions JL, Peetre J. Sur une classe d'espaces d'interpolation. Inst Hautes Etudes Sci Publ Math. 1964;19:5–68.
  • Peetre J. Sur la transformation de Fourier des fonctions a valeurs vectorielles. Rend Sem Mat Univ Padova. 1969;42:15–26.

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