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Existence Results for a Nonlinear Evolution Equation Arising in ElastoplasticMicrostructure Models

Year 2019, Volume: 7 Issue: 2, 312 - 315, 25.05.2019
https://doi.org/10.21541/apjes.477603

Abstract

We establish global existence results for a nonlinear evolution equation which arises in elastoplastic-microstructure models on
a bounded domain, employing potential well method. A functional is defined for the potential well method, and global
existence is proved by use of sign invariance of this functional in the case of high initial energy.

References

  • [1] L. J. An, A. Peirce, A weakly nonlinear analysis of elasto-plastic-microstructure models, SIAM J. Appl. Math. 55, 136-155, (1995).
  • [2] G. Chen, Z. Yang, Existence and nonexistence of global solutions for a class of nonlinear wave equations, Math. Meth. Appl. Sci. 23, 615-631, (2000).
  • [3] H. Zhang, G. Chen, Potential well method for a class of nonlinear wave equations of fourth order, Acta Math. Sci. Series A 23(6), 758-768, (2003). (In Chinese).
  • [4] J.A. Esquivel-Avila, Dynamics around the ground state of a nonlinear evolution equation, Nonlinear Anal. 63 (5-7), 331-343, (2005).
  • [5] L. Yacheng, X. Runzhang, A class of fourth-order wave equations with dissipative and nonlinear strain terms, J. Differential Equations 244, 200-228, (2008).
  • [6] Z. Yang, Global existence, asymptotic behavior and blow up of solutions for a class of nonlinear wave equations with dissipative term, J. Differential Equations 187, 520-540, (2003).
  • [7] Kutev, N., Kolkovska, N., Dimova, M., Christov, C.I., Theoretical and numerical aspects for global existence and blow up for the solutions to Boussinesq paradigm equation, AIP Conf. Proc. 1404, 68—76, (2011).
  • [8] N. Kutev, N. Kolkovska, M. Dimova, Global existence of Cauchy problem for Boussinesq paradigm equation, Comput. Math. Appl., 65, 500-511, (2013).
  • [9] Taskesen, H., Polat, N., Ertaş, A. On Global Solutions for the Cauchy Problem of a Boussinesq-Type Equation. Abst. Appl. Anal. 10 pages, Doi:10.1155/2012/535031, (2012).
  • [10] H. Taskesen, N. Polat Existence of global solutions for a multidimensional Boussinesq-type equation with supercritical initial energy, First International Conference on Analysis and Applied Mathematics:ICAAM, AIP Conf. Proc. 1470, pp:159-162, (2012).
  • [11] N. Polat, H. Taskesen On the existence of global solutions for a nonlinear Klein-Gordon equation, Filomat, 28(5), 1073-1079, (2014).
  • [12] E. H. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Annals of Mathematics, 118, 349—374, (1983).
  • [13] N. Polat, D. Kaya, H.·I. Tutalar, in: Dynamical Systems and Applications, GBS Publishers and Distributers, India 572, (2005).
  • [14] Levine, H.A. Instability and nonexistence of global solutions to nonlinear wave equations of the form , Transactions of the American Mathematical Society, 192, 1-21, (1974).

Elastoplastik-Mikro Yapı Modellerinde Ortaya Çıkan Doğrusal Olmayan Evolüsyon Denklemi İçin Varlık Sonuçları

Year 2019, Volume: 7 Issue: 2, 312 - 315, 25.05.2019
https://doi.org/10.21541/apjes.477603

Abstract

Bu çalışmada, sınırlı bir alanda elastoplastik-mikroyapı modellerinde ortaya çıkan doğrusal olmayan bir evrim denklemi için
global varlık sonuçları potential well metodu kullanılarak oluşturulmuştur. Potential well yöntemi için bir fonksiyonel
tanımlanmış ve

References

  • [1] L. J. An, A. Peirce, A weakly nonlinear analysis of elasto-plastic-microstructure models, SIAM J. Appl. Math. 55, 136-155, (1995).
  • [2] G. Chen, Z. Yang, Existence and nonexistence of global solutions for a class of nonlinear wave equations, Math. Meth. Appl. Sci. 23, 615-631, (2000).
  • [3] H. Zhang, G. Chen, Potential well method for a class of nonlinear wave equations of fourth order, Acta Math. Sci. Series A 23(6), 758-768, (2003). (In Chinese).
  • [4] J.A. Esquivel-Avila, Dynamics around the ground state of a nonlinear evolution equation, Nonlinear Anal. 63 (5-7), 331-343, (2005).
  • [5] L. Yacheng, X. Runzhang, A class of fourth-order wave equations with dissipative and nonlinear strain terms, J. Differential Equations 244, 200-228, (2008).
  • [6] Z. Yang, Global existence, asymptotic behavior and blow up of solutions for a class of nonlinear wave equations with dissipative term, J. Differential Equations 187, 520-540, (2003).
  • [7] Kutev, N., Kolkovska, N., Dimova, M., Christov, C.I., Theoretical and numerical aspects for global existence and blow up for the solutions to Boussinesq paradigm equation, AIP Conf. Proc. 1404, 68—76, (2011).
  • [8] N. Kutev, N. Kolkovska, M. Dimova, Global existence of Cauchy problem for Boussinesq paradigm equation, Comput. Math. Appl., 65, 500-511, (2013).
  • [9] Taskesen, H., Polat, N., Ertaş, A. On Global Solutions for the Cauchy Problem of a Boussinesq-Type Equation. Abst. Appl. Anal. 10 pages, Doi:10.1155/2012/535031, (2012).
  • [10] H. Taskesen, N. Polat Existence of global solutions for a multidimensional Boussinesq-type equation with supercritical initial energy, First International Conference on Analysis and Applied Mathematics:ICAAM, AIP Conf. Proc. 1470, pp:159-162, (2012).
  • [11] N. Polat, H. Taskesen On the existence of global solutions for a nonlinear Klein-Gordon equation, Filomat, 28(5), 1073-1079, (2014).
  • [12] E. H. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Annals of Mathematics, 118, 349—374, (1983).
  • [13] N. Polat, D. Kaya, H.·I. Tutalar, in: Dynamical Systems and Applications, GBS Publishers and Distributers, India 572, (2005).
  • [14] Levine, H.A. Instability and nonexistence of global solutions to nonlinear wave equations of the form , Transactions of the American Mathematical Society, 192, 1-21, (1974).
There are 14 citations in total.

Details

Primary Language English
Subjects Engineering
Journal Section Articles
Authors

Hatice Taşkesen 0000-0003-1058-0507

Publication Date May 25, 2019
Submission Date November 1, 2018
Published in Issue Year 2019 Volume: 7 Issue: 2

Cite

IEEE H. Taşkesen, “Existence Results for a Nonlinear Evolution Equation Arising in ElastoplasticMicrostructure Models”, APJES, vol. 7, no. 2, pp. 312–315, 2019, doi: 10.21541/apjes.477603.