Existence Results for a Nonlinear Evolution Equation Arising in ElastoplasticMicrostructure Models

Yıl 2019, Cilt: 7 Sayı: 2, 312 - 315, 25.05.2019

Hatice Taşkesen

https://doi.org/10.21541/apjes.477603

Öz

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.

Anahtar Kelimeler

Evolution equation, High initial energy level, Global solution, Initial-boundary value problem

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

Öz

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

Anahtar Kelimeler

Evolüsyon denklemi, Yüksek başlangıç enerjisi, Global çözüm, Başlangıç-sınır değer problemi

Kaynakça

• [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).