Geodesic
A nonexistence result for the semi-linear Moore–Gibson–Thompson equation with nonlinear memory on the Heisenberg group
Vladikavkazskij matematičeskij žurnal, Tome 24 (2022) no. 1, pp. 24-35 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

The Moore–Gibson–Thompson theory was developed starting from a third order differential equation, built in the context of some consideration related fluid mechanics. Subsequently the equation was considered as a heat conduction equation because it has been obtained by considering a relaxation parameter into the type III heat conduction. Since the advent of the Moore–Gibson–Thompson theory, the number of dedicated studies to this theory has increased considerably. The Moore–Gibson–Thompson equation modifies and defines equations for thermal conduction and mass diffusion that occur in solids. In this paper we investigate a class of Moore–Gibson–Thompson equation with nonlinear memory on the Heisenberg group.The problem of nonexistence of global weak solutions in the Heisenberg group has received specific attention in the recent years. In the present paper we use the method of test functions to prove nonexistence of global weak solutions. The results obtained in this paper extend several contributions and we focus on new nonexistence results which are due to the presence of the fractional Laplacian operator of order $\frac{\sigma}{2}$. 
@article{VMJ_2022_24_1_a2,
     author = {S. Georgiev and A. Hakem},
     title = {A nonexistence result for the semi-linear {Moore{\textendash}Gibson{\textendash}Thompson} equation with nonlinear memory on the {Heisenberg} group},
     journal = {Vladikavkazskij matemati\v{c}eskij \v{z}urnal},
     pages = {24--35},
     year = {2022},
     volume = {24},
     number = {1},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/VMJ_2022_24_1_a2/}
}
TY  - JOUR
AU  - S. Georgiev
AU  - A. Hakem
TI  - A nonexistence result for the semi-linear Moore–Gibson–Thompson equation with nonlinear memory on the Heisenberg group
JO  - Vladikavkazskij matematičeskij žurnal
PY  - 2022
SP  - 24
EP  - 35
VL  - 24
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/VMJ_2022_24_1_a2/
LA  - en
ID  - VMJ_2022_24_1_a2
ER  - 
%0 Journal Article
%A S. Georgiev
%A A. Hakem
%T A nonexistence result for the semi-linear Moore–Gibson–Thompson equation with nonlinear memory on the Heisenberg group
%J Vladikavkazskij matematičeskij žurnal
%D 2022
%P 24-35
%V 24
%N 1
%U http://geodesic.mathdoc.fr/item/VMJ_2022_24_1_a2/
%G en
%F VMJ_2022_24_1_a2
S. Georgiev; A. Hakem. A nonexistence result for the semi-linear Moore–Gibson–Thompson equation with nonlinear memory on the Heisenberg group. Vladikavkazskij matematičeskij žurnal, Tome 24 (2022) no. 1, pp. 24-35. http://geodesic.mathdoc.fr/item/VMJ_2022_24_1_a2/

[1] Chen W., Palmieri A., “A Blow-up Result for the Semilinear Moore–Gibson–Thompson Equation with Nonlinearity of Derivative Type in the Conservative Case”, Evolution Equations and Control Theory, 10:4 (2021), 673–687 | DOI | MR | Zbl

[2] Caixeta A. H., Lasiecka I., Domingos Cavalcanti V. N., “On Long Time Behavior of Moore–Gibson–Thompson Equation with Molecular Relaxation”, Evolution Equations and Control Theory, 5:4 (2016), 661–676 | DOI | MR | Zbl

[3] Lecaros R., Mercado A., Zamorano S., An Inverse Problem for Moore–Gibson–Thompson Equation Arising in High Intensity Ultrasound, 2020, arXiv: 2001.07673 | DOI | Zbl

[4] Lai N. A., Takamura H., “Nonexistence of Global Solutions of Nonlinear Wave Equations with Weak Time-Dependent Damping Related to Glassey's Conjecture”, Differential Integral Equations, 32:1, 2 (2019), 37–48 | MR | Zbl

[5] Dao T. A., Fino A. Z., “Blow up Results for Semi-Linear Structural Damped Wave Model with Nonlinear Memory”, Mathematische Nachrichten, 295:2 (2022), 309–322 | DOI | MR

[6] Benibrir F., Hakem A., “Nonexistence Results for a Semi-Linear Equation with Fractional Derivatives on the Heisenberg Group”, J. Adv. Math. Stud., 11:3 (2018), 587–596 | MR | Zbl

[7] Folland G. B., Stein E. M., “Estimates for the $\partial_h$ Complex and Analysis on the Heisenberg Group”, Comm. Pure Appl. Math., 27 (1974), 492–522 | DOI | MR

[8] Garofalo N., Lanconelli E., “Existence and non Existence Results for Semilinear Equations on the Heisenberg Group”, Indiana Univ. Math. Journ., 41 (1992), 71–97 | DOI | MR

[9] Goldstein J. A., Kombe I., “Nonlinear Degenerate Parabolic Equations on the Heisenberg Group”, Int. J. Evol. Equ., 1:1 (2005), 122 | MR

[10] Folland G. B., “Fondamental Solution for Subelliptic Operators”, Bull. Amer. Math. Soc., 79 (1979), 373–376 | DOI | MR

[11] Pohozaev S., Veron L., “Nonexistence Results of Solutions of Semilinear Differential Inequalities on the Heisenberg Group”, Manuscript Math., 102 (2000), 85–99 | DOI | MR | Zbl

[12] Samko S. G., Kilbas A. A., Marichev O. I., Fractional Integrals and Derivatives, Theory and Applications, Gordon and Breach Science Publishers, 1987 | MR

[13] Kilbas A. A., Srivastava H. M., Trujillo J. J., Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, 204, 2006, 523 pp. | DOI | MR | Zbl