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Global Solution and Blow-up for a Thermoelastic System of $p$-Laplacian Type with Logarithmic Source

Year 2023, Volume: 11 Issue: 3, 112 - 128, 02.09.2023

Carlos Raposo , Adriano Cattai , Octavio Vera , Ganesh Ch. Goraın , Ducival Pereira

https://doi.org/10.36753/mathenot.1084371
Cited By: 1

Abstract

This manuscript deals with global solution, polynomial stability and blow-up behavior at a finite time for the nonlinear system $$ \left\{ \begin{array}{rcl} & u'' - \Delta_{p} u + \theta + \alpha u' = \left\vert u\right\vert ^{p-2}u\ln \left\vert u\right\vert \\ &\theta' - \Delta \theta = u' \end{array} \right. $$ where $\Delta_{p}$ is the nonlinear $p$-Laplacian operator, $ 2 \leq p < \infty$. Taking into account that the initial data is in a suitable stability set created from the Nehari manifold, the global solution is constructed by means of the Faedo-Galerkin approximations. Polynomial decay is proven for a subcritical level of initial energy. The blow-up behavior is shown on an instability set with negative energy values.

Keywords

Global solution blow-up thermoelastic system of p-Laplacian type logarithmic source

References

  • [1] Dafermos, C. M.: On the existence and the asymptotic stability of solutions to the equations of linear thermoelasticity. Arch. Rational Mech. Anal. 29, 241-271 (1968).
  • [2] Chen, W.: Cauchy problem for thermoelastic plate equations with different damping mechanisms. Commun. Math. Sci. 18, 429-457 (2020).
  • [3] Fareh, A., Messaoudi, S. A.: Energy decay for a porous thermoelastic system with thermoelasticity of second sound and with a non-necessary positive definite energy. Appl. Math. Comput. 293, 493-507 (2017).
  • [4] Feng, B.: On a thermoelastic laminated Timoshenko beam: well posedness and stability. Complexity 2020, 5139419 (2020).
  • [5] Kafini, M., Messaoudi, S. A., Mustafa, M. I.: Energy decay result in a Timoshenko-type system of thermoelasticity of type III with distributive delay. J. Math. Phys. 54, 101503 (2013).
  • [6] Lasiecka, I., Pokojovy, M., Wan, X.: Global existence and exponential stability for a nonlinear thermoelastic Kirchhoff- Love plate. Nonlinear Anal. RealWorld Appl. 38, 184-221 (2017).
  • [7] Lebeau, G., Zuazua, E.: Decay rates for the three-dimensional linear system of thermoelasticity. Arch. Ration. Mech. Anal. 148, 179-231 (1999).
  • [8] Nonato, C., Raposo, C. A., Feng, B.: Exponential stability for a thermoelastic laminated beam with nonlinear weights and time-varying delay. Asymptot. Anal. 2021, 1-29 (2021).
  • [9] Racke, R., Ueda, Y.: Nonlinear thermoelastic plate equations - global existence and decay rates for the Cauchy problem. Journal of Differential Equations 263, 8138-8177 (2017).
  • [10] Raposo, C. A., Villagran, O. P. V., Ferreira, J., Pi¸skin, E.: Rao-Nakra sandwich beam with second sound. Partial Differ. Equ. Appl. Math. 4, 100053 (2021).
  • [11] Rivera, J. M.: Energy decay rates in linear thermoelasticity. Funkcial. Ekvac. 35, 19-30 (1992).
  • [12] Lian,W., Xu, R.: Global well-posedness of nonlinear wave equation with weak and strong damping terms and logarithmic source term. Adv. Nonlinear Anal. 9, 613-632 (2020).
  • [13] Ha, T. G., Park, S. H.: Blow-up phenomena for a viscoelastic wave equation with strong damping and logarithmic nonlinearity. Adv. Differ. Equ. 2020, 235 (2020).
  • [14] Barrow, J. D., Parsons, P.: Inflationary models with logarithmic potentials. Phys. Rev. D. 52, 5576-5587 (1995).
  • [15] Enqvist, K., McDonald, J.: Q-balls and baryogenesis in the MSSM. Phys. Lett. 425, 309-321 (1998).
  • [16] Gorka, P.: Logarithmic Klein-Gordon equation. Acta Phys. Polon. B. 40, 59-66 (2009).
  • [17] Zloshchastiev, K. G.: Applications of wave equations with logarithmic nonlinearity in fluid mechanics. J. Phys. Conf. Ser. 1, 012-051 (2018).
  • [18] Dreher, M.: The wave equation for the p-Laplacian. Hokkaido Math. J. 36, 21-52 (2007).
  • [19] Greenberg, J. M., MacCamy, R. C., Vizel, V. J.: On the existence, uniqueness, and stability of solution of the equation $\sigma^{'}(u_{x})u_{xx} + \lambda u_{xtx} = \rho_{0} u_{tt}$ J. Math. Mech. 17, 707-728 (1968).
  • [20] Ang, D. D., Dinh, A. P. N.: Strong solutions of a quasilinear wave equation with nonlinear damping. SIAM J. Math. Anal. 19, 337-347 (1988).
  • [21] Benaissa, A., Mokeddem, S.: Decay estimates for the wave equation of p-Laplacian type with dissipation of m-Laplacian type. Math. Methods Appl. Sci. 30, 237-247 (2007).
  • [22] Biazutti, A. C.: On a nonlinear evolution equation and its applications. Nonlinear Anal. Theory Methods Appl. 24, 1221-1234 (1995).
  • [23] D’Ancona, P., Spagnolo, S.: On the life span of the analytic solutions to quasilinear weakly hyperbolic equations. Indiana Univ. Math. J. 40, 71-99 (1991).
  • [24] Ma, T. F., Soriano, J. A.: On weak solutions for an evolution equation with exponential nonlinearities. Nonlinear Analysis: Theory, Methods & Applications 37, 1029-1038 (1999).
  • [25] Pei, P., Rammaha, M. A., Toundykov, D.: Weak solutions and blow-up for wave equations of p-Laplacian type with supercritical sources. J. Math. Phys. 56, 081503 (2015).
  • [26] Rammaha, M., Toundykov, D., Wilstein, Z.: Global existence and decay of energy for a nonlinear wave equation with p-Laplacian damping. Discrete Contin. Dyn. Syst. 32 4361-4390 (2012).
  • [27] Ye, Y.: Global existence and asymptotic behavior of solutions for a class of nonlinear degenerate wave equations. Differential Equations and Nonlinear Mechanics. 2007, 19685 (2007).
  • [28] Chueshov, I., Lasiecka, I.: Existence, uniqueness of weak solution and global attactors for a class of nonlinear 2D Kirchhoff-Boussinesq models. Discrete Contin. Dyn. Syst. 15, 777-809 (2006).
  • [29] Gao, H., Ma, T. F.: Global solutions for a nonlinear wave equation with the p–Laplacian operator. Electronic Journal of Qualitative Theory of Differential Equations. 11, 1-13 (1999).
  • [30] Choi, H., Kim, H., Laforest, M.: Relaxation model for the -Laplacian problem with stiffness. J. Comput. Appl. Math. 344, 173-189 (2018).
  • [31] Li, Y.: Global boundedness of weak solution in an attraction–repulsion chemotaxis system with p-Laplacian diffusion. Nonlinear Analysis: RealWorld Applications. 51, 102933 (2020).
  • [32] Zhang, X., Liu, L., Wu, Y., Cui, Y.: Entire blow-up solutions for a quasilinear p-Laplacian Schrödinger equation with a non-square diffusion term. Appl. Math. Lett. 74, 85-93 (2017).
  • [33] Kalleji, M. K.: Weighted Hardy–Sobolev inequality and global existence result of thermoelastic system on manifolds with corner-edge singularities. Math. Methods Appl. Sci. (2021). https://doi.org/10.1002/mma.7916
  • [34] Abdulla, U. G., Jeli, R.: Evolution of interfaces for the nonlinear parabolic p-Laplacian-type reaction-diffusion equations. II. Fast diffusion vs. absorption. European Journal of Applied Mathematics. 31, 385-406 (2020).
  • [35] Boudjeriou, T.: Stability of solutions for a parabolic problem involving fractional p-Laplacian with logarithmic nonlinearity. Mediterr. J. Math. 17 (2020). https://doi.org/10.1007/s00009-020-01584-6
  • [36] Raposo, C. A., Ribeiro, J. O., Cattai, A. P.: Global solution for a thermoelastic system with p-Laplacian. Appl. Math. Lett. 86, 119-125 (2018).
  • [37] Ding, H., Zhou, J.: Global existence and blow-up for a thermoelastic system with p-Laplacian. Appl. Anal. (2021). https://doi.org/10.1080/00036811.2021.1941906.
  • [38] Kim, J. U.: A boundary thin obstacle problem for a wave equation. Commun. Partial Differ. Equ. 14, 1011-1026 (1989).
  • [39] Lions, J. L.: Quelques méthodes de résolution des problèmes aux limites non linéaires. Dunod-Gauthier. Paris (1969).
  • [40] Martinez, P.: A new method to obtain decay rate estimates for dissipative systems. ESAIM Control, Optimisation and Calculus of Variations. 4, 419-444 (1999).
  • [41] Levine, H. A.: Instability and nonexistence of global solutions to nonlinear wave equations of the form Putt = 􀀀Au + F(u). Trans. Amer. Math. Soc. 192, 1–21 (1974).
  • [42] Qin, Y., Rivera, J. M.: Blow-up of solutions to the Cauchy problem in nonlinear one-dimensional thermoelasticity. J. Math. Anal. Appl. 292, 160–193 (2004).
  • [43] Payne, L. E., Sattinger, D. H.: Saddle points and instability of nonlinear hyperbolic equations. Israel Journal of Mathematics. 22, 273-303 (1975).
  • [44] Ambrosetti, A., Rabinowitz, P. H.: Dual variational methods in critical point theory and applications. Journal of Functional Analysis. 14, 349-381 (1973).
  • [45] Willem, M.: Minimax Theorems. Progress in Nonlinear Differential Equations and their Applications. Birkhöuser Boston Inc. Boston (1996).

Year 2023, Volume: 11 Issue: 3, 112 - 128, 02.09.2023

Carlos Raposo , Adriano Cattai , Octavio Vera , Ganesh Ch. Goraın , Ducival Pereira

https://doi.org/10.36753/mathenot.1084371
Cited By: 1

Abstract

References

  • [1] Dafermos, C. M.: On the existence and the asymptotic stability of solutions to the equations of linear thermoelasticity. Arch. Rational Mech. Anal. 29, 241-271 (1968).
  • [2] Chen, W.: Cauchy problem for thermoelastic plate equations with different damping mechanisms. Commun. Math. Sci. 18, 429-457 (2020).
  • [3] Fareh, A., Messaoudi, S. A.: Energy decay for a porous thermoelastic system with thermoelasticity of second sound and with a non-necessary positive definite energy. Appl. Math. Comput. 293, 493-507 (2017).
  • [4] Feng, B.: On a thermoelastic laminated Timoshenko beam: well posedness and stability. Complexity 2020, 5139419 (2020).
  • [5] Kafini, M., Messaoudi, S. A., Mustafa, M. I.: Energy decay result in a Timoshenko-type system of thermoelasticity of type III with distributive delay. J. Math. Phys. 54, 101503 (2013).
  • [6] Lasiecka, I., Pokojovy, M., Wan, X.: Global existence and exponential stability for a nonlinear thermoelastic Kirchhoff- Love plate. Nonlinear Anal. RealWorld Appl. 38, 184-221 (2017).
  • [7] Lebeau, G., Zuazua, E.: Decay rates for the three-dimensional linear system of thermoelasticity. Arch. Ration. Mech. Anal. 148, 179-231 (1999).
  • [8] Nonato, C., Raposo, C. A., Feng, B.: Exponential stability for a thermoelastic laminated beam with nonlinear weights and time-varying delay. Asymptot. Anal. 2021, 1-29 (2021).
  • [9] Racke, R., Ueda, Y.: Nonlinear thermoelastic plate equations - global existence and decay rates for the Cauchy problem. Journal of Differential Equations 263, 8138-8177 (2017).
  • [10] Raposo, C. A., Villagran, O. P. V., Ferreira, J., Pi¸skin, E.: Rao-Nakra sandwich beam with second sound. Partial Differ. Equ. Appl. Math. 4, 100053 (2021).
  • [11] Rivera, J. M.: Energy decay rates in linear thermoelasticity. Funkcial. Ekvac. 35, 19-30 (1992).
  • [12] Lian,W., Xu, R.: Global well-posedness of nonlinear wave equation with weak and strong damping terms and logarithmic source term. Adv. Nonlinear Anal. 9, 613-632 (2020).
  • [13] Ha, T. G., Park, S. H.: Blow-up phenomena for a viscoelastic wave equation with strong damping and logarithmic nonlinearity. Adv. Differ. Equ. 2020, 235 (2020).
  • [14] Barrow, J. D., Parsons, P.: Inflationary models with logarithmic potentials. Phys. Rev. D. 52, 5576-5587 (1995).
  • [15] Enqvist, K., McDonald, J.: Q-balls and baryogenesis in the MSSM. Phys. Lett. 425, 309-321 (1998).
  • [16] Gorka, P.: Logarithmic Klein-Gordon equation. Acta Phys. Polon. B. 40, 59-66 (2009).
  • [17] Zloshchastiev, K. G.: Applications of wave equations with logarithmic nonlinearity in fluid mechanics. J. Phys. Conf. Ser. 1, 012-051 (2018).
  • [18] Dreher, M.: The wave equation for the p-Laplacian. Hokkaido Math. J. 36, 21-52 (2007).
  • [19] Greenberg, J. M., MacCamy, R. C., Vizel, V. J.: On the existence, uniqueness, and stability of solution of the equation $\sigma^{'}(u_{x})u_{xx} + \lambda u_{xtx} = \rho_{0} u_{tt}$ J. Math. Mech. 17, 707-728 (1968).
  • [20] Ang, D. D., Dinh, A. P. N.: Strong solutions of a quasilinear wave equation with nonlinear damping. SIAM J. Math. Anal. 19, 337-347 (1988).
  • [21] Benaissa, A., Mokeddem, S.: Decay estimates for the wave equation of p-Laplacian type with dissipation of m-Laplacian type. Math. Methods Appl. Sci. 30, 237-247 (2007).
  • [22] Biazutti, A. C.: On a nonlinear evolution equation and its applications. Nonlinear Anal. Theory Methods Appl. 24, 1221-1234 (1995).
  • [23] D’Ancona, P., Spagnolo, S.: On the life span of the analytic solutions to quasilinear weakly hyperbolic equations. Indiana Univ. Math. J. 40, 71-99 (1991).
  • [24] Ma, T. F., Soriano, J. A.: On weak solutions for an evolution equation with exponential nonlinearities. Nonlinear Analysis: Theory, Methods & Applications 37, 1029-1038 (1999).
  • [25] Pei, P., Rammaha, M. A., Toundykov, D.: Weak solutions and blow-up for wave equations of p-Laplacian type with supercritical sources. J. Math. Phys. 56, 081503 (2015).
  • [26] Rammaha, M., Toundykov, D., Wilstein, Z.: Global existence and decay of energy for a nonlinear wave equation with p-Laplacian damping. Discrete Contin. Dyn. Syst. 32 4361-4390 (2012).
  • [27] Ye, Y.: Global existence and asymptotic behavior of solutions for a class of nonlinear degenerate wave equations. Differential Equations and Nonlinear Mechanics. 2007, 19685 (2007).
  • [28] Chueshov, I., Lasiecka, I.: Existence, uniqueness of weak solution and global attactors for a class of nonlinear 2D Kirchhoff-Boussinesq models. Discrete Contin. Dyn. Syst. 15, 777-809 (2006).
  • [29] Gao, H., Ma, T. F.: Global solutions for a nonlinear wave equation with the p–Laplacian operator. Electronic Journal of Qualitative Theory of Differential Equations. 11, 1-13 (1999).
  • [30] Choi, H., Kim, H., Laforest, M.: Relaxation model for the -Laplacian problem with stiffness. J. Comput. Appl. Math. 344, 173-189 (2018).
  • [31] Li, Y.: Global boundedness of weak solution in an attraction–repulsion chemotaxis system with p-Laplacian diffusion. Nonlinear Analysis: RealWorld Applications. 51, 102933 (2020).
  • [32] Zhang, X., Liu, L., Wu, Y., Cui, Y.: Entire blow-up solutions for a quasilinear p-Laplacian Schrödinger equation with a non-square diffusion term. Appl. Math. Lett. 74, 85-93 (2017).
  • [33] Kalleji, M. K.: Weighted Hardy–Sobolev inequality and global existence result of thermoelastic system on manifolds with corner-edge singularities. Math. Methods Appl. Sci. (2021). https://doi.org/10.1002/mma.7916
  • [34] Abdulla, U. G., Jeli, R.: Evolution of interfaces for the nonlinear parabolic p-Laplacian-type reaction-diffusion equations. II. Fast diffusion vs. absorption. European Journal of Applied Mathematics. 31, 385-406 (2020).
  • [35] Boudjeriou, T.: Stability of solutions for a parabolic problem involving fractional p-Laplacian with logarithmic nonlinearity. Mediterr. J. Math. 17 (2020). https://doi.org/10.1007/s00009-020-01584-6
  • [36] Raposo, C. A., Ribeiro, J. O., Cattai, A. P.: Global solution for a thermoelastic system with p-Laplacian. Appl. Math. Lett. 86, 119-125 (2018).
  • [37] Ding, H., Zhou, J.: Global existence and blow-up for a thermoelastic system with p-Laplacian. Appl. Anal. (2021). https://doi.org/10.1080/00036811.2021.1941906.
  • [38] Kim, J. U.: A boundary thin obstacle problem for a wave equation. Commun. Partial Differ. Equ. 14, 1011-1026 (1989).
  • [39] Lions, J. L.: Quelques méthodes de résolution des problèmes aux limites non linéaires. Dunod-Gauthier. Paris (1969).
  • [40] Martinez, P.: A new method to obtain decay rate estimates for dissipative systems. ESAIM Control, Optimisation and Calculus of Variations. 4, 419-444 (1999).
  • [41] Levine, H. A.: Instability and nonexistence of global solutions to nonlinear wave equations of the form Putt = 􀀀Au + F(u). Trans. Amer. Math. Soc. 192, 1–21 (1974).
  • [42] Qin, Y., Rivera, J. M.: Blow-up of solutions to the Cauchy problem in nonlinear one-dimensional thermoelasticity. J. Math. Anal. Appl. 292, 160–193 (2004).
  • [43] Payne, L. E., Sattinger, D. H.: Saddle points and instability of nonlinear hyperbolic equations. Israel Journal of Mathematics. 22, 273-303 (1975).
  • [44] Ambrosetti, A., Rabinowitz, P. H.: Dual variational methods in critical point theory and applications. Journal of Functional Analysis. 14, 349-381 (1973).
  • [45] Willem, M.: Minimax Theorems. Progress in Nonlinear Differential Equations and their Applications. Birkhöuser Boston Inc. Boston (1996).
There are 45 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Research Article
Authors

Carlos Raposo 0000-0001-8014-7499

Adriano Cattai 0000-0002-6171-6585

Octavio Vera 0000-0001-7304-0976

Ganesh Ch. Goraın 0000-0002-5326-3635

Ducival Pereira 0000-0003-4511-0185

Submission Date March 8, 2022
Acceptance Date July 29, 2022
Publication Date September 2, 2023
Published in Issue Year 2023 Volume: 11 Issue: 3

Cite

APA Raposo, C., Cattai, A., Vera, O., … Ch. Goraın, G. (2023). Global Solution and Blow-up for a Thermoelastic System of $p$-Laplacian Type with Logarithmic Source. Mathematical Sciences and Applications E-Notes, 11(3), 112-128. https://doi.org/10.36753/mathenot.1084371
AMA Raposo C, Cattai A, Vera O, Ch. Goraın G, Pereira D. Global Solution and Blow-up for a Thermoelastic System of $p$-Laplacian Type with Logarithmic Source. Math. Sci. Appl. E-Notes. September 2023;11(3):112-128. doi:10.36753/mathenot.1084371
Chicago Raposo, Carlos, Adriano Cattai, Octavio Vera, Ganesh Ch. Goraın, and Ducival Pereira. “Global Solution and Blow-up for a Thermoelastic System of $p$-Laplacian Type With Logarithmic Source”. Mathematical Sciences and Applications E-Notes 11, no. 3 (September 2023): 112-28. https://doi.org/10.36753/mathenot.1084371.
EndNote Raposo C, Cattai A, Vera O, Ch. Goraın G, Pereira D (September 1, 2023) Global Solution and Blow-up for a Thermoelastic System of $p$-Laplacian Type with Logarithmic Source. Mathematical Sciences and Applications E-Notes 11 3 112–128.
IEEE C. Raposo, A. Cattai, O. Vera, G. Ch. Goraın, and D. Pereira, “Global Solution and Blow-up for a Thermoelastic System of $p$-Laplacian Type with Logarithmic Source”, Math. Sci. Appl. E-Notes, vol. 11, no. 3, pp. 112–128, 2023, doi: 10.36753/mathenot.1084371.
ISNAD Raposo, Carlos et al. “Global Solution and Blow-up for a Thermoelastic System of $p$-Laplacian Type With Logarithmic Source”. Mathematical Sciences and Applications E-Notes 11/3 (September2023), 112-128. https://doi.org/10.36753/mathenot.1084371.
JAMA Raposo C, Cattai A, Vera O, Ch. Goraın G, Pereira D. Global Solution and Blow-up for a Thermoelastic System of $p$-Laplacian Type with Logarithmic Source. Math. Sci. Appl. E-Notes. 2023;11:112–128.
MLA Raposo, Carlos et al. “Global Solution and Blow-up for a Thermoelastic System of $p$-Laplacian Type With Logarithmic Source”. Mathematical Sciences and Applications E-Notes, vol. 11, no. 3, 2023, pp. 112-28, doi:10.36753/mathenot.1084371.
Vancouver Raposo C, Cattai A, Vera O, Ch. Goraın G, Pereira D. Global Solution and Blow-up for a Thermoelastic System of $p$-Laplacian Type with Logarithmic Source. Math. Sci. Appl. E-Notes. 2023;11(3):112-28.

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