Exponential growth of solution for a couple of semi-linear pseudo-parabolic equations with memory and source terms

Main Article Content



This work is concerned with coupled semi-linear pseudo-parabolic equations with memory terms in both equations, associated with the homogeneous Dirichlet boundary condition. We show that the solution grows exponentially under specific conditions regarding the relaxation functions and initial energy. In order to prove the result, we use the energy method based on the construction of a suitable Lyapunov function.
The most important behavior of the evolution system is the exponential growth phenomena because of its wide range of applications in modern science, such as chemistry, biology, ecology, and other areas of engineering and physical sciences.


Download data is not yet available.


Metrics Loading ...

Article Details

How to Cite
OUAOUA, A. and BOUGHAMSA, W. 2022. Exponential growth of solution for a couple of semi-linear pseudo-parabolic equations with memory and source terms. Journal of Innovative Applied Mathematics and Computational Sciences. 2, 1 (May 2022), 43–52. DOI:https://doi.org/10.58205/jiamcs.v2i1.23.
Research Articles


T. B. Benjamin, J. L. Bona, and J. J. Mahony, Model equations for long waves in nonlinear dispersive systems, Phil. Trans. R. Soc. Lond. A. 272(1220) (1972), 47–78.

J. Dang, Q. Hu and H. Zhang, Asymptotic stability and blow-up of solutions for the generalized Boussinesq equation with nonlinear boundary condition. Open J. Math. Anal. 2(2) (2018), 93–113.

K. Deng and H. A. Levine, The role of critical exponents in blow-up theorems: the sequel, J. Math. Anal. Appl. 243(1) (2000), 85–126.

J. Escerh and Z. Yin, Stable equilibra to parabolic systems in unbounded domains. J. Nonlinear Math. Phys. 11(2) (2004), 243–255.

M. Escobedo and MA. Herrero, A semilinear reaction diffusion system in a bounded domain, Ann. Mat. Pura Appl. 165(4) (1993), 315–336.

M. Escobedo and HA. Levine, Critical blowup and global existence numbers for a weakly coupled system of reaction-diffusion equations, Arch. Ration. Mech. Anal. 129 (1995), 47–100.

V. K. Kalantarov and O. A. Ladyzhenskaya, The occurrence of collapse for quasilinear equations of parabolic and hyperbolic types, J. Sov. math. 10(1) (1978), 53–70.

G. Karch, Asymptotic behavior of solutions to some pseudoparabolic equations, Mathematical Methods in the Applied Sciences. 20(3) (1997), 271–289.

M. O. Korpusov and A. G. Sveshnikov, Sufficient conditions, which are close to necessary conditions, for the blow-up of the solution of a strongly nonlinear generalized Boussinesq equation, Comput. Math. Math. Phys. 48(9) (2008), 1629–1637.

H. A. Levine, Some nonexistence and instability theorems for solutions of formally parabolic equations of the form Pu_t = -Au + F(u), Arch. Ration. Mech. Anal. 51 (1973), 371–386.

S. Messaoudi, Blow-up of solutions of a semilinear heat equation with a memory term, Abstr. Appl. Anal. 2 (2005), 87–94.

S. Messaoudi and B. Said-Houari, Global nonexistence of positive initial-energy solutions of a system of nonlinear viscoelastic wave equations with damping and source terms, J. Math. Anal. Appl. 365(1) (2010), 277–287.

A. Ouaoua, A. Khaldi and M. Maouni, Stabilisation of solutions for a Kirchhoff type reaction-diffusion equation, Canad. J. Appl. Math. 2 (2) (2020), 71–80.

A. Ouaoua and M. Maouni, Blow-Up, Exponential growth of solution for a nonlinear parabolic equation with p(x)-Laplacian, Int. J. Anal. Appl. 17(4) (2019), 620–929.

E. Pi¸skin and F. Ekinci, Exponential growth of solutions for a parabolic system, Journal of Engineering and Tecnology. 3(2) (2019), 29–34.

N. Polat, Blow up of solution for a nonlinear reaction-diffusion equation with multiple nonlinearities, Int. J. Sci. Technol. 2(2) (2007), 123–128.

A. Sveshnikov, G. Alshin, A. B, Korpusov, and Y. D. Pletner, Linear and nonlinear equations of Sobolev type, Fizmatlit, Moscow, (2007).

L. X. Truong and N. Van Y, Exponential growth with Lp-norm of solutions for nonlinear heat equations with viscoelastic term, Appl. Math. Comput. 27(C) (2016), 656–663.

L. X. Truong and N. Van Y, On a class of nonlinear heat equations with viscoelastic term, Comput. Math. Appl. 72(1) (2016), 216–232.

H. Zhang, J. Lu and Q. Y. Hu, Exponential growth of solution of a strongly nonlinear generalized Boussinesq equation, Comput. Math. Appl. 68(12) (2014), 1787–1793.

H. W. Zhang, G. X. Zhang and Q. Y. Hu, Global nonexistence of solutions for a class of doubly nonlinear parabolic equations, Acta Anal. Funct. Appl. 18(2) (2016), 204–211.