Nonexistence of Solutions for a Logarithmic m-Laplacian Type Equation with Delay Term

Year 2021, Volume: 9 Issue: 2, 238 - 244, 15.10.2021

Hazal Yüksekkaya Erhan Pişkin

Abstract

In this work, we consider a logarithmic m-Laplacian type equation with delay term with initial and boundary conditions. Under suitable conditions on the initial data, we study the nonexistence of solutions in a finite time with negative initial energy $E\left( 0\right) <0$ in a bounded domain.

Keywords

Decay term, logarithmic source term, m-Laplacian equation, nonexistence of solutions

References

• K. Bartkowski, P. Gorka, One dimensional Klein-Gordon equation with logarithmic nonlinearities, J. Phys. A 41 (2008).
• I. Bialynicki-Birula, J. Mycielski, Wave equations with logarithmic nonlinearities, Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys., 23(4) (1975) 461-466.
• I. Bialynicki-Birula, J. Mycielski, Nonlinear wave mechanics, Ann. Physics, 100(1-2) (1976) 62-93.
• T. Cazenave, A. Haraux, Equations d'evolution avec non-linearite logarithmique, Ann. Fac. Sci. Toulouse Math., 2(1) (1980) 21-51.
• R. Datko, J. Lagnese, M. P. Polis, An example on the effect of time delays in boundary feedback stabilization of wave equations, SIAM J. Control Optim., 24(1) (1986) 152-156.
• P. Gorka, Logarithmic Klein-Gordon equation, Acta Phys. Polon. B 40 (2009) 59--66.
• M. Kafini, S. A. Messaoudi, A blow-up result in a nonlinear wave equation with delay, Mediterr. J. Math., 13 (2016) 237-247. M. Kafini, S. Messaoudi, Local existence and blow up of solutions to a logarithmic nonlinear wave equation with delay, Appl. Anal., (2018) 1-18.
• C.N. Le, X. T. Le, Global solution and blow up for a class of Pseudo p-Laplacian evolution equations with logarithmic nonlinearity, Comput. Math. Appl., 73(9) (2017) 2076.
• N. Mezouar, S.M. Boulaaras, A. Allahem, Global existence of solutions for the viscoelastic Kirchhoff equation with logarithmic source terms, J. Complex, (2020), 1-25. S. Nicaise, C. Pignotti, Stabilization of the wave equation with boundary or internal distributed delay, Differ. Integral Equ., 21 (2008) 935-958.
• S. Nicaise, C. Pignotti, Stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks, SIAM J. Control Optim., 45(5) (2006) 1561-1585.
• S.H. Park, Global existence, energy decay and blow-up of solutions for wave equations with time delay and logariithmic source, Adv. Differ. Equ., 2020:631 (2020) 1-17.
• E. Pişkin, N. Irkıl, Mathematical behavior of solutions of p-Laplacian equation with logarithmic source term, Sigma J. Eng. & Nat. Sci., 10(2) (2019) 213-220.
• E. Pişkin, H. Yüksekkaya, Nonexistence of solutions of a delayed wave equation with variable-exponents, C-POST, 3(1), 97--101, 2020. E. Pişkin, H. Yüksekkaya, Decay and blow up of solutions for a delayed wave equation with variable-exponents, C-POST, 3(1), 91--96, 2020.
• E. Pişkin, H. Yüksekkaya, Local existence and blow up of solutions for a logarithmic nonlinear viscoelastic wave equation with delay, Comput. Methods Differ. Equ., 1-14, 2020, doi:10.22034/cmde.2020.35546.1608. (In press)
• E. Pişkin, H. Yüksekkaya, Nonexistence of global solutions of a delayed wave equation with variable-exponents, Miskolc Math. Notes, 1-19. (Accepted)
• E. Pişkin, H. Yüksekkaya, Blow-up of solutions for a logarithmic quasilinear hyperbolic equation with delay term, J. Math. Anal., 12(1), 56-64, 2021.