Document Type : Research Paper


Instituto de Investigacion ,FCM-UNMSM, Av. Venezuela S/N, Lima- Peru.


The current paper discusses the global existence and asymptotic behavior of solutions of the following new nonlocal problem
$$ u_{tt}- M\left(\displaystyle \int_{\Omega}|\nabla u|^{2}\, dx\right)\triangle u + \delta u_{t}= |u|^{\rho-2}u \log|u|, \quad \text{in}\ \Omega \times ]0,\infty[,  $$
{a-bs,}&{\text{for}\ s \in [0,\frac{a}{b}[,}\\
{0,}&{\text{for}\ s \in [\frac{a}{b}, +\infty[.}
If the initial data are appropriately small, we derive existence of global strong solutions and the  exponential decay of the energy.


Main Subjects

1. M. Aassila and A. Benaissa, Existence globale et comportement asymptotique des solutions des équations de Kirchhoff moyennement dégénérées avec un terme nonlinear dissipatif, Funkcial. Ekvac, 44 (2001), pp. 309-333.
2. F.D. Araruna, F.O. Matias, M.L. Oliveira and S.M.S. Souza, Well-Posedness and Asymptotic Behavior for a nonlinear wave equation, Recent Advances in PDEs:Analysis, Numerics and Control, Springer, 17 ( 2018), pp. 17-32.
3. J.Barrow and P. Parsons, Inflationary models with logarithmic potentials, Phys. Rev. D, 52 (1995), pp. 5576-5587.
4. K. Bartkowski and P. Gorka, One-dimensional Klein-Gordon equation with logarithmic nonlinearities, J. Phys. A, 41 (2008), 355201.
5. A. Benaissa and L. Rahmani, Global existence and energy decay of solutions for Kirchhoff-Carrier equations with weakly nonlinear dissipation, Bull. Belg. Math. Soc., 11 (2004), pp. 547-574.
6. K. Boukerrioua, Note on some nonlinear integral inequalities and applications to differential equations,Int. J. Differ. Equations, (2011), pp.1-15.
7. S. Boulaaras, A. Draifia and M. Alnegga, Polynomial decay rate for kirchhoff type in viscoelasticity with logarithmic nonlinearity and not necessarily decreasing kernel, Symmetry, 11 (2019), pp.1-24.
8. S.M.S. Cordeiro, D.C. Pereira, J. Ferreira and C.A. Raposo, Global solutions and exponential decay to a Klein–Gordon equation of Kirchhoff-Carrier type with strong damping and nonlinear logarithmic source term, Partial Differential Equations Appl. Math., 3 (2021) 100018.
9. K. Enqvist and J. McDonald, Q-balls and baryogenesis in the MSSM, Phys. Lett. B , 425 (1998), pp. 309-321
10. M. Ghisi, Global solutions to some nonlinear dissipative mildly degenerate Kirchhoff equations, Port. Math., 60 (1) (2003), pp.1-22.
11. M. Ghisi, Some remarks on global solutions to nonlinear dissipative mildly degenerate Kirchhoff strings, Rend. Sem. Mat. Univ. Padova, 106 (2001), pp. 185-205.
12. M. Ghisi and M. Gobbino, Global existence and asymptotic behaviour for a mildly degenerate dissipative hyperbolic equation of Kirchhoff type, Asymptot. Anal., 40 (2004), pp. 25-36.
13. D. Gilbarg and N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd edn, Springer, Berlin, 1983.
14. P. Gorka, Logarithmic quantum mechanics: Existence of the ground state, Found. Phys. Lett., 19 (2006), pp. 591-601.
15. J. Hadamard Le problèmes de Cauchy et les equations aux derivées partielles lineaires hiperboliques, Paris, Hermann, 1932.
16. M.K. Hamdani, N.T. Chung and D.D. Repovš, New class of sixthorder nonhomogeneous p(x)-Kirchhoff problems with sign-changing weight functions, Adv. Nonlinear Anal, 10 (2021), pp. 1117-1131.
17. G. Kirchhoff, Vorlesungen über Mechanik, Leipzig, Teubner, 1883.
18. J.E. Lions, “On some questions in boundary value problems of mathematical physics”, in Proceedings of the International Symposium on Continuum Mechanics and Partial Differential Equations, North Holland, Amsterdam, The Netherlands, 1978.
19. J.L. Lions, Some remarks on the optimal control on singular distributed systems, INRIA, France, 1984.
20. M. Milla Miranda, A.T. Louredo and L.A. Medeiros, Nonlinear perturbations of the Kirchhof equation, Electron J. Differential Equations, 77 (2017), pp. 1-21.
21. M. Nakao, Convergence of solutions of the wave equation with a nonlinear dissipative term to the steady state, Mem. Fac. Sci. Kyushu Univ., 30 (1976), pp. 257-265.
22. K. Ono, Global existence and decay properties of solutions for coupled degenerate dissipative hyperbolic systems of Kirchhoff type, Funkcial. Ekvac. , 57 (2014), pp. 319-337.
23. X. Qian, Multiplicity of positive solutions for a class of nonlocal problem involving critical exponent, Electron. J. Qual. Theory Differ. Equ., 57 (2021), pp. 1-14.
24. L.Tartar, Topics in Nonlinear Analysis, Publications Mathématiques d’Orsay, Uni.Paris Sud. Dep. Math., Orsay, France, 1978.
25. Y. Ye, Global existence of solutions and energy decay for a Kirchhoff-type equation with nonlinear dissipation, J. Inequal. Appl., 195 (2013).
26. Y. Yang, J. Li and T. Yu, The qualitative analysis of solutions for a class of Kirchhoff equation with linear strong damping term, nonlinear weak damping term and power-type logarithmic source term, Appl Numer Math, 141 (2019), pp. 263-285.
27. G.S. Yin and J.S. Liu, Existence and multiplicity of nontrivial solutions for a nonlocal problem, Bound. Value Probl., 2015 (2015) 26, pp. 1–7.
28. Y. Wang and X. Yang, Infinitely many solutions for a new Kirchhoff-type equation with subcritical exponent, Appl. Anal., published online: 25 May 2020.