A Nonlinear $r(x)$-Kirchhoff Type Hyperbolic Equation: Stability Result and Blow up of Solutions with Positive Initial Energy

Year 2021, Volume: 4 Issue: 4, 208 - 216, 27.12.2021

Mohammad Shahrouzi Jorge Ferreıra

https://doi.org/10.33434/cams.941324

Cited By: 1

Abstract

In this paper we consider $r(x)-$Kirchhoff type equation with variable-exponent nonlinearity of the form $$ u_{tt}-\Delta u-\big(a+b\int_{\Omega}\frac{1}{r(x)}|\nabla u|^{r(x)}dx\big)\Delta_{r(x)}u+\beta u_{t}=|u|^{p(x)-2}u, $$ associated with initial and Dirichlet boundary conditions. Under appropriate conditions on $r(.)$ and $p(.)$, stability result along the solution energy is proved. It is also shown that regarding arbitrary positive initial energy and suitable range of variable exponents, solutions blow-up in a finite time.

Keywords

Kirchhoff equation, stability result, variable exponents, blow-up

References

• [1] G. Kirchhoff, Vorlesungen uber Mechanik, Teubner, Leipzig , (1883).
• [2] T. Matsuyama, R. Ikehata, On global solutions and energy decay for the wave equations of Kirchhoff type with nonlinear damping terms, J. Math. Anal. Appl. 204, 729–753 (1996).
• [3] K. Ono, On global solutions and blow-up solutions of nonlinear Kirchhoff strings with nonlinear dissipation, J. Math. Anal. Appl. 216, 321–342 (1997).
• [4] E. Pis¸kin, Existence, decay and blow up of solutions for the extensible beam equation with nonlinear damping and source terms, Open Math. 13, 408–420 (2015).
• [5] J. Zhou, Global existence and blow-up of solutions for a Kirchhoff type plate equation with damping, Appl. Math. Comput. 265, 807–818 (2015).
• [6] R. Ikehata, A note on the global solvability of solutions to some nonlinear wave equations with dissipative terms, Differ. Integral Equ. 8, 607–616 (1995).
• [7] S. A. Messaoudi, Blow-up of solutions for the Kirchhoff equation of q􀀀Laplacian type with nonlinear dissipation, Colloq. Math. 94, 103–109 (2002).
• [8] K. Ono, On global existence, asymptotic stability and blowing up of solutions for some degenerate nonlinear wave equations of Kirchhoff type with a strong dissipation, Math. Meth. Appl. Sci. 20, 151–177 (1997).
• [9] C. O. Alves, F. J. S. A. Corr ˆ ea, T. F. Ma, Positive Solutions for a Quasilinear Elliptic Equation of Kirchhoff Type, Comput. Math. Appl. 49, 85–93 (2005).
• [10] A. Yang, Z. Gong, Blow-up of solutions for some nonlinear wave equations of Kirchhoff type with arbitrary positive initial energy, Electron. J. Differ. Equ. 332, 1–8 (2016).
• [11] M. Shahrouzi, F. Kargarfard, Blow-up of solutions for a Kirchhoff type equation with variable-exponent nonlinearities, J. App. Anal. 27(1), 97–105 (2021).
• [12] S. N. Antontsev, J. Ferreira, E. Pis¸kin, S. M. S. Cordeiro, Existence and non-existence of solutions for Timoshenko-type equations with variable exponents, Nonlinear Anal. Real World Appl. 61, 103341, (2021).
• [13] E. Pis¸kin, Finite time blow up of solutions of the Kirchhoff-type equation with variable exponents, Int. J. Nonlinear Anal. Appl. 11(1), 37–45 (2020).
• [14] M. Shahrouzi, Blow up of solutions for a r(x)􀀀Laplacian Lam´e equation with variable-exponent nonlinearities and arbitrary initial energy level, Int. J. Nonlinear Anal. Appl. 13(1), 441–450 (2022).
• [15] M. Shahrouzi, General decay and blow up of solutions for a class of inverse problem with elasticity term and variableexponent nonlinearities, Math. Meth. Appl. Sci. 2021. DOI: https://doi.org/10.1002/mma.7891.
• [16] G. Dai, R. Hao, Existence of solutionsfor a p(x)􀀀Kirchhoff-type equation, J. Math. Anal. Appl. 359, 275–284 (2009). [17] M. K. Hamdani, A. Harrabi, F. Mtiri, D. D. Repovˇe, Existence and multiplicity results for a new p(x)􀀀Kirchhoff problem, Nonlinear Analysis 190, 111598 (2020).
• [18] G. Dai, R. Ma, Solutions for a p(x)􀀀Kirchhoff type equation with Neumann boundary data, Nonlinear Analysis: Real World Applications 12 , 2666–2680 (2011).
• [19] A. Ghanmi, Nontrivial solutions for Kirchhoff-type problems involving the p(x)􀀀Laplace operator, Rocky Mount. J. Math. 48(4), 1145–1158 (2018).
• [20] E. J. Hurtado, O. H. Miyagaki, R. D. S. Rodrigues, Existence and Asymptotic Behaviour for a Kirchhoff Type Equation With Variable Critical Growth Exponent, Milan J. Math. 85, 71–102 (2017).
• [21] S. Heidarkhani, A. L. A. De Araujo, G. A. Afrouzi, S. Moradi, Multiple solutions for Kirchhoff-type problems with variable exponent and nonhomogeneous Neumann conditions, Math. Nachr. 291(2), 326–342 (2018).
• [22] M. Shahrouzi, On behaviour of solutions for a nonlinear viscoelastic equation with variable-exponent nonlinearities, Comp. Math. Appl. 75, 3946–3956 (2018).
• [23] M. Shahrouzi, Global nonexistence of solutions for a class of viscoelastic Lam´e system, Indian J. Pure Appl. Math. 51(4), 1383–1397 (2020).
• [24] L. Diening, P. Hasto, et al. Lebesgue and Sobolev spaces with variable exponents, Springer-Verlag, (2011).
• [25] D. Edmunds, J. Rakosnik, Sobolev embeddings with variable exponent, Stud. Math. 143, 267–293 (2000).
• [26] D. Edmunds, J. Rakosnik, Sobolev embeddings with variable exponent II, Math. Nachr. 246, 53–67 (2002).
• [27] X. Fan, D. Zhao, On the spaces Lp(x) andWm;p(x)(W), J. Math. Anal. Appl. 263, 424–446 (2001).
• [28] X. Fan, Q. H. Zhang, Existence of solutions for p(x)􀀀Laplacian Dirichlet problem, Nonlinear Anal. 52, 1843–1852 (2003).