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Finite-time property of a mechanical viscoelastic system with nonlinear boundary conditions on corner-Sobolev spaces

Year 2024, Volume: 53 Issue: 4, 1085 - 1101, 27.08.2024
https://doi.org/10.15672/hujms.1286267

Abstract

In this article, we deal with the initial boundary value problem for a viscoelastic system related to the quasilinear parabolic equation with nonlinear boundary source term on a manifold $\mathbb{M}$ with corner singularities. We prove that, under certain conditions on relaxation function $g$, any solution $u$ in the corner-Sobolev space $\mathcal{H}^{1,(\frac{N-1}{2},\frac{N}{2})}_{\partial^{0}\mathbb{M}}(\mathbb{M})$ blows up in finite time. The estimates of the life-span of solutions are also given.

References

  • [1] M. Alimohamady, M.K. Kalleji and Gh. Karamali, Global results for semilinear hyperbolic equations with damping term on manifolds with conical singularity, Math. Meth. Appl. Sci. 40 (11), 4160–4178, 2017.
  • [2] D. Andrade, M.M. Cavalcanti, V.N. Domingos Cavalcanti and H. Portillo Oquendo, Existence and Asymptotic Stability for Viscoelastic Evolution Problems on Compact Manifolds, J. Comput. Anal. Appl. 8 (2), 173–193 2006.
  • [3] A.B. Al’shin and O.M. Korpusov, Blow-up in Nonlinear Sobolev Type Equations, in : De Gruyter Series in Nonlinear Analysis and Applications, Vol. 15, 2011.
  • [4] J. Ball, Remarks on blow up and nonexistence theorems for nonlinear evolutions equations, Q. J. Math. 28, 473–486, 1977.
  • [5] J.T. Beale, T. Kato and A.J. Majda, Remarks on the breakdown of smooth solutions for the 3D Euler equations, Commun. Math. Phys. 94 (61), 1984.
  • [6] M.M. Cavalcanti, V.N. Domingos Cavalcanti and J. Ferreira, Existence and uniform decay for nonlinear viscoelastic equation with strong damping, Math. Meth. Appl.Sci. 24, 1043–1053, 2001.
  • [7] M.M. Cavalcanti, V.N. Domingos Cavalcanti, J.S. P. Filho and J.A. Soriano, Existence and uniform decay rates for viscoelastic problems with nonlinear boundary damping , Differ. Integ. Equ. 14, 85–116, 2001.
  • [8] D.C. Chang, T. Qian and B.W. Schulze, Corner Boundary Value Problems, Complex Anal. Oper. Theory 9, 1157–1210, 2015.
  • [9] H. Chen and N. Liu, Asymptotic stability and Blow-up of solutions for semi-linear edge-degenerate parabolic equations with singular potentials, Discrete Contin. Dyn. Syst. 36 (2), 661–682, 2016.
  • [10] H. Chen, X. Liu and Y. Wei, Multiple solutions for semi-linear corner degenerate elliptic equations, J. Funct. Anal. 266, 3815–3839, 2014.
  • [11] B.C. Coleman and W. Noll, Foundations of linear viscoelasticity, Rev.Modern Phys. 33, 239–249, 1961.
  • [12] P. Constantin, C. Fefferman and A.J. Majda, Geometric constraints on potentially singular solutions for the 3D Euler equations, Comm. Partial Diff. Eqns. 21 (3-4), 1996.
  • [13] W.R. Dean, P.E. Montagnon, On the steady motion of viscous liquid in a corner, Math. Proc. Cambridge Philos. Soc. 45 (3), 389–394, 1949.
  • [14] H. Di, Y. Shang and X. Peng, Global existence and nonexistence of solutions for a viscoelastic wave equation with nonlinear boundary source term, Math. Nachr. 289 (11-12), 1408–1432, 2016.
  • [15] M. Fabrizio and A. Morro, Mathematical Problems in Linear Viscoelasticity, SIAM Studies in Applied Mathematics, Vol. 12, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1992.
  • [16] F. Gazzola and M. Squassina, Global solutions and finite time blow up for damped semilinear wave equations, Ann. Inst. H. Poincaré Anal. Non Lineaire 23, 185–207, 2006.
  • [17] Y. Guo, M.A. Rammaha, S. Sakuntasathien, E.S. Titi and D. Toundykov, Hadamard well-posedness for a hyperbolic equation of viscoelasticity with supercritical sources and damping, J. Differ. Equ. 257, 3778–3812, 2014.
  • [18] Y. Guo, M.A. Rammaha and S. Sakuntasathien, Blow-up of a hyperbolic equation of viscoelasticity withsupercritical nonlinearities, J. Differ. Equ. 262, 1956–1979, 2017.
  • [19] L. Guo, Z. Yuan and G. Lin, Blow Up and Global Existence for a Nonlinear Viscoelastic Wave Equation with Strong Damping and Nonlinear Damping and Source terms, Appl. Math. 6, 806–816, 2015.
  • [20] N. Irkil, E. Pişkin and P. Agarwal, Global existence and decay of solutions for a system of viscoelastic wave equations of Kirchhoff type with logarithmic nonlinearity, Math. Meth. Appl.Sci. 45, 2921–2948, 2022.
  • [21] M. Kafini and S.A. Messaoudi, A blow-up result for a viscoelastic system in $\mathbb{R}^{N}$, Electron. J. Differ. Equ. 2007, No. 113, 1–7, 2007.
  • [22] M. Kafini and S.A. Messaoudi, A blow-up result in a Cauchy viscoelastic problem, Appl. Math. Lett. 21, 549–553, 2008.
  • [23] M.K. Kalleji, Invariance and existence analysis of viscoelastic equations with nonlinear damping and source terms on corner singularity, Complex Var. Elliptic Equ. 67 (9), 2198–2225, 2022.
  • [24] J.A. Kim and Y.H. Han, Blow up of solutions of a nonlinear viscoelastic wave equation, Acta. Appl. Math. 111, 1–6, 2010.
  • [25] O.M. Korpursov, Non-existence of global solutions to generalized dissipative Klein- Gordon equations with positive energy, Electron. J. Differ. Equ. 2012, No. 119, 1–10, 2012.
  • [26] M.R. Li and L.Y. Tsai, Existence and nonexistence of global solutions of some systems of semilinear wave equations, Nonlinear Anal. 54, 1397–1415, 2003.
  • [27] J. Ma , C. Mu and R. Zeng, A blow up result for viscoelastic equations with arbitrary positive initial energy, Bound. Value Probl. 6, 2011.
  • [28] H.K. Moffatt and Y. Kimura , Towards a finite-time singularity of the Navier-Stokes equations. Part 1. Derivation and analysis of dynamical system, J. Fluid Mech. 861, 930–967, 2019.
  • [29] N.I. Muskhelishvili, Some Basic Problems of the Mathematical Theory of Elasticity, P. Noordroff Ltd., Groningen, Holland, 1953.
  • [30] N.I. Muskhelishvili, Singular Integral Equations, P. Noordroff Ltd., Groningen, Holland, 1953.
  • [31] E. Pişkin and A. Fidan, Blow up of solutins for viscoelastic wave equations of Kirchhoff type with arbitrary positive inital energy, Electron. J. Differ. Equ. 2017, No. 242, 1-10, 2017.
  • [32] J.W.S. Rayleigh, Scientific Papers, Vol. 6 (18), Cambridge University Press, 1920.
  • [33] W. Rungrottheera and B.W. Schulze, Weighted spaces on corner manifolds, Complex Var. Elliptic Equ. 59 (12), 1706–1738, 2014.
  • [34] E. Vitillaro, Global nonexistence theorems for a class of evolution equations with dissipation, Arch. Ration. Mech. Anal. 149, 155–182, 1999.
  • [35] S.T. Wu, Blow-up results for systems of nonlinear Klein-Gordon equations with arbitrary positive initial energy, Electron. J. Differ. Equ. 2012, No. 92, 1–13, 2012.
  • [36] G. Xu and J. Zhou, Upper bounds of blow-up time and blow-up rate for a semi-linear edge-degenerate parabolic equation, Appl. Math. Lett. 73, 1–7, 2017.
  • [37] Y. Zhou, Global existence and nonexistence for a nonlinear wave equation with damping and source terms, Math. Nachr. 278, 1341–1358, 2005.
Year 2024, Volume: 53 Issue: 4, 1085 - 1101, 27.08.2024
https://doi.org/10.15672/hujms.1286267

Abstract

References

  • [1] M. Alimohamady, M.K. Kalleji and Gh. Karamali, Global results for semilinear hyperbolic equations with damping term on manifolds with conical singularity, Math. Meth. Appl. Sci. 40 (11), 4160–4178, 2017.
  • [2] D. Andrade, M.M. Cavalcanti, V.N. Domingos Cavalcanti and H. Portillo Oquendo, Existence and Asymptotic Stability for Viscoelastic Evolution Problems on Compact Manifolds, J. Comput. Anal. Appl. 8 (2), 173–193 2006.
  • [3] A.B. Al’shin and O.M. Korpusov, Blow-up in Nonlinear Sobolev Type Equations, in : De Gruyter Series in Nonlinear Analysis and Applications, Vol. 15, 2011.
  • [4] J. Ball, Remarks on blow up and nonexistence theorems for nonlinear evolutions equations, Q. J. Math. 28, 473–486, 1977.
  • [5] J.T. Beale, T. Kato and A.J. Majda, Remarks on the breakdown of smooth solutions for the 3D Euler equations, Commun. Math. Phys. 94 (61), 1984.
  • [6] M.M. Cavalcanti, V.N. Domingos Cavalcanti and J. Ferreira, Existence and uniform decay for nonlinear viscoelastic equation with strong damping, Math. Meth. Appl.Sci. 24, 1043–1053, 2001.
  • [7] M.M. Cavalcanti, V.N. Domingos Cavalcanti, J.S. P. Filho and J.A. Soriano, Existence and uniform decay rates for viscoelastic problems with nonlinear boundary damping , Differ. Integ. Equ. 14, 85–116, 2001.
  • [8] D.C. Chang, T. Qian and B.W. Schulze, Corner Boundary Value Problems, Complex Anal. Oper. Theory 9, 1157–1210, 2015.
  • [9] H. Chen and N. Liu, Asymptotic stability and Blow-up of solutions for semi-linear edge-degenerate parabolic equations with singular potentials, Discrete Contin. Dyn. Syst. 36 (2), 661–682, 2016.
  • [10] H. Chen, X. Liu and Y. Wei, Multiple solutions for semi-linear corner degenerate elliptic equations, J. Funct. Anal. 266, 3815–3839, 2014.
  • [11] B.C. Coleman and W. Noll, Foundations of linear viscoelasticity, Rev.Modern Phys. 33, 239–249, 1961.
  • [12] P. Constantin, C. Fefferman and A.J. Majda, Geometric constraints on potentially singular solutions for the 3D Euler equations, Comm. Partial Diff. Eqns. 21 (3-4), 1996.
  • [13] W.R. Dean, P.E. Montagnon, On the steady motion of viscous liquid in a corner, Math. Proc. Cambridge Philos. Soc. 45 (3), 389–394, 1949.
  • [14] H. Di, Y. Shang and X. Peng, Global existence and nonexistence of solutions for a viscoelastic wave equation with nonlinear boundary source term, Math. Nachr. 289 (11-12), 1408–1432, 2016.
  • [15] M. Fabrizio and A. Morro, Mathematical Problems in Linear Viscoelasticity, SIAM Studies in Applied Mathematics, Vol. 12, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1992.
  • [16] F. Gazzola and M. Squassina, Global solutions and finite time blow up for damped semilinear wave equations, Ann. Inst. H. Poincaré Anal. Non Lineaire 23, 185–207, 2006.
  • [17] Y. Guo, M.A. Rammaha, S. Sakuntasathien, E.S. Titi and D. Toundykov, Hadamard well-posedness for a hyperbolic equation of viscoelasticity with supercritical sources and damping, J. Differ. Equ. 257, 3778–3812, 2014.
  • [18] Y. Guo, M.A. Rammaha and S. Sakuntasathien, Blow-up of a hyperbolic equation of viscoelasticity withsupercritical nonlinearities, J. Differ. Equ. 262, 1956–1979, 2017.
  • [19] L. Guo, Z. Yuan and G. Lin, Blow Up and Global Existence for a Nonlinear Viscoelastic Wave Equation with Strong Damping and Nonlinear Damping and Source terms, Appl. Math. 6, 806–816, 2015.
  • [20] N. Irkil, E. Pişkin and P. Agarwal, Global existence and decay of solutions for a system of viscoelastic wave equations of Kirchhoff type with logarithmic nonlinearity, Math. Meth. Appl.Sci. 45, 2921–2948, 2022.
  • [21] M. Kafini and S.A. Messaoudi, A blow-up result for a viscoelastic system in $\mathbb{R}^{N}$, Electron. J. Differ. Equ. 2007, No. 113, 1–7, 2007.
  • [22] M. Kafini and S.A. Messaoudi, A blow-up result in a Cauchy viscoelastic problem, Appl. Math. Lett. 21, 549–553, 2008.
  • [23] M.K. Kalleji, Invariance and existence analysis of viscoelastic equations with nonlinear damping and source terms on corner singularity, Complex Var. Elliptic Equ. 67 (9), 2198–2225, 2022.
  • [24] J.A. Kim and Y.H. Han, Blow up of solutions of a nonlinear viscoelastic wave equation, Acta. Appl. Math. 111, 1–6, 2010.
  • [25] O.M. Korpursov, Non-existence of global solutions to generalized dissipative Klein- Gordon equations with positive energy, Electron. J. Differ. Equ. 2012, No. 119, 1–10, 2012.
  • [26] M.R. Li and L.Y. Tsai, Existence and nonexistence of global solutions of some systems of semilinear wave equations, Nonlinear Anal. 54, 1397–1415, 2003.
  • [27] J. Ma , C. Mu and R. Zeng, A blow up result for viscoelastic equations with arbitrary positive initial energy, Bound. Value Probl. 6, 2011.
  • [28] H.K. Moffatt and Y. Kimura , Towards a finite-time singularity of the Navier-Stokes equations. Part 1. Derivation and analysis of dynamical system, J. Fluid Mech. 861, 930–967, 2019.
  • [29] N.I. Muskhelishvili, Some Basic Problems of the Mathematical Theory of Elasticity, P. Noordroff Ltd., Groningen, Holland, 1953.
  • [30] N.I. Muskhelishvili, Singular Integral Equations, P. Noordroff Ltd., Groningen, Holland, 1953.
  • [31] E. Pişkin and A. Fidan, Blow up of solutins for viscoelastic wave equations of Kirchhoff type with arbitrary positive inital energy, Electron. J. Differ. Equ. 2017, No. 242, 1-10, 2017.
  • [32] J.W.S. Rayleigh, Scientific Papers, Vol. 6 (18), Cambridge University Press, 1920.
  • [33] W. Rungrottheera and B.W. Schulze, Weighted spaces on corner manifolds, Complex Var. Elliptic Equ. 59 (12), 1706–1738, 2014.
  • [34] E. Vitillaro, Global nonexistence theorems for a class of evolution equations with dissipation, Arch. Ration. Mech. Anal. 149, 155–182, 1999.
  • [35] S.T. Wu, Blow-up results for systems of nonlinear Klein-Gordon equations with arbitrary positive initial energy, Electron. J. Differ. Equ. 2012, No. 92, 1–13, 2012.
  • [36] G. Xu and J. Zhou, Upper bounds of blow-up time and blow-up rate for a semi-linear edge-degenerate parabolic equation, Appl. Math. Lett. 73, 1–7, 2017.
  • [37] Y. Zhou, Global existence and nonexistence for a nonlinear wave equation with damping and source terms, Math. Nachr. 278, 1341–1358, 2005.
There are 37 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Mathematics
Authors

Morteza Koozehgar Kalleji 0000-0002-7052-7241

Early Pub Date January 10, 2024
Publication Date August 27, 2024
Published in Issue Year 2024 Volume: 53 Issue: 4

Cite

APA Koozehgar Kalleji, M. (2024). Finite-time property of a mechanical viscoelastic system with nonlinear boundary conditions on corner-Sobolev spaces. Hacettepe Journal of Mathematics and Statistics, 53(4), 1085-1101. https://doi.org/10.15672/hujms.1286267
AMA Koozehgar Kalleji M. Finite-time property of a mechanical viscoelastic system with nonlinear boundary conditions on corner-Sobolev spaces. Hacettepe Journal of Mathematics and Statistics. August 2024;53(4):1085-1101. doi:10.15672/hujms.1286267
Chicago Koozehgar Kalleji, Morteza. “Finite-Time Property of a Mechanical Viscoelastic System With Nonlinear Boundary Conditions on Corner-Sobolev Spaces”. Hacettepe Journal of Mathematics and Statistics 53, no. 4 (August 2024): 1085-1101. https://doi.org/10.15672/hujms.1286267.
EndNote Koozehgar Kalleji M (August 1, 2024) Finite-time property of a mechanical viscoelastic system with nonlinear boundary conditions on corner-Sobolev spaces. Hacettepe Journal of Mathematics and Statistics 53 4 1085–1101.
IEEE M. Koozehgar Kalleji, “Finite-time property of a mechanical viscoelastic system with nonlinear boundary conditions on corner-Sobolev spaces”, Hacettepe Journal of Mathematics and Statistics, vol. 53, no. 4, pp. 1085–1101, 2024, doi: 10.15672/hujms.1286267.
ISNAD Koozehgar Kalleji, Morteza. “Finite-Time Property of a Mechanical Viscoelastic System With Nonlinear Boundary Conditions on Corner-Sobolev Spaces”. Hacettepe Journal of Mathematics and Statistics 53/4 (August 2024), 1085-1101. https://doi.org/10.15672/hujms.1286267.
JAMA Koozehgar Kalleji M. Finite-time property of a mechanical viscoelastic system with nonlinear boundary conditions on corner-Sobolev spaces. Hacettepe Journal of Mathematics and Statistics. 2024;53:1085–1101.
MLA Koozehgar Kalleji, Morteza. “Finite-Time Property of a Mechanical Viscoelastic System With Nonlinear Boundary Conditions on Corner-Sobolev Spaces”. Hacettepe Journal of Mathematics and Statistics, vol. 53, no. 4, 2024, pp. 1085-01, doi:10.15672/hujms.1286267.
Vancouver Koozehgar Kalleji M. Finite-time property of a mechanical viscoelastic system with nonlinear boundary conditions on corner-Sobolev spaces. Hacettepe Journal of Mathematics and Statistics. 2024;53(4):1085-101.