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

Year 2024, , 1085 - 1101, 27.08.2024

Morteza Koozehgar Kalleji

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.

Keywords

higher-order hyperbolic viscoelastic equations, singular potential wells of higher-order hyperbolic, corner Sobolev space, singularities, blow up

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.