Blow-up of Solutions for a Problem with Balakrishnan-Taylor Damping and Nonlocal Singular Viscoelastic Equations
Abstract
In this paper, we study the nonlinear one-dimensional viscoelastic nonlocal problem with Balakrishnan-Taylor damping terms and nonlinear source of polynomial type. We demonstrate that the nonlinear source of polynomial type is able to force solutions to blow up infinite time even in presence of stronger damping with non positive initial energy combined with a positive initial energy.
References
Cahlon, D., & Shi, P. (1995). Stepwise stability for the heat equation with a nonlocal constraint. SIAM Journal on Numerical Analysis, 32, 571-593. https://doi.org/10.1137/0732025
Mesloub, S., & Lekrine, N. (2004). On a nonlocal hyperbolic mixed problem. Acta Scientiarum Mathematicarum (Szeged), 70, 65-75.
Ewing, R., & Lin, T. (1991). A class of parameter estimation techniques for fluid flow in porous media. Advances in Water Resources, 14, 89-97. https://doi.org/10.1016/0309-1708(91)90055-S
Shi, P. (1993). Weak solution to an evolution problem with a non-local constraint. SIAM Journal on Mathematical Analysis, 24(1), 46-58. https://doi.org/10.1137/0524004
Choi, Y., & Chan, K. (1992). A parabolic equation with nonlocal boundary conditions arising from electrochemistry. Nonlinear Analysis, 18, 317-331. https://doi.org/10.1016/0362-546X(92)90148-8
Cannon, R. (1963). The solution of the heat equation subject to the specification of energy. Quarterly of Applied Mathematics, 21, 155-160. https://doi.org/10. 1090/qam/160437
Capasso, V., & Kunisch, K. (1988). A reaction-diffusion system arising in modeling man-environment diseases. Quarterly of Applied Mathematics, 46, 431-449. https://doi.org/10.1090/qam/963580
Yurchuk, N. (1986). Mixed problem with an integral condition for certain parabolic equations. Differentsial'nye Uravneniya, 22(19), 2117-2126.
Shi, P., & Shilor, M. (1992). Design of contact patterns in one-dimensional thermoelasticity. In Theoretical aspects of industrial design. Philadelphia: SIAM.
Ionkin, N., & Moiseev, E. I. (1979). A problem for the heat conduction equation with two-point boundary condition. Differentsial'nye Uravneniya, 15(7), 1284-1295.
Kamynin, L. (1964). A boundary-value problem in the theory of heat conduction with non-classical boundary conditions. (Russian) Zhurnal Vychislitel'noi Matematiki i Matematicheskoi Fiziki, 4, 1006-1024.
Mesloub, S. (2008). On a singular two-dimensional nonlinear evolution equation with nonlocal conditions. Nonlinear Analysis, 68, 2594-2607. https://doi.org/10.1016/j.na.2007.02.006
Mesloub, S. (2006). A nonlinear nonlocal mixed problem for a second-order parabolic equation. Journal of Mathematical Analysis and Applications, 316, 189-209. https://doi.org/10.1016/j.jmaa.2005.04.072
Ionkin, N. (1977). Solution of boundary value problem in heat conduction theory with nonclassical boundary conditions. Differentsial'nye Uravneniya, 13(2), 1177-1182.
Mesloub, S., & Messaoudi, S. (2010). Global existence, decay, and blow up of solutions of a singular nonlocal viscoelastic problem. Acta Applicandae Mathematicae, 110, 705-724. https://doi.org/10.1007/s10440-009-9469-6
Mesloub, S., & Messaoudi, S. (2003). A nonlocal mixed semilinear problem for second-order hyperbolic equations. Electronic Journal of Differential Equations, 2003(30), 1-17.
Kartynnik, A. (1990). Three-point boundary value problem with an integral space-variable condition for a second-order parabolic equation. Differential Equations, 26, 1160-1162.
Pulkina, L. (1999). A non-local problem with integral conditions for hyperbolic equations. Electronic Journal of Differential Equations, 45, 1-6.
Pul'kina, L. (2000). The L² solvability of a nonlocal problem with integral conditions for a hyperbolic equation. Diff. Equat., 36, 316-318. https://doi.org/10.1007/BF02754219
Mesloub, S., & Bouziani, A. (1998). Problème mixte avec conditions aux limites intégrales pour une classe d'équations paraboliques bidimensionnelles. Bulletin de la Classe des Sciences. Académie Royale de Belgique, 6, 59-69. https://doi.org/10. 3406/barb.1998.27880
Mesloub, S., Bouziani, A. (2002). Mixed problem with a weighted integral condition for a parabolic equation with Bessel operator. Journal of Applied Mathematics and Stochastic Analysis, 15(3), 291-300. https://doi.org/10.1155/S1048953302000242
Mu, C., & Ma, J. (2014). On a system of nonlinear wave equations with Balakrishnan-Taylor damping. Z. Angew. Math. Phys., 65, 91-113. https://doi.org/10.1007/s00033-013-0324-2
Tatar, N. & Zaraï, A. (2011). Exponential stability and blow up for a problem with Balakrishnan-Taylor damping. Demonstratio Mathematica, 44(1), 67-90. https://doi.org/10.1515/dema-2013-0297
Zaraï, A., Tatar, N., & Abdelmalek, S. (2013). Elastic membrane equation with memory term and nonlinear boundary damping: Global existence, decay, and blow-up of the solution. Acta Mathematica Scientia B, 33(1), 84-106. https://doi.org/10.1016/S0252-9602(12)60196-9
Mesloub, S., & Mesloub, F. (2010). Solvability of a mixed nonlocal problem for a nonlinear singular viscoelastic equation. Acta Applicandae Mathematicae, 110, 109-129. https://doi.org/10.1007/s10440-008-9388-y
Wu, S. (2011). Blow-up of solutions for a singular nonlocal viscoelastic equation. Journal of Partial Differential Equations, 24(2), 140-149. https://doi.org/10. 4208/jpde.v24.n2.3
Liu, W., Sun, Y., & Li, G. (2017). On decay and blow-up of solutions for a singular nonlocal viscoelastic problem with a nonlinear source term. Topol. Methods Nonlinear Anal., 49(1), 299-323. https://doi.org/10.12775/TMNA.2016.077
Zarai, A., Draifia, A., & Boulaaras, S. (2018). Blow-up of solutions for a system of nonlocal singular viscoelastic equations. Applicable Analysis, 97, 2231-2245.
Li, M. R., & Tsai, L. (2003). Existence and nonexistence of global solutions of some systems of semilinear wave equations. Nonlinear Analysis: Theory, Methods & Applications, 54, 1397-1415. https://doi.org/10.1016/S0362-546X(03)00192-5
Draifia, A., Zarai, A. & Boulaaras, S. (2020). Global existence and decay of solutions of a singular nonlocal viscoelastic system. Rend. Circ. Mat. Palermo, II. Ser 69, 125-149. https://doi.org/10.1007/s12215-018-00391-z
Boulaaras, S., Draifia, A., & Alnegga, M. (2019). Polynomial decay rate for Kirchhoff type in viscoelasticity with logarithmic nonlinearity and not necessarily decreasing kernel. Symmetry, 11(2), 226. https://doi.org/10.3390/sym11020226
This work is licensed under a Creative Commons Attribution 4.0 International License.
.jpg){kind=logo}
