Global existence and decay of solutions of a singular nonlocal viscoelastic system with damping terms

Nadia Mezouar, Salah Boulaaras

DOI: http://dx.doi.org/10.12775/TMNA.2020.014

Abstract


In this paper, a singular one-dimensional viscoelastic system with a nonlinear source term, nonlocal boundary condition and damping terms is considered. We prove the existence of a global solution using the potential-well theory. Furthermore, by constructing Lyapunov functional combined with the perturbed energy method, the general decay result is proved.

Keywords


Viscoelastic equations; global existence; general decay; damping terms

Full Text:

PREVIEW FULL TEXT

References


S. Boulaaras, R. Guefaifia and S. Kabli, An asymptotic behavior of positive solutions for a new class of elliptic systems involving of (p(x), q(x))-Laplacian systems, Bol. Soc. Mat. Mex. 25 (2019), 145–162.

D.M. Cahlon and P. Shi, Stepwise stability for the heat equation with a nonlocal constraint, SIAM. J. Numer. Anal. 32 (1995), 571–593.

R. Cannon, The solution of heat equation subject to the specification of energy, Quart. Appl. Math. 21 (1963), 155–160.

V. Capasso and K. Kunisch, A reaction-diffusion system arising in modeling manenvironment diseases, Quart. Appl. Math. 46 (1988), 431–449.

Y.S. Choi and K.Y. Chan, A parabolic equation with nonlocal boundary conditions arising from electrochemistry, Nonlinear Anal. 18 (1992), 317–331.

Mu. Chunlai and J. Ma, On a system of nonlinear wave equations with Balakrishnan–Taylor damping, Z. Angew. Math. Phys. 65 (2014), 91–113.

A. Draifia, A. Zarai and S. Boulaaras, Global existence and decay of solutions of a singular nonlocal viscoelastic system, Rend. Circ. Mat. Palermo Ser. II 69 (2020), 125–149.

R.E. Ewing and T. Lin, A class of parameter estimation techniques for fluid flow in porous media, Adv. Water Resour. 14 (1991), 89–97.

N.I. Ionkin, Solution of boundary value problem in heat conduction theory with nonclassical boundary conditions, Differ. Uravn. 13 (1977), 1177–1182.

N.I. Ionkin and E.I. Moiseev, A problem for the heat conduction equation with two-point boundary condition, Differ. Uravn. 15 (1979), 1284–1295.

M. Kafini and S. Messaoudi, A blow up result for a viscoelastic system in Rn , Electron. J. Differential Equations 7 (2007), 1–9.

L.I. Kamynin, A boundary-value problem in the theory of heat conduction with nonclassical boundary conditions Zh. Vycisl. Mat. Mat. Fiz. 4 (1964), 1006–1024. (in Russian)

A.V. Kartynnik, Three-point boundary value problem with an integral space-variable condition for a second order parabolic equation, Differential Equations 26 (1990), 1160–1162.

M.R. Li and L.Y. Tsai, Existence and nonexistence of global solutions of some systems of semilinear wave equations, Nonlinear Anal., 54 (2003), 1397–1415.

S. Mesloub and F. Mesloub, Solvability of a mixed nonlocal problem for a nonlinear singular viscoelastic equation, Acta. Appl. Math. 110 (2010), 109–129.

S. Mesloub and S. Messaoudi, Global existence, decay, and blow up of solutions of a singular nonlocal viscoelastic problem, Acta Appl. Math. 110 (2010), 705–724.

S. Messaoudi, Blow up and global existence in a nonlinear viscoelastic wave equation, Math. Nachr. 260 (2003), 58–66.

L.S. Pulkina, A nonlocal problem with integral conditions for hyperbolic equations, Electron. J. Differ. Equ. 45 (1999), 1–6.

L.S. Pulkina, On solvability in L2 of nonlocal problem with integral conditions for a hyperbolic equation, Differ. Uravn. 36 (2000), 316–318.

P. Shi, Weak solution to an evolution problem with a non local constraint, S.I.A.M. J. Math. Anal. 24 (1993), 46–58.

P. Shi and M. Shilor, Design of Contact Patterns in One Dimensional Thermoelasticity, Theoretical Aspects of Industrial Design, SIAM, Philadelphia, 1992.

S. Wu, Blow-up of solutions for a singular nonlocal viscoelastic equation, J. Partial Differential Equations 24 (2011), 140–149.

N.I. Yurchuk, Mixed problem with an integral condition for certain parabolic equations, Differ. Uravn. 22 (1986), 2117–2126.

A. Zarai, A. Draifia and S. Boulaaras, Blow up of solutions for a system of nonlocal singular viscoelatic equations, Appl. Anal. 97 (2018), 2231–2245.


Refbacks

  • There are currently no refbacks.

Partnerzy platformy czasopism