Asymptotic Behavior of the Non-Autonomous Reaction-Diffusion Equation with Robin Boundary Condition

Year 2019, Volume: 68 Issue: 1, 422 - 440, 01.02.2019

Eylem Öztürk

https://doi.org/10.31801/cfsuasmas.425500

Abstract

In this paper, we investigate the long-time behavior of the time-dependent reaction-diffusion equation u_{t}-Δu+a(x)|u|^{ρ}u-b(x)|u|^{ν}u=h(x,t) with Robin boundary condition. We begin this paper with the existence and uniqueness results of the solution to the problem. For the asymptotic behavior, we firstly prove the existence of an absorbing set in W₂¹(Ω)∩L_{ρ+2}(Ω). The existence of a uniform attractor is obtained in W₂¹(Ω)∩L_{ρ+2}(Ω).

Keywords

Reaction-diffusion equation, Robin boundary value, existence and uniqueness theorems, uniform attractor, process

References

• Arrieta, J., Carvalho, A.N. and Bernal, A.R., Attractors of parabolic problems with nonlinear boundary conditions. Uniform bounds, Comm. Partial Differential equations 25 (1/2), 2000, 1-37.
• Babin, A.V. and Vishik, M.I., Regular attractors of semigroups and evolution equations. J. Math. Pures Appl.(9)62 (1983) no.4 441-491
• Babin, A.V. and Vishik, M.I., Attractors of Evolution Equations, North-Holland, Amsterdam, 1992.
• Bernal, A. R. and Tajdine, A., Nonlinear balance for reaction-diffusion equations under Nonlinear boundary conditions: Dissipativity and Blow up, Journal of Differential Equations 169, 332-372, 2001.
• Chen, G. and Zhong, C.K., Uniform attractors for non-autonomous p-Laplacian equation, Nonlinear Analysis 68, 3349-3363, 2008.
• Chepyzhov, V.V. and Vishik, M.I., Attractors for Equations of Mathematical Physics, Amer. Math. Soc., Providence, RI, 2002.
• Chepyzhov, V.V. and Vishik, M.I., Attractors of non-autonomous dynamical systems and their dimension, J. Math. Pures Appl., 73(1994) 279-333.
• Chepyzhov, V.V. Vishik, M.I., Non-autonomous evolutionary equation with translation-compact symbols and their attractors, C.R. Acad. Sci. Paris Ser. I, 321(1995) 153-158.
• Hale, J. and Rocha, C., Interaction of diffusion and boundary conditions, Nonlinear Anal. 11(5), 633-649 (1987)
• Haraux, A., Systemes dynamiques dissipatifs et applications, Paris, Masson, 1991.
• Ladyzhenskaya, O.A., On the determination of minimal global attractors for the Navier-Stokes equations and other partial differential equations, Uspekhi Mat. Nauk 42 (6) (1987) 25-60, Russian Math. Surveys 42 (6) (1987) 27-73 (English Transl.)
• Ladyzhenskaya, O.A., Attractors for Semigroups and Evolution Equations, Leizioni Lincei, Cambridge University Press, Cambridge, New York, 1991.
• Lions, J. L., Quelques methodes de resolution des problémes aux Limities non lineaires, Dunod, Gauthier-Villars, Paris (1969).
• Lu, S.S., Attractors for non-autonomous reaction-diffusion systems with symbols without strong translation compactness, Asymptot. Anal. 54 (2007) 197-210.
• Marion, M., Attractors for reactions-diffusion equations: existence and estimate of their dimension, Appl. Anal. 25 (1987) 101-147.
• Öztürk, E. and Soltanov, K.N., Solvability and long time behavior of nonlinear reaction-diffusion equations with Robin boundary condition, Nonlinear Analysis, Theory Methods Application, Volume:108, 1-13, (2014). Öztürk, E. and Kalli, K., Existence and non-existence of solutions of Reaction-Diffusion equation with Robin boundary condition, Ukrainian Mathematical Journal 68 (3), October 2016.
• Öztürk, E. and Soltanov, K. N., On Some Yamabe Type Equations, AIP Conference Proceedings (2009) Volume: 1168 Pages: 377-380
• Öztürk, E. and Soltanov, K. N., Robin and Initial Value Problem for a Yamabe Type Parabolic Equation, The 8th Congress of the ISAAC 2011.
• Payne, L.E. and Schaefer, P.W., Blow-up in parabolic problems under Robin boundary conditions, Applicable Analysis Vol. 87, No. 6, 699-707, (2008).
• Payne, L.E. and Philippin, G.A., Blow-up in a class of non-linear parabolic problems with time-dependent coefficients under Robin type boundary conditions, Applicable Analysis Vol. 91, No. 12, 2245-2256 (2012).
• Rault, J. F.,The Fujita phenomenon in exterior domains under the Robin boundary conditions, C. R. Sci. Paris, Ser. I 349(2011) 1059-1061.
• Robinson, J.C., Rodríguez-Bernal, A. and Vidal-Lopez, A., Pullback attractors and extremal complete trajectories for non-autonomous reaction-diffusion problems, J. Differential Equations 238 (2007) 289-337.
• Robinson, J. C., Infinite-Dimensional Dynamical Systems, Cambridge University Press, Cambridge, (2001).
• Sell, G. R. and You, Y., Dynamics of Evolutionary Equations, Springer, New York, 2002.
• Song, H.T. Ma, S. and Zhong, C.K., Attractors of non-autonomous reaction-diffusion equations, Nonlinearity 22 (2009) 667-681.
• Song, H.T., and Zhong, C.K., Attractors of non-autonomous reaction-diffusion equations in Lp, Nonlinear Anal. 68 (2008) 1890-1897.
• Sun, C., Asymptotic regularity for some dissipative equations, J. Differential Equations 248 (2010), 342-362
• Temam, R., Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1998. Yang, L. and Yang, M.H., Attractors of non-autonomous reaction diffusion equation with nonlinear boundary condition, Nonlinear Analysis: Real World Applic. 11(2010)3946-3954.
• Yang, L., Asymptotic regularity and attractors of the reaction-diffusion equation with nonlinear boundary condition, Nonlinear Analysis: Real World Applic. 13(2012)1069-1079
• Zhong, C.K., Yang, M.H. and Sun, C.Y., The existence of global attractors for the norm-to-weak continuous semigroup and application to the nonlinear reaction-diffusion equations, J. Differential Equations 223 (2006) 367-399.