Well-Posedness and asymptotic behavior of partly dissipative reaction diffusion systems with memory
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In this paper, we consider the asymptotic behavior of solutions for the partly dissipative reaction diffusion systems of the FitzHugh-Nagumo type with hereditary memory and a very large class of nonlinearities, which have no restriction on the upper growth of the nonlinearity. We first prove the existence and uniqueness of weak solutions to the initial boundary value problem for the above-mentioned model. Next, we investigate the existence of a uniform attractor of this problem, where the time-dependent forcing term $h \in L^2_b(\mathbb{R}; H^{-1}(\mathbb{R}^N))$ is the only translation bounded instead of translation compact. Finally, we prove the regularity of the uniform attractor $\mathcal{A}$, i.e., $\mathcal{A}$ is a bounded subset of $ H^2(\mathbb{R}^N)\times H^1(\mathbb{R}^N)\times L^2_\mu(\mathbb{R}^+, H^2(\mathbb{R}^N))$. The results in this paper will extend and improve some previously obtained results, which have not been studied before in the case of non-autonomous, exponential growth nonlinearity and contain memory kernels.
Abstract
In this paper, we consider the asymptotic behavior of solutions for the partly dissipative reaction diffusion systems of the FitzHugh-Nagumo type with hereditary memory and a very large class of nonlinearities, which have no restriction on the upper growth of the nonlinearity. We first prove the existence and uniqueness of weak solutions to the initial boundary value problem for the above-mentioned model. Next, we investigate the existence of a uniform attractor of this problem, where the time-dependent forcing term $h \in L^2_b(\mathbb{R}; H^{-1}(\mathbb{R}^N))$ is the only translation bounded instead of translation compact. Finally, we prove the regularity of the uniform attractor $\mathcal{A}$, i.e., $\mathcal{A}$ is a bounded subset of $ H^2(\mathbb{R}^N)\times H^1(\mathbb{R}^N)\times L^2_\mu(\mathbb{R}^+, H^2(\mathbb{R}^N))$. The results in this paper will extend and improve some previously obtained results, which have not been studied before in the case of non-autonomous, exponential growth nonlinearity and contain memory kernels.
참고문헌 (24)
- S. Borini, "Uniform attractors for a strongly damped wave equation with linear memory", Asymptot. Anal, 20(3-4), pp.263~277 (1999)
- G. Cantin, "Non-existence of the global attractor for a partly dissipative reaction-diffusion system with hysteresis", J. Differential Equations, 299, pp.333~361 (2021)[DOI]
- V. V. Chepyzhov, "Providence", American Mathematical Society Colloquium Publications, pp.49~ (2002), Amer. Math. Soc.
- R. FitzHugh, "Impulses and physiological states in theoretical models of nerve membrane", Biophys. J, 1, pp.445~466 (1961)
- S. Gatti, "Attractors for semi-linear equations of viscoelasticity with very low dissipation", Rocky Mountain J. Math, 38(4), pp.1117~1138 (2008)[DOI]
- P. G. Geredeli, "Long-time dynamics of the parabolic p-Laplacian equation", Commun. Pure Appl. Anal, 12(2), pp.735~754 (2013)[DOI]
- C. Giorgi, "Asymptotic behavior of a semilinear problem in heat conduction with memory", NoDEA Nonlinear Differential Equations Appl, 5(3), pp.333~354 (1998)[DOI]
- M. Grasselli, "Uniform attractors of nonautonomous dynamical systems with memory, in Evolution equations, semigroups and functional analysis (Milano, 2000)", Progr. Nonlinear Differential Equations Appl., 50, pp.155~178 (2002)
- M. Krasnoselskii, "Monographs and Textbooks on Mechanics of Solids and Fluids, Mechanics: Analysis" (1976), Noordhoff
- J. Lee, "Global attractors and exponential stability of partly dissipative reaction diffusion systems with exponential growth nonlinearity", Appl. Anal, 100(4), pp.735~751 (2021)[DOI]
- J. -L. Lions, "Die Grundlehren der mathematischen Wissenschaften, Band 181" (1972), Springer
- Y. Lu, "Determining nodes for partly dissipative reaction diffusion systems", Nonlinear Anal, 54(5), pp.873~884 (2003)[DOI]
- M. Marion, "Inertial manifolds associated to partly dissipative reaction-diffusion systems", J. Math. Anal. Appl, 143(2), pp.295~326 (1989)[DOI]
- M. Marion, "Finite-dimensional attractors associated with partly dissipative reactiondiffusion systems", SIAM J. Math. Anal, 20(4), pp.816~844 (1989)[DOI]
- J. Nagumo, "An active pulse transmission line simulating nerve axon", Proc. J. R. E, 50, pp.2061~2070 (1964)
- X. Pu, "A remark on partly dissipative reaction diffusion systems on Rn", Acta Math. Appl. Sin. Engl. Ser, 24(4), pp.583~588 (2008)[DOI]
- J. C. Robinson, "Infinite-Dimensional Dynamical Systems, Cambridge Texts in Applied Mathematics" (2001), Cambridge Univ. Press
- A. Rodr´ıguez-Bernal, "Attractors for partly dissipative reaction diffusion systems in Rn", J. Math. Anal. Appl, 252(2), pp.790~803 (2000)[DOI]
- Z. D. Shao, "Existence of inertial manifolds for partly dissipative reaction diffusion systems in higher space dimensions", J. Differential Equations, 144(1), pp.1~43 (1998)[DOI]
- Z. D. Shao, "Existence and continuity of strong solutions of partly dissipative reaction diffusion systems", Discrete Contin. Dyn. Syst, 2011, pp.1319~1328 (2011)
- W. A. Strauss, "On continuity of functions with values in various Banach spaces", Pacific J. Math, 19, pp.543~551 (1966)
- R. M. Temam, "Navier-Stokes equations and nonlinear functional analysis, second edition", CBMS-NSF Regional Conference Series in Applied Mathematics, pp.66~ (1995)
- L. T. Thuy, "Uniform attractors of nonclassical diffusion equations on RN with memory and singularly oscillating external forces", Math. Methods Appl. Sci, 44(1), pp.820~852 (2021)[DOI]
- M. Yang, "The existence and uniqueness of the solutions for partly dissipative reaction diffusion systems in Rn", J. Lanzhou Univ. Nat. Sci, 42(5), pp.130~136 (2006)