Abstract
In this paper we study the bi-spatial dynamics of non-autonomous parabolic equations with nonlinear Laplacian on \(n+1\)-dimensional unbounded thin domains, where the nonlinearity has a (p, q)-growth exponents. We first prove the existence and uniqueness of tempered pullback attractors in \( L^2\) on \(n+1\)-dimensional unbounded thin domains, and then obtain the upper semi-continuity of these attractors in \(L^2\) when the \(n+1\)-dimensional thin domains degenerates onto a n-dimensional entire space \({\mathbb {R}}^n\). Finally by borrowing an inductive method and a bootstrap technique, we show that the difference of solutions near the initial time is higher-order integrable for any space dimension \(n\ge 1\), which further shows the existence of pullback attractors in \( L^p\cap L^q\). Based on the higher-order integrability, we find that the obtained pullback attractor is attracting under the topology of \(L^\delta \) with \(\delta \in [2,\infty )\).
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References
Antoci, F., Prizzi, M.: Reaction-diffusion equations on unbounded thin domains. Topol. Methods Nonlinear Anal. 18, 283–302 (2001)
Arrieta, J.M., Carvalho, A.N., Lozada-Cruz, G.: Dynamics in dumbbell domains III. Continuity of attractors. J. Differ. Equ. 247, 225–259 (2009)
Arrieta, J.M., Carvalho, A.N., Silva, R.P., Pereira, M.C.: Semilinear parabolic problems in thin domains with a highly oscillatory boundary. Nonlinear Anal. 74, 5111–5132 (2011)
Arrieta, J.M., Nakasato, J.C., Pereira, M.C.: The \(p\)-Laplacian equation in thin domains: the unfolding approach. J. Differ. Equ. 274, 1–34 (2021)
Arrieta, J.M., Villanueva-Pesqueira, M.: Elliptic and parabolic problems in thin domains with doubly oscillatory boundary. Commun. Pure Appl. Anal. 19, 1891–1914 (2020)
Cao, D., Sun, C., Yang, M.: Dynamics for a stochastic reaction-diffusion equation with additive noise. J. Differ. Equ. 259, 838–872 (2015)
Caraballo, T., Chueshov, I.D., Kloeden, P.E.: Synchronization of a stochastic reaction-diffusion system on a thin two-layer domain. SIAM J. Math. Anal. 38, 1489–1507 (2007)
Carvalho, A.N., Langa, J.A., Robinson, J.C.: Attractor for Infinite-Dimensional Nonautonomous Dynamical Systems. In: Appl. Math. Sciences, vol. 182. Springer, Berlin (2013)
Chen, P., Zhang, X., Zhang, X.: Asymptotic behavior of non-autonomous fractional stochastic p-Laplacian equations with delay on \({\mathbb{R} }^n\). J. Dyn. Differ. Equ. (2021). https://doi.org/10.1007/s10884-021-10076-4
Chueshov, I.: Monotone Random Systems Theory and Applications. Springer-Verlag, Berlin (2002)
Elsken, T.: Attractors for reaction-diffusion equations on thin domains whose linear part is non-self-adjoint. J. Differ. Equ. 206, 94–126 (2004)
Gess, B., Liu, W., Rockner, M.: Random attractors for a class of stochastic partial differential equations driven by general additive noise. J. Differ. Equ. 251, 1225–1253 (2011)
Hale, J.K., Raugel, G.: Reaction-diffusion equation on thin domains. J. Math. Pure. Appl. 71, 33–95 (1992)
Hale, J.K., Raugel, G.: A damped hyperbolic equation on thin domains. Trans. Am. Math. Soc. 329, 185–219 (1992)
Hale, J.K., Raugel, G.: A reaction-diffusion equation on a thin L-shaped domain. Proc. R. Soc. Edinb. Sect. A 125, 283–327 (1995)
Johnson, R., Kamenskii, M., Nistri, P.: Existence of periodic solutions of an autonomous damped wave equation in thin domains. J. Dyn. Differ. Equ. 10, 409–424 (1998)
Krause, A., Lewis, M., Wang, B.: Dynamics of the non-autonomous stochastic p-Laplace equation driven by multiplicative noise. Appl. Math. Comput. 246(1), 365–376 (2014)
Krause, A., Wang, B.: Pullback attractors of non-autonomous stochastic degenerate parabolic equations on unbounded domains. J. Math. Anal. Appl. 417, 1018–1038 (2014)
Li, D., Lu, K., Wang, B., Wang, X.: Limiting behavior of dynamics for stochastic reaction-diffusion equations with additive noise on thin domains. Discret. Contin. Dyn. Syst. 38, 187–208 (2018)
Li, D., Lu, K., Wang, B., Wang, X.: Limiting dynamics for non-autonomous stochastic retarded reaction-diffusion equations on thin domains. Discret. Contin. Dyn. Syst. 39, 3717–3747 (2019)
Li, D., Shi, L., Wang, X., Zhao, J.: Random dynamics for non-autonomous stochastic evolution equations without uniqueness on unbounded narrow domains. Stoch. Anal. Appl. 38, 1019–1044 (2020)
Li, D., Wang, B., Wang, X.: Limiting behavior of non-autonomous stochastic reaction-diffusion equations on thin domains. J. Differ. Equ. 262, 1575–1602 (2017)
Li, F.: Dynamics for stochastic Fitzhugh–Nagumo systems with general multiplicative noise on thin domains. Math. Methods Appl. Sci. 44, 5050–5078 (2021)
Li, F., Li, Y., Wang, R.: Regular measurable dynamics for reaction-diffusion equations on narrow domains with rough noise. Discret. Contin. Dyn. Syst. 38, 3663–3685 (2018)
Li, F., Li, Y., Wang, R.: Strong convergence of bi-spatial random attractors for parabolic on thin domains with rough noise. Topol. Methods Nonlinear Anal. 53, 659–682 (2019)
Li, F., Li, Y., Wang, R.: Limiting dynamics for stochastic reaction diffusion equations on the Sobolev space with thin domains. Comput. Math. Appl. 79, 457–475 (2020)
Li, Y., Gu, A., Li, J.: Existence and continuity of bi-spatial random attractors and application to stochastic semilinear Laplacian equations. J. Differ. Equ. 258, 504–534 (2015)
Nakasato, J.C., Pereira, M.C.: The \(p\)-Laplacian in thin channels with locally periodic roughness and different scales. Nonlinearity 35, 2474–2512 (2022)
Prizzi, M., Rybakowski, K.P.: Recent results on thin domain problems II. Topol. Methods Nonlinear Anal. 19, 199–219 (2002)
Pu, Z., Gong, T., Li, D.: Asymptotic properties in non-autonomous stochastic parabolic problems dominated by \(p\)-Laplacian operator on thin domains. Discret. Contin. Dyn. Syst. Ser. B 28, 2294–2315 (2023)
Raugel, G., Sell, G.R.: Navier-Stokes equations on thin 3D domains. I. Global attractors and global regularity of solutions. J. Am. Math. Soc. 6, 503–568 (1993)
Shi, L., Li, D., Li, X., Wang, X.: Dynamics of stochastic FitzHugh–Nagumo systems with additive noise on unbounded thin domains. Stoch. Dyn. 20, 2050018 (2020)
Shi, L., Li, X.: Limiting behavior of non-autonomous stochastic reaction-diffusion equations on unbounded thin domains. J. Math. Phys. 60, 082702 (2019)
Shi, L., Wang, R., Lu, K., Wang, B.: Asymptotic behavior of stochastic FitzHugh–Nagumo systems on unbounded thin domains. J. Differ. Equ. 267, 4373–4409 (2019)
Silva, R.P.: Behavior of the p-Laplacian on thin domains. Int. J. Differ. Equ. 2013, 210270 (2013)
Silva, R.P.: Upper semicontinuity of global attractors for quasilinear parabolic equations on unbounded thin domains. São Paulo J. Math. Sci. 9, 251–262 (2015)
Wang, B.: Attractors for reaction-diffusion equations in unbounded domains. Physica D 128, 41–52 (1999)
Wang, R., Li, Y., Wang, B.: Bi-spatial pullback attractors of fractional nonclassical diffusion equations on unbounded domains with \((p, q)\)-growth nonlinearities. Appl. Math. Optim. 84, 425–461 (2021)
Wang, R., Wang, B.: Random dynamics of non-autonomous fractional stochastic p-Laplacian equations on \({\mathbb{R} }^N\). Banach J. Math. Anal. 15, 19 (2021)
Wang, R., Wang, B.: Asymptotic behavior of non-autonomous fractional \(p\)-Laplacian equations driven by additive noise on unbounded domains. Bull. Math. Sci. 11, 2050020 (2021)
Wang, R., Wang, B.: Asymptotic behavior of non-autonomous fractional stochastic \(p\)-laplacian equations. Comput. Math. Appl. 78, 3527–3543 (2019)
Yin, J., Li, Y.: Two types of upper semi-continuity of bi-spatial attractors for non-autonomous stochastic \(p\)-Laplacian equations on \({\mathbb{R} }^N\). Math. Methods Appl. Sci. 40, 4863–4879 (2017)
Zhang, X., Zhang, X.: Upper semi-continuity of non-autonomous fractional stochastic \(p\)-Laplacian equation driven by additive noise on \({\mathbb{R} }^n\). Discret. Contin. Dyn. Syst. Ser. B 28, 385–407 (2023)
Zhang, X.: Pullback random attractors for fractional stochastic \(p\)-Laplacian equation with delay and multiplicative noise. Discret. Contin. Dyn. Syst. Ser. B 27, 1695–1724 (2022)
Zhao, W.: Random dynamics of stochistic \(p\)-Laplacian equations on \({\mathbb{R} }^N\) with an unbounded additive noise. J. Math. Anal. Appl. 455, 1178–1203 (2017)
Zhao, W.: Long-time random dynamics of stochastic parabolic \(p\)-Laplacian equations on \({\mathbb{R} }^N\). Nonlinear Anal. 152, 196–219 (2017)
Zhu, K., Zhou, F.: Continuity and pullback attractors for a non-autonomous reaction-diffusion equation in \({\mathbb{R} }^N\). Comput. Math. Appl. 71, 2089–2105 (2016)
Funding
Fuzhi Li was supported by the NSFC (12201415) and the Jiangxi Provincial Natural Science Foundation (20224BAB201009, 20202BABL211006). M. M. Freitas thank the CNPq for financial support through the project Attractors and asymptotic behavior of nonlinear evolution equations by Grant 313081/2021-2. Jiali Yu was supported by the Natural Science Research Project of the Educational Department of Liaoning Province (JDL2020027).
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Li, F., Freitas, M.M. & Yu, J. Bi-spatial Pullback Attractors of Non-autonomous p-Laplacian Equations on Unbounded Thin Domains. Appl Math Optim 88, 18 (2023). https://doi.org/10.1007/s00245-023-10001-9
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DOI: https://doi.org/10.1007/s00245-023-10001-9
Keywords
- Laplacian equations
- Unbounded thin domains
- Bi-spatial pullback attractors
- Upper semi-continuity
- Higher-order integrability