- Previous Article
- ERA Home
- This Issue
-
Next Article
The boundedness and upper semicontinuity of the pullback attractors for a 2D micropolar fluid flows with delay
Strong $ (L^2,L^\gamma\cap H_0^1) $-continuity in initial data of nonlinear reaction-diffusion equation in any space dimension
1. | School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, China |
2. | Mathematisches Institut, Universität Tübingen, D-72076 Tübingen, Germany |
3. | School of Mathematics and Statistics, Chongqing Technology and Business University, Chongqing 400067, China |
In this paper we study the continuity in initial data of a classical reaction-diffusion equation with arbitrary $ p>2 $ order nonlinearity and in any space dimension $ N \geqslant 1 $. It is proved that the weak solutions can be $ (L^2, L^\gamma\cap H_0^1) $-continuous in initial data for arbitrarily large $ \gamma \geqslant 2 $ (independent of the physical parameters of the system), i.e., can converge in the norm of any $ L^\gamma\cap H_0^1 $ as the corresponding initial values converge in $ L^2 $. In fact, the system is shown to be $ (L^2, L^\gamma\cap H_0^1) $-smoothing in a H$ \ddot{\rm o} $lder way. Applying this to the global attractor we find that, with external forcing only in $ L^2 $, the attractor $ \mathscr{A} $ attracts bounded subsets of $ L^2 $ in the norm of any $ L^\gamma\cap H_0^1 $, and that every translation set $ \mathscr{A}-z_0 $ of $ \mathscr{A} $ for any $ z_0\in \mathscr{A} $ is a finite dimensional compact subset of $ L^\gamma\cap H_0^1 $. The main technique we employ is a combination of a Moser iteration and a decomposition of the nonlinearity, by which the interpolation inequalities are avoided and the new continuity result is obtained without any restrictions on the order $ p>2 $ of the nonlinearity and the space dimension $ N \geqslant 1 $.
References:
[1] |
A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, Studies in Mathematics and its Applications, 25, North-Holland Publishing Co., Amsterdam, 1992. |
[2] |
T. Bartsch and Z. Liu,
On a superlinear elliptic $p$-Laplacian equation, J. Differential Equations, 198 (2004), 149-175.
doi: 10.1016/j.jde.2003.08.001. |
[3] |
D. Cao, C. Sun and M. Yang,
Dynamics for a stochastic reaction-diffusion equation with additive noise, J. Differential Equations, 259 (2015), 838-872.
doi: 10.1016/j.jde.2015.02.020. |
[4] |
T. Caraballo and S. Sonner,
Random pullback exponential attractors: General existence results for random dynamical systems in Banach spaces, Discrete Contin. Dyn. Syst., 37 (2017), 6383-6403.
doi: 10.3934/dcds.2017277. |
[5] |
M. Coti Zelati and P. Kalita,
Minimality properties of set-valued processes and their pullback attractors, SIAM J. Math. Anal., 47 (2015), 1530-1561.
doi: 10.1137/140978995. |
[6] |
H. Cui, J. A. Langa and Y. Li,
Measurability of random attractors for quasi strong-to-weak continuous random dynamical systems, J. Dynam. Differential Equations, 30 (2018), 1873-1898.
doi: 10.1007/s10884-017-9617-z. |
[7] |
H. Cui, Y. Li and J. Yin,
Existence and upper semicontinuity of bi-spatial pullback attractors for smoothing cocycles, Nonlinear Anal., 128 (2015), 303-324.
doi: 10.1016/j.na.2015.08.009. |
[8] |
A. Eden, C. Foias, B. Nicolaenko and R. Temam, Exponential Attractors for Dissipative Evolution Equations, RAM: Research in Applied Mathematics, 37, Masson, Paris; John Wiley & Sons, Ltd., Chichester, 1994. |
[9] |
L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, 19, American Mathematical Society, Providence, RI, 2010.
doi: 10.1090/gsm/019. |
[10] |
Y. Li, A. Gu and J. Li,
Existence and continuity of bi-spatial random attractors and application to stochastic semilinear Laplacian equations, J. Differential Equations, 258 (2015), 504-534.
doi: 10.1016/j.jde.2014.09.021. |
[11] |
Y. Li and B. Guo,
Random attractors for quasi-continuous random dynamical systems and applications to stochastic reaction-diffusion equations, J. Differential Equations, 245 (2008), 1775-1800.
doi: 10.1016/j.jde.2008.06.031. |
[12] |
A. Miranville and S. Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains, in Handbook of Differential Equations: Evolutionary Equations. Vol. IV, Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, 103–200.
doi: 10.1016/S1874-5717(08)00003-0. |
[13] |
V. Pata and S. Zelik,
A result on the existence of global attractors for semigroups of closed operators, Commun. Pure Appl. Anal., 6 (2007), 481-486.
doi: 10.3934/cpaa.2007.6.481. |
[14] |
J. C. Robinson, Infinite-Dimensional Dynamical Systems. An Introduction to Dissipative
Parabolic PDEs and the Theory of Global Attractors, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2001.
doi: 10.1007/978-94-010-0732-0. |
[15] |
J. C. Robinson, Dimensions, Embeddings, and Attractors, Cambridge Tracts in Mathematics,
186, Cambridge University Press, Cambridge, 2011.
doi: 10.1017/CBO9780511933912. |
[16] |
A. Shirikyan and S. Zelik,
Exponential attractors for random dynamical systems and applications, Stoch. Partial Differ. Equ. Anal. Comput., 1 (2013), 241-281.
doi: 10.1007/s40072-013-0007-1. |
[17] |
C. Sun,
Asymptotic regularity for some dissipative equations, J. Differential Equations, 248 (2010), 342-362.
doi: 10.1016/j.jde.2009.08.007. |
[18] |
T. Trujillo and B. Wang,
Continuity of strong solutions of the reaction-diffusion equation in initial data, Nonlinear Anal., 69 (2008), 2525-2532.
doi: 10.1016/j.na.2007.08.032. |
[19] |
S. Zelik, The attractor for a nonlinear reaction-diffusion system with a supercritical nonlinearity and its dimension, Rend. Accad. Naz. Sci. XL Mem. Mat. Appl. (5), 24 (2000),
1–25. |
[20] |
W. Zhao,
Random dynamics of stochastic $p$-Laplacian equations on $\mathbb{R}^N$ with an unbounded additive noise, J. Math. Anal. Appl., 455 (2017), 1178-1203.
doi: 10.1016/j.jmaa.2017.06.025. |
[21] |
W. Zhao and Y. Li,
$ (L^2, L^p)$-random attractors for stochastic reaction-diffusion equation on unbounded domains, Nonlinear Anal., 75 (2012), 485-502.
doi: 10.1016/j.na.2011.08.050. |
[22] |
C.-K. Zhong, M.-H. Yang and C.-Y. Sun,
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.
doi: 10.1016/j.jde.2005.06.008. |
[23] |
K. Zhu and F. Zhou,
Continuity and pullback attractors for a non-autonomous reaction-diffusion equation in $\mathbb{R}^N$, Comput. Math. Appl., 71 (2016), 2089-2105.
doi: 10.1016/j.camwa.2016.04.004. |
show all references
References:
[1] |
A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, Studies in Mathematics and its Applications, 25, North-Holland Publishing Co., Amsterdam, 1992. |
[2] |
T. Bartsch and Z. Liu,
On a superlinear elliptic $p$-Laplacian equation, J. Differential Equations, 198 (2004), 149-175.
doi: 10.1016/j.jde.2003.08.001. |
[3] |
D. Cao, C. Sun and M. Yang,
Dynamics for a stochastic reaction-diffusion equation with additive noise, J. Differential Equations, 259 (2015), 838-872.
doi: 10.1016/j.jde.2015.02.020. |
[4] |
T. Caraballo and S. Sonner,
Random pullback exponential attractors: General existence results for random dynamical systems in Banach spaces, Discrete Contin. Dyn. Syst., 37 (2017), 6383-6403.
doi: 10.3934/dcds.2017277. |
[5] |
M. Coti Zelati and P. Kalita,
Minimality properties of set-valued processes and their pullback attractors, SIAM J. Math. Anal., 47 (2015), 1530-1561.
doi: 10.1137/140978995. |
[6] |
H. Cui, J. A. Langa and Y. Li,
Measurability of random attractors for quasi strong-to-weak continuous random dynamical systems, J. Dynam. Differential Equations, 30 (2018), 1873-1898.
doi: 10.1007/s10884-017-9617-z. |
[7] |
H. Cui, Y. Li and J. Yin,
Existence and upper semicontinuity of bi-spatial pullback attractors for smoothing cocycles, Nonlinear Anal., 128 (2015), 303-324.
doi: 10.1016/j.na.2015.08.009. |
[8] |
A. Eden, C. Foias, B. Nicolaenko and R. Temam, Exponential Attractors for Dissipative Evolution Equations, RAM: Research in Applied Mathematics, 37, Masson, Paris; John Wiley & Sons, Ltd., Chichester, 1994. |
[9] |
L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, 19, American Mathematical Society, Providence, RI, 2010.
doi: 10.1090/gsm/019. |
[10] |
Y. Li, A. Gu and J. Li,
Existence and continuity of bi-spatial random attractors and application to stochastic semilinear Laplacian equations, J. Differential Equations, 258 (2015), 504-534.
doi: 10.1016/j.jde.2014.09.021. |
[11] |
Y. Li and B. Guo,
Random attractors for quasi-continuous random dynamical systems and applications to stochastic reaction-diffusion equations, J. Differential Equations, 245 (2008), 1775-1800.
doi: 10.1016/j.jde.2008.06.031. |
[12] |
A. Miranville and S. Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains, in Handbook of Differential Equations: Evolutionary Equations. Vol. IV, Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, 103–200.
doi: 10.1016/S1874-5717(08)00003-0. |
[13] |
V. Pata and S. Zelik,
A result on the existence of global attractors for semigroups of closed operators, Commun. Pure Appl. Anal., 6 (2007), 481-486.
doi: 10.3934/cpaa.2007.6.481. |
[14] |
J. C. Robinson, Infinite-Dimensional Dynamical Systems. An Introduction to Dissipative
Parabolic PDEs and the Theory of Global Attractors, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2001.
doi: 10.1007/978-94-010-0732-0. |
[15] |
J. C. Robinson, Dimensions, Embeddings, and Attractors, Cambridge Tracts in Mathematics,
186, Cambridge University Press, Cambridge, 2011.
doi: 10.1017/CBO9780511933912. |
[16] |
A. Shirikyan and S. Zelik,
Exponential attractors for random dynamical systems and applications, Stoch. Partial Differ. Equ. Anal. Comput., 1 (2013), 241-281.
doi: 10.1007/s40072-013-0007-1. |
[17] |
C. Sun,
Asymptotic regularity for some dissipative equations, J. Differential Equations, 248 (2010), 342-362.
doi: 10.1016/j.jde.2009.08.007. |
[18] |
T. Trujillo and B. Wang,
Continuity of strong solutions of the reaction-diffusion equation in initial data, Nonlinear Anal., 69 (2008), 2525-2532.
doi: 10.1016/j.na.2007.08.032. |
[19] |
S. Zelik, The attractor for a nonlinear reaction-diffusion system with a supercritical nonlinearity and its dimension, Rend. Accad. Naz. Sci. XL Mem. Mat. Appl. (5), 24 (2000),
1–25. |
[20] |
W. Zhao,
Random dynamics of stochastic $p$-Laplacian equations on $\mathbb{R}^N$ with an unbounded additive noise, J. Math. Anal. Appl., 455 (2017), 1178-1203.
doi: 10.1016/j.jmaa.2017.06.025. |
[21] |
W. Zhao and Y. Li,
$ (L^2, L^p)$-random attractors for stochastic reaction-diffusion equation on unbounded domains, Nonlinear Anal., 75 (2012), 485-502.
doi: 10.1016/j.na.2011.08.050. |
[22] |
C.-K. Zhong, M.-H. Yang and C.-Y. Sun,
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.
doi: 10.1016/j.jde.2005.06.008. |
[23] |
K. Zhu and F. Zhou,
Continuity and pullback attractors for a non-autonomous reaction-diffusion equation in $\mathbb{R}^N$, Comput. Math. Appl., 71 (2016), 2089-2105.
doi: 10.1016/j.camwa.2016.04.004. |
[1] |
Daoyuan Fang, Ting Zhang. Compressible Navier-Stokes equations with vacuum state in one dimension. Communications on Pure & Applied Analysis, 2004, 3 (4) : 675-694. doi: 10.3934/cpaa.2004.3.675 |
[2] |
Alexandre B. Simas, Fábio J. Valentim. $W$-Sobolev spaces: Higher order and regularity. Communications on Pure & Applied Analysis, 2015, 14 (2) : 597-607. doi: 10.3934/cpaa.2015.14.597 |
[3] |
Rafael Luís, Sandra Mendonça. A note on global stability in the periodic logistic map. Discrete & Continuous Dynamical Systems - B, 2020, 25 (11) : 4211-4220. doi: 10.3934/dcdsb.2020094 |
[4] |
Lakmi Niwanthi Wadippuli, Ivan Gudoshnikov, Oleg Makarenkov. Global asymptotic stability of nonconvex sweeping processes. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1129-1139. doi: 10.3934/dcdsb.2019212 |
[5] |
Philippe G. Lefloch, Cristinel Mardare, Sorin Mardare. Isometric immersions into the Minkowski spacetime for Lorentzian manifolds with limited regularity. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 341-365. doi: 10.3934/dcds.2009.23.341 |
[6] |
Carlos Gutierrez, Nguyen Van Chau. A remark on an eigenvalue condition for the global injectivity of differentiable maps of $R^2$. Discrete & Continuous Dynamical Systems - A, 2007, 17 (2) : 397-402. doi: 10.3934/dcds.2007.17.397 |
[7] |
Bernold Fiedler, Carlos Rocha, Matthias Wolfrum. Sturm global attractors for $S^1$-equivariant parabolic equations. Networks & Heterogeneous Media, 2012, 7 (4) : 617-659. doi: 10.3934/nhm.2012.7.617 |
[8] |
Manoel J. Dos Santos, Baowei Feng, Dilberto S. Almeida Júnior, Mauro L. Santos. Global and exponential attractors for a nonlinear porous elastic system with delay term. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2805-2828. doi: 10.3934/dcdsb.2020206 |
[9] |
A. Aghajani, S. F. Mottaghi. Regularity of extremal solutions of semilinaer fourth-order elliptic problems with general nonlinearities. Communications on Pure & Applied Analysis, 2018, 17 (3) : 887-898. doi: 10.3934/cpaa.2018044 |
[10] |
Hyeong-Ohk Bae, Hyoungsuk So, Yeonghun Youn. Interior regularity to the steady incompressible shear thinning fluids with non-Standard growth. Networks & Heterogeneous Media, 2018, 13 (3) : 479-491. doi: 10.3934/nhm.2018021 |
[11] |
Irena PawŃow, Wojciech M. Zajączkowski. Global regular solutions to three-dimensional thermo-visco-elasticity with nonlinear temperature-dependent specific heat. Communications on Pure & Applied Analysis, 2017, 16 (4) : 1331-1372. doi: 10.3934/cpaa.2017065 |
[12] |
Ying Yang. Global classical solutions to two-dimensional chemotaxis-shallow water system. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2625-2643. doi: 10.3934/dcdsb.2020198 |
[13] |
Luigi C. Berselli, Jishan Fan. Logarithmic and improved regularity criteria for the 3D nematic liquid crystals models, Boussinesq system, and MHD equations in a bounded domain. Communications on Pure & Applied Analysis, 2015, 14 (2) : 637-655. doi: 10.3934/cpaa.2015.14.637 |
Impact Factor: 0.263
Tools
Article outline
[Back to Top]