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Asymptotic behavior of time periodic solutions for extended Fisher-Kolmogorov equations with delays
BMO type space associated with Neumann operator and application to a class of parabolic equations
1. | School of Statistics and Mathematics, Zhejiang Gongshang University, Hangzhou 310018, China |
2. | Department of Mathematics, Jiangxi University of Finance and Economics, Nanchang 330032, China |
$ {\rm BMO}_{\Delta_{N}}(\mathbb{R}^{n}) $ |
$ \mathbb{R}^{n} $ |
$ \mathcal{L}: = -\Delta_{N} $ |
$ f\in {\rm BMO}_{\Delta_{N}}(\mathbb{R}^{n}) $ |
$ {\mathbb L}u = u_{t}+ \mathcal{L} u = 0, u(x, 0) = f(x), $ |
$ u $ |
$ \begin{eqnarray*} \sup\limits_{x_B, r_B} r_B^{-n}\int_0^{r_B^2}\int_{B(x_B, r_B)} \left\{t| \partial_t u(x, t) |^2+ | \nabla_x u(x, t) |^2 \right\}{dx dt } \leq C <\infty, \end{eqnarray*} $ |
$ C>0 $ |
$ {\mathbb L} $ |
$ {\rm BMO}_{\Delta_{N}}(\mathbb{R}^{n}) $ |
$ {\rm BMO}_{\Delta_{N}}(\mathbb{R}^{n}) $ |
$ u_{0}\in {{\rm BMO}_{\Delta_{N}}^{-1}(\mathbb{R}^{n})} $ |
References:
[1] |
P. Auscher and D. Frey,
On the Well-Posedness of parabolic equations of Navier-Stokes type with ${\rm {BMO}}^{-1}$ data, J. Inst. Math. Jussieu., 16 (2017), 947-985.
doi: 10.1017/S1474748015000158. |
[2] |
P. Auscher, S. Monniaux and P. Portal,
The maximal regularity operator on tent spaces, Commun. Pure Appl. Anal., 11 (2012), 2213-2219.
doi: 10.3934/cpaa.2012.11.2213. |
[3] |
P. Auscher and P. Tchamitchian., Square Root Problem for Divergence Operators and Related Topics, Ast$\acute{e}$risqueno., 249, 1998. |
[4] |
D. Deng, X. T. Duong, A. Sikora and L. Yan,
Comparison of the classical BMO with the BMO spaces associated with operators and applications, Rev. Mat. Iberoamericana., 24 (2008), 267-296.
doi: 10.4171/RMI/536. |
[5] |
M. Dindos, C. Kenig and J. Pipher, BMO solvability and the $A_{\infty}$ condition for elliptic operators, J. Geom. Anal., 21 (2011), 78-95.
doi: 10.1007/s12220-010-9142-3. |
[6] |
Y. Du and K. Wang, Regularity of the solutions to the liquid crystal equations with small rough data, J. Differential Equations, 256 (2014), 65-81.
doi: 10.1016/j.jde.2013.07.066. |
[7] |
X. T. Duong, L. Yan and C. Zhang,
On characterization of Poisson integrals of Schrödinger operators with BMO traces, J. Funct. Anal., 266 (2014), 2053-2085.
doi: 10.1016/j.jfa.2013.09.008. |
[8] |
X. T. Duong and L. Yan,
New function spaces of $ {\rm{BMO}} $ type, the John-Nirenberg inequality, interpolation and applications, Comm. Pure Appl. Math., 58 (2005), 1375-1420.
doi: 10.1002/cpa.20080. |
[9] |
X. T. Duong and L. Yan,
Duality of Hardy and BMO spaces associated with operators with heat kernel bounds, J. Amer. Math. Soc., 18 (2005), 943-973.
doi: 10.1090/S0894-0347-05-00496-0. |
[10] |
E. B. Fabes, R. L. Johnson and U. Neri, Spaces of harmonic functions representable by Poisson integrals of functions in BMO and $L_{p, \lambda}$, Indiana Univ. Math. J., 25 (1976), 159-170.
doi: 10.1512/iumj.1976.25.25012. |
[11] |
E. B. Fabes and U. Neri, Characterization of temperatures with initial data in BMO, Duke Math. J., 42 (1975), 725-734.
doi: 10.1215/S0012-7094-75-04260-X. |
[12] |
E. B. Fabes and U. Neri, Dirichlet problem in Lipschitz domains with BMO data, Proc. Amer. Math. Soc., 78 (1980), 33-39.
doi: 10.1090/S0002-9939-1980-0548079-8. |
[13] |
C. Fefferman and E. M. Stein,
$H^p$ spaces of several variables, Acta Math., 129 (1972), 137-193.
doi: 10.1007/BF02392215. |
[14] |
S. Hofmann, G. Lu, D. Mitrea, M. Mitrea and L. Yan, Hardy spaces associated to non-negative self-adjoint operators satisfying Davies-Gaffney estimates, Memoirs of the Amer. Math. Soc., 214 (2011), no. 1007.
doi: 10.1090/S0065-9266-2011-00624-6. |
[15] |
R. Jiang, J. Xiao and D. Yang, Towards spaces of harmonic functions with traces in square Campanato spaces and their scaling invariants, Anal. Appl. (Singap.), 14 (2016), 679-703.
doi: 10.1142/S0219530515500190. |
[16] |
H. Koch and D. Tataru,
Well-posedness for the Navier-Stokes equations, Adv. Math., 157 (2001), 22-35.
doi: 10.1006/aima.2000.1937. |
[17] |
J. Li and B. D. Wick,
Characterizations of $\rm H^{1}_{\Delta_{N}}(\mathbb{R}^{n})$ and $\rm BMO_{\Delta_{N}}(\mathbb{R}^{n})$ via weak factorizations and commutators, J. Funct. Anal., 272 (2017), 5384-5416.
doi: 10.1016/j.jfa.2017.03.007. |
[18] |
L. Song, X. X. Tian and L. X. Yan,
On characterization of Poisson integrals of Schrödinger operators with Morry traces, Acta Math. Sin. (Engl. Ser.), 34 (2018), 787-800.
doi: 10.1007/s10114-018-7368-3. |
[19] |
E. M. Stein, Topics in Harmonic Analysis Related to the Littlewood-Paley Theory, Annals of
Mathematics Studies, 63, Princeton Univ. Press, Princeton, N.J., 1970. |
[20] |
M. Yang and C. Zhang, Characterization of temperatures associated to Schrödinger operators with initial data in BMO spaces, to appear in Math. Nachr. arXiv: 1710.01160 |
[21] |
W. Yuan, W. Sickel and D. Yang, Morrey and Campanato meet Besov, Lizorkin and Triebel, Lecture Notes in Mathematics, 2005. Springer-Verlag, Berlin, 2010.
doi: 10.1007/978-3-642-14606-0. |
show all references
References:
[1] |
P. Auscher and D. Frey,
On the Well-Posedness of parabolic equations of Navier-Stokes type with ${\rm {BMO}}^{-1}$ data, J. Inst. Math. Jussieu., 16 (2017), 947-985.
doi: 10.1017/S1474748015000158. |
[2] |
P. Auscher, S. Monniaux and P. Portal,
The maximal regularity operator on tent spaces, Commun. Pure Appl. Anal., 11 (2012), 2213-2219.
doi: 10.3934/cpaa.2012.11.2213. |
[3] |
P. Auscher and P. Tchamitchian., Square Root Problem for Divergence Operators and Related Topics, Ast$\acute{e}$risqueno., 249, 1998. |
[4] |
D. Deng, X. T. Duong, A. Sikora and L. Yan,
Comparison of the classical BMO with the BMO spaces associated with operators and applications, Rev. Mat. Iberoamericana., 24 (2008), 267-296.
doi: 10.4171/RMI/536. |
[5] |
M. Dindos, C. Kenig and J. Pipher, BMO solvability and the $A_{\infty}$ condition for elliptic operators, J. Geom. Anal., 21 (2011), 78-95.
doi: 10.1007/s12220-010-9142-3. |
[6] |
Y. Du and K. Wang, Regularity of the solutions to the liquid crystal equations with small rough data, J. Differential Equations, 256 (2014), 65-81.
doi: 10.1016/j.jde.2013.07.066. |
[7] |
X. T. Duong, L. Yan and C. Zhang,
On characterization of Poisson integrals of Schrödinger operators with BMO traces, J. Funct. Anal., 266 (2014), 2053-2085.
doi: 10.1016/j.jfa.2013.09.008. |
[8] |
X. T. Duong and L. Yan,
New function spaces of $ {\rm{BMO}} $ type, the John-Nirenberg inequality, interpolation and applications, Comm. Pure Appl. Math., 58 (2005), 1375-1420.
doi: 10.1002/cpa.20080. |
[9] |
X. T. Duong and L. Yan,
Duality of Hardy and BMO spaces associated with operators with heat kernel bounds, J. Amer. Math. Soc., 18 (2005), 943-973.
doi: 10.1090/S0894-0347-05-00496-0. |
[10] |
E. B. Fabes, R. L. Johnson and U. Neri, Spaces of harmonic functions representable by Poisson integrals of functions in BMO and $L_{p, \lambda}$, Indiana Univ. Math. J., 25 (1976), 159-170.
doi: 10.1512/iumj.1976.25.25012. |
[11] |
E. B. Fabes and U. Neri, Characterization of temperatures with initial data in BMO, Duke Math. J., 42 (1975), 725-734.
doi: 10.1215/S0012-7094-75-04260-X. |
[12] |
E. B. Fabes and U. Neri, Dirichlet problem in Lipschitz domains with BMO data, Proc. Amer. Math. Soc., 78 (1980), 33-39.
doi: 10.1090/S0002-9939-1980-0548079-8. |
[13] |
C. Fefferman and E. M. Stein,
$H^p$ spaces of several variables, Acta Math., 129 (1972), 137-193.
doi: 10.1007/BF02392215. |
[14] |
S. Hofmann, G. Lu, D. Mitrea, M. Mitrea and L. Yan, Hardy spaces associated to non-negative self-adjoint operators satisfying Davies-Gaffney estimates, Memoirs of the Amer. Math. Soc., 214 (2011), no. 1007.
doi: 10.1090/S0065-9266-2011-00624-6. |
[15] |
R. Jiang, J. Xiao and D. Yang, Towards spaces of harmonic functions with traces in square Campanato spaces and their scaling invariants, Anal. Appl. (Singap.), 14 (2016), 679-703.
doi: 10.1142/S0219530515500190. |
[16] |
H. Koch and D. Tataru,
Well-posedness for the Navier-Stokes equations, Adv. Math., 157 (2001), 22-35.
doi: 10.1006/aima.2000.1937. |
[17] |
J. Li and B. D. Wick,
Characterizations of $\rm H^{1}_{\Delta_{N}}(\mathbb{R}^{n})$ and $\rm BMO_{\Delta_{N}}(\mathbb{R}^{n})$ via weak factorizations and commutators, J. Funct. Anal., 272 (2017), 5384-5416.
doi: 10.1016/j.jfa.2017.03.007. |
[18] |
L. Song, X. X. Tian and L. X. Yan,
On characterization of Poisson integrals of Schrödinger operators with Morry traces, Acta Math. Sin. (Engl. Ser.), 34 (2018), 787-800.
doi: 10.1007/s10114-018-7368-3. |
[19] |
E. M. Stein, Topics in Harmonic Analysis Related to the Littlewood-Paley Theory, Annals of
Mathematics Studies, 63, Princeton Univ. Press, Princeton, N.J., 1970. |
[20] |
M. Yang and C. Zhang, Characterization of temperatures associated to Schrödinger operators with initial data in BMO spaces, to appear in Math. Nachr. arXiv: 1710.01160 |
[21] |
W. Yuan, W. Sickel and D. Yang, Morrey and Campanato meet Besov, Lizorkin and Triebel, Lecture Notes in Mathematics, 2005. Springer-Verlag, Berlin, 2010.
doi: 10.1007/978-3-642-14606-0. |
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