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Boundedness and asymptotic stability in a two-species chemotaxis-competition model with signal-dependent sensitivity
Asymptotic behaviors of Green-Sch potentials at infinity and its applications
School of of Mathematics and Information Science, Henan University of Economics and Law, Zhengzhou 450046, China |
The first aim in this paper is to deal with asymptotic behaviors of Green-Sch potentials in a cylinder. As an application we prove the integral representation of nonnegative weak solutions of the stationary Schrödinger equation in a cylinder. Next we give asymptotic behaviors of them outside an exceptional set. Finally we obtain a quantitative property of rarefied sets with respect to the stationary Schrödinger operator at $+\infty$ in a cylinder. Meanwhile we show that the reverse of this property is not true.
References:
| [1] |
L. Ahlfors and M. Heins, Questions of regularity connected with the Phragmén-Lindelöf principle, Ann. of Math., 50 (1949), 341-346. |
| [2] |
H. Aikawa, On the behavior at infinity of nonnegative superharmonic functions in a half space, Hiroshima Math. J., 11 (1981), 425-441. |
| [3] |
H. Aikawa and M. Essén, Potential theory-selected topics. Lecture Notes in Mathematics, 1633, Springer-Verlag, Berlin, 1996. |
| [4] |
V. S. Azarin, Generalization of a theorem of Hayman's on a subharmonic function in an n-dimensional cone (Russian), Mat. Sb. (N.S.), 66 (1965), 248-264. |
| [5] |
R. Courant and D. Hilbert, Methods of Mathematical Physics. Vol. I. Interscience Publishers, Inc. , New York, N. Y. , 1953. |
| [6] |
M. Cranston, Conditional Brownian motion, Whitney squares and the conditional gauge theorem, Seminar on Stochastic Processes, 1988 (Gainesville, FL, 1988), 109-119, Progr. Probab. , 17, Birkhäuser Boston, Boston, MA, 1989. |
| [7] |
M. Cranston, E. Fabes and Z. Zhao, Conditional gauge and potential theory for the Schrödinger operator, Trans. Amer. Math. Soc., 307 (1988), 171-194. |
| [8] |
M. Essén and H. L. Jackson, On the covering properties of certain exceptional sets in a half-space, Hiroshima Math. J., 10 (1980), 233-262. |
| [9] |
D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 2001. |
| [10] |
P. Hartman, Ordinary Differential Equations, John Wiley and Sons, Inc. , New York-LondonSydney, 1964. |
| [11] |
J. Lelong-Ferrand, Étude au voisinage de la frontière des fonctions surharmoniques positives dans un demi-espace, Ann. Sci. École Norm. Sup., 66 (1949), 125-159. |
| [12] |
B. Ya. Levin and A. I. Kheyfits, Asymptotic behavior of subfunctions of time-independent Schrödinger operator, in Some Topics on Value Distribution and Differentiability in Complex and P-adic Analysis (eds. A. Escassut, W. Tutschke and C. C. Yang), Science Press, 11 (2008), 323-397. |
| [13] |
I. Miyamoto and H. Yoshida, Two criterions of Wiener type for minimally thin sets and rarefied sets in a cone, J. Math. Soc. Japan., 54 (2002), 487-512. |
| [14] |
I. Miyamoto, Two criteria of Wiener type for minimally thin sets and rarefied sets in a cylinder, Hokkaido Math. J., 36 (2007), 507-534. |
| [15] |
Y. Mizuta, Potential theory in Euclidean spaces. GAKUTO International Series. Mathematical Sciences and Applications, 6, Gakkötosho Co. , Ltd. , Tokyo, 1996. |
| [16] |
L. Qiao, Weak solutions for the stationary Schrödinger equation and its application, Appl. Math. Lett., 63 (2017), 34-39. |
| [17] |
L. Qiao and G. Deng, Growth properties of modified α-potentials in the upper-half space, Filomat, 27 (2013), 703-712. |
| [18] |
L. Qiao and G. Deng, Minimally thin sets at infinity with respect to the Schrödinger operator, Sci. Sin. Math., 44 (2014), 1247-1256. |
| [19] |
L. Qiao and G. Pan, Integral representations of generalized harmonic functions, Taiwanese J. Math., 17 (2013), 1503-1521. |
| [20] |
L. Qiao and G. Pan, Lower-bound estimates for a class of harmonic functions and applications to Masaev's type theorem, Bull. Sci. Math., 140 (2016), 70-85. |
| [21] |
L. Qiao and Y. Ren, ntegral representations for the solutions of infinite order of the stationary Schrödinger equation in a cone, Monats. Math., 173 (2014), 593-603. |
| [22] |
B. Simon, Schrödinger semigroups, Bull. Amer. Math. Soc., 7 (1982), 447-526. |
| [23] |
H. Yoshida and I. Miyamoto, Solutions of the Dirichlet problem on a cone with continuous data, J. Math. Soc. Japan, 50 (1998), 71-93. |
| [24] |
Y. Zhang, G. Deng and K. Kou, Asymptotic behavior of fractional Laplacians in the half space, Appl. Math. Comput., 254 (2015), 125-132. |
| [25] |
Y. Zhang, G. Deng and T. Qian, Integral representations of a class of harmonic functions in the half space, J. Differential Equations, 260 (2016), 923-936. |
show all references
References:
| [1] |
L. Ahlfors and M. Heins, Questions of regularity connected with the Phragmén-Lindelöf principle, Ann. of Math., 50 (1949), 341-346. |
| [2] |
H. Aikawa, On the behavior at infinity of nonnegative superharmonic functions in a half space, Hiroshima Math. J., 11 (1981), 425-441. |
| [3] |
H. Aikawa and M. Essén, Potential theory-selected topics. Lecture Notes in Mathematics, 1633, Springer-Verlag, Berlin, 1996. |
| [4] |
V. S. Azarin, Generalization of a theorem of Hayman's on a subharmonic function in an n-dimensional cone (Russian), Mat. Sb. (N.S.), 66 (1965), 248-264. |
| [5] |
R. Courant and D. Hilbert, Methods of Mathematical Physics. Vol. I. Interscience Publishers, Inc. , New York, N. Y. , 1953. |
| [6] |
M. Cranston, Conditional Brownian motion, Whitney squares and the conditional gauge theorem, Seminar on Stochastic Processes, 1988 (Gainesville, FL, 1988), 109-119, Progr. Probab. , 17, Birkhäuser Boston, Boston, MA, 1989. |
| [7] |
M. Cranston, E. Fabes and Z. Zhao, Conditional gauge and potential theory for the Schrödinger operator, Trans. Amer. Math. Soc., 307 (1988), 171-194. |
| [8] |
M. Essén and H. L. Jackson, On the covering properties of certain exceptional sets in a half-space, Hiroshima Math. J., 10 (1980), 233-262. |
| [9] |
D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 2001. |
| [10] |
P. Hartman, Ordinary Differential Equations, John Wiley and Sons, Inc. , New York-LondonSydney, 1964. |
| [11] |
J. Lelong-Ferrand, Étude au voisinage de la frontière des fonctions surharmoniques positives dans un demi-espace, Ann. Sci. École Norm. Sup., 66 (1949), 125-159. |
| [12] |
B. Ya. Levin and A. I. Kheyfits, Asymptotic behavior of subfunctions of time-independent Schrödinger operator, in Some Topics on Value Distribution and Differentiability in Complex and P-adic Analysis (eds. A. Escassut, W. Tutschke and C. C. Yang), Science Press, 11 (2008), 323-397. |
| [13] |
I. Miyamoto and H. Yoshida, Two criterions of Wiener type for minimally thin sets and rarefied sets in a cone, J. Math. Soc. Japan., 54 (2002), 487-512. |
| [14] |
I. Miyamoto, Two criteria of Wiener type for minimally thin sets and rarefied sets in a cylinder, Hokkaido Math. J., 36 (2007), 507-534. |
| [15] |
Y. Mizuta, Potential theory in Euclidean spaces. GAKUTO International Series. Mathematical Sciences and Applications, 6, Gakkötosho Co. , Ltd. , Tokyo, 1996. |
| [16] |
L. Qiao, Weak solutions for the stationary Schrödinger equation and its application, Appl. Math. Lett., 63 (2017), 34-39. |
| [17] |
L. Qiao and G. Deng, Growth properties of modified α-potentials in the upper-half space, Filomat, 27 (2013), 703-712. |
| [18] |
L. Qiao and G. Deng, Minimally thin sets at infinity with respect to the Schrödinger operator, Sci. Sin. Math., 44 (2014), 1247-1256. |
| [19] |
L. Qiao and G. Pan, Integral representations of generalized harmonic functions, Taiwanese J. Math., 17 (2013), 1503-1521. |
| [20] |
L. Qiao and G. Pan, Lower-bound estimates for a class of harmonic functions and applications to Masaev's type theorem, Bull. Sci. Math., 140 (2016), 70-85. |
| [21] |
L. Qiao and Y. Ren, ntegral representations for the solutions of infinite order of the stationary Schrödinger equation in a cone, Monats. Math., 173 (2014), 593-603. |
| [22] |
B. Simon, Schrödinger semigroups, Bull. Amer. Math. Soc., 7 (1982), 447-526. |
| [23] |
H. Yoshida and I. Miyamoto, Solutions of the Dirichlet problem on a cone with continuous data, J. Math. Soc. Japan, 50 (1998), 71-93. |
| [24] |
Y. Zhang, G. Deng and K. Kou, Asymptotic behavior of fractional Laplacians in the half space, Appl. Math. Comput., 254 (2015), 125-132. |
| [25] |
Y. Zhang, G. Deng and T. Qian, Integral representations of a class of harmonic functions in the half space, J. Differential Equations, 260 (2016), 923-936. |
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