An inhomogeneous nonlocal diffusion problem with unbounded steps
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- 作者：Carmen Cortázar ; Manuel Elgueta ; Jorge García-Melián…
- 刊名：Journal of Evolution Equations
- 出版年：2016
- 出版时间：March 2016
- 年：2016
- 卷：16
- 期：1
- 页码：209-232
- 全文大小：594 KB
- 参考文献：1.Bates P., Chen X., A. Chmaj: Heteroclinic solutions of a van der Waals model with indefinite nonlocal interactions. Calc. Var. 24, 261–281 (2005)MathSciNet CrossRef MATH
2.Bates P., Fife P., Ren X., Wang X.: Traveling waves in a convolution model for phase transitions. Arch. Rat. Mech. Anal. 138, 105–136 (1997)MathSciNet CrossRef MATH
3.Carr J., Chmaj A.: Uniqueness of traveling waves for nonlocal monostable equations. Proc. Amer. Math. Soc. 132, 2433–2439 (2004)MathSciNet CrossRef MATH
4.Chasseigne E., Chaves M., Rossi J.D.: Asymptotic behavior for nonlocal diffusion equations. J. Math. Pures Appl. 86, 271–291 (2006)MathSciNet CrossRef MATH
5.Chen X.: Existence, uniqueness and asymptotic stability of traveling waves in nonlocal evolution equations. Adv. Diff. Eqns. 2, 125–160 (1997)MathSciNet MATH
6.Chmaj A., Ren X.: Homoclinic solutions of an integral equation: existence and stability. J. Diff. Eqns. 155, 17–43 (1999)MathSciNet CrossRef MATH
7.A. Chmaj, X. Ren, The nonlocal bistable equation: stationary solutions on a bounded interval, Electronic J. Diff. Eqns. 2002 (2002), no. 2, 1–12.
8.Cortázar C., Coville J., Elgueta M., Martínez S.: A non local inhomogeneous dispersal process. J. Diff. Eqns. 241, 332–358 (2007)CrossRef MATH
9.C. Cortázar, M. Elgueta, J. García-Melián, S. Martínez, Existence and asymptotic behavior of solutions to some inhomogeneous nonlocal diffusion problems. SIAM J. Math. Anal. 41 (2009), no. 5, 2136–2164.
10.C. Cortázar, M. Elgueta, J. García-Melián, S. Martínez, Stationary sign changing solutions for an inhomogeneous nonlocal problem. Indiana Univ. Math. J. 60 (2011), no. 1, 209–232.
11.Cortázar C., Elgueta M., García-Melián J., Martínez S.: Finite mass solutions for a nonlocal inhomogeneous dispersal equation. Discrete Cont. Dyn. Syst. 35, 1409–1419 (2015)MathSciNet MATH
12.C. Cortázar, M. Elgueta, S. Martínez, J. Rossi. Random walks and the porous medium equation. Rev. Un. Mat. Argentina 50 (2009), no. 2, 149–155.
13.Cortázar C., Elgueta M., Rossi J.D., Wolanski N.: Boundary fluxes for nonlocal diffusion. J. Diff. Eqns. 234, 360–390 (2007)MathSciNet CrossRef MATH
14.J. Coville. On a simple criterion for the existence of a principal eigenfunction of some nonlocal operators. J. Differential Equations 249 (2010), no. 11, 2921–2953.
15.J. Coville, Harnack type inequality for positive solution of some integral equation. Ann. Mat. Pura Appl. (4) 191 (2012), no. 3, 503–528.
16.J. Coville, J. Dávila and S. Martínez. Existence and uniqueness of solutions to a nonlocal equation with monostable nonlinearity. SIAM J. Math. Anal. 39 (2008), no. 5, 1693–1709.
17.J. Coville, J. Dávila and S. Martínez. Pulsating fronts for nonlocal dispersion and KPP nonlinearity. Ann. Inst. H. Poincaré Anal. Non Linéaire 30 (2013), no. 2, 179–223.
18.Coville J., Dupaigne L.: Propagation speed of travelling fronts in nonlocal reaction diffusion equations. Nonl. Anal. 60, 797–819 (2005)MathSciNet CrossRef MATH
19.Coville J., Dupaigne L.: On a nonlocal equation arising in population dynamics. Proc. Roy. Soc. Edinburgh 137, 1–29 (2007)MathSciNet CrossRef MATH
20.E. B. Davies, “Linear Operators and their Spectra”. Cambridge studies in advanced mathematics 106, Cambridge University Press, Cambridge, 2007.
21.P. Fife, Some nonclassical trends in parabolic and parabolic-like evolutions, in “Trends in nonlinear analysis”, pp. 153–191, Springer-Verlag, Berlin, 2003.
22.Hutson V., Martínez S., Mischaikow K., Vickers G. T.: The evolution of dispersal. J. Math. Biol. 47, 483–517 (2003)MathSciNet CrossRef MATH
23.Ignat L. I., Rossi J. D.: Refined asymptotic expansions for nonlocal evolution equations. J. Evol. Eqns. 8, 617–629 (2008)MathSciNet CrossRef MATH
24.M. G. Krein, R. Rutman, Linear operators leaving invariant a cone in a Banach space. Amer. Math. Soc. Translation 1950 (1950) no. 26, 128 pp.
25.L. Kong and W. Shen. Positive stationary solutions and spreading speeds of KPP equations in locally spatially inhomogeneous media. Methods Appl. Anal. 18 (2011), no. 4, 427–456.
26.W. T. Li, J. W. Sun, and F. Y. Yang, Approximate the Fokker-Planck equation by a class of nonlocal dispersal problems. Nonlinear Anal. 74 (2011), no. 11, 3501–3509.
27.Schumacher K.: Travelling-front solutions for integro-differential equations I. J. Reine Angew. Math. 316, 54–70 (1980)MathSciNet MATH
28.W. Shen and A. Zhang. Spreading speeds for monostable equations with nonlocal dispersal in space periodic habitats. J. Differential Equations 249 (2010), no. 4, 747–795.
29.W. Shen and A. Zhang. Stationary solutions and spreading speeds of nonlocal monostable equations in space periodic habitats. Proc. Amer. Math. Soc. 140 (2012), no. 5, 1681–1696.
30.Zhang L.: Existence, uniqueness and exponential stability of traveling wave solutions of some integral differential equations arising from neural networks. J. Diff. Eqns. 197, 162–196 (2004)CrossRef MATH - 作者单位：Carmen Cortázar (1)
Manuel Elgueta (1)
Jorge García-Melián (2) (3)
Salomé Martínez (4) (5)
1. Departamento de Matemáticas, Facultad de Matemáticas, Pontificia, Universidad Católica de Chile, Casilla 306, Correo 22, Santiago, Chile
2. Departamento de Análisis Matemático, Universidad de La Laguna, C/. Astrofísico Francisco Sánchez s/n, 38271, La Laguna, Spain
3. Instituto Universitario de Estudios Avanzados, (IUdEA) en Física Atómica, Molecular y Fotónica, Universidad de La Laguna, C/. Astrofísico Francisco Sánchez s/n, 38203, La Laguna, Spain
4. Departamento de Ingeniería Matemática, Universidad de Chile, Blanco Encalada 2120, 5° piso, Santiago, Chile
5. Centro de Modelamiento Matemático, UMI 2807 CNRS-UChile, Universidad de Chile, Casilla 170, Correo 3, Santiago, Chile - 刊物类别：Mathematics and Statistics
- 刊物主题：Mathematics
Analysis - 出版者：Birkh盲user Basel
- ISSN：1424-3202
文摘
We consider the following nonlocal equation $$\int J\left(\frac{x-y}{g(y)} \right) \frac{u(y)}{g(y)} dy -u(x)=0\qquad x\in \mathbb{R},$$where J is an even, compactly supported, Hölder continuous kernel with unit integral and g is a continuous positive function. Our main concern will be with unbounded functions g, contrary to previous works. More precisely, we study the influence of the growth of g at infinity on the integrability of positive solutions of this equation, therefore determining the asymptotic behavior as \({t\to +\infty}\) of the solutions to the associated evolution problem in terms of the growth of g. Page %P Close Plain text Look Inside Reference tools Export citation EndNote (.ENW) JabRef (.BIB) Mendeley (.BIB) Papers (.RIS) Zotero (.RIS) BibTeX (.BIB) Add to Papers Other actions Register for Journal Updates About This Journal Reprints and Permissions Share Share this content on Facebook Share this content on Twitter Share this content on LinkedIn Related Content Supplementary Material (0) References (30) References1.Bates P., Chen X., A. Chmaj: Heteroclinic solutions of a van der Waals model with indefinite nonlocal interactions. Calc. Var. 24, 261–281 (2005)MathSciNetCrossRefMATH2.Bates P., Fife P., Ren X., Wang X.: Traveling waves in a convolution model for phase transitions. Arch. Rat. Mech. Anal. 138, 105–136 (1997)MathSciNetCrossRefMATH3.Carr J., Chmaj A.: Uniqueness of traveling waves for nonlocal monostable equations. Proc. Amer. Math. Soc. 132, 2433–2439 (2004)MathSciNetCrossRefMATH4.Chasseigne E., Chaves M., Rossi J.D.: Asymptotic behavior for nonlocal diffusion equations. J. Math. Pures Appl. 86, 271–291 (2006)MathSciNetCrossRefMATH5.Chen X.: Existence, uniqueness and asymptotic stability of traveling waves in nonlocal evolution equations. Adv. Diff. Eqns. 2, 125–160 (1997)MathSciNetMATH6.Chmaj A., Ren X.: Homoclinic solutions of an integral equation: existence and stability. J. Diff. Eqns. 155, 17–43 (1999)MathSciNetCrossRefMATH7.A. Chmaj, X. Ren, The nonlocal bistable equation: stationary solutions on a bounded interval, Electronic J. Diff. Eqns. 2002 (2002), no. 2, 1–12.8.Cortázar C., Coville J., Elgueta M., Martínez S.: A non local inhomogeneous dispersal process. J. Diff. Eqns. 241, 332–358 (2007)CrossRefMATH9.C. Cortázar, M. Elgueta, J. García-Melián, S. Martínez, Existence and asymptotic behavior of solutions to some inhomogeneous nonlocal diffusion problems. SIAM J. Math. Anal. 41 (2009), no. 5, 2136–2164.10.C. Cortázar, M. Elgueta, J. García-Melián, S. Martínez, Stationary sign changing solutions for an inhomogeneous nonlocal problem. Indiana Univ. Math. J. 60 (2011), no. 1, 209–232.11.Cortázar C., Elgueta M., García-Melián J., Martínez S.: Finite mass solutions for a nonlocal inhomogeneous dispersal equation. Discrete Cont. Dyn. Syst. 35, 1409–1419 (2015)MathSciNetMATH12.C. Cortázar, M. Elgueta, S. Martínez, J. Rossi. Random walks and the porous medium equation. Rev. Un. Mat. Argentina 50 (2009), no. 2, 149–155.13.Cortázar C., Elgueta M., Rossi J.D., Wolanski N.: Boundary fluxes for nonlocal diffusion. J. Diff. Eqns. 234, 360–390 (2007)MathSciNetCrossRefMATH14.J. Coville. On a simple criterion for the existence of a principal eigenfunction of some nonlocal operators. J. Differential Equations 249 (2010), no. 11, 2921–2953.15.J. Coville, Harnack type inequality for positive solution of some integral equation. Ann. Mat. Pura Appl. (4) 191 (2012), no. 3, 503–528.16.J. Coville, J. Dávila and S. Martínez. Existence and uniqueness of solutions to a nonlocal equation with monostable nonlinearity. SIAM J. Math. Anal. 39 (2008), no. 5, 1693–1709.17.J. Coville, J. Dávila and S. Martínez. Pulsating fronts for nonlocal dispersion and KPP nonlinearity. Ann. Inst. H. Poincaré Anal. Non Linéaire 30 (2013), no. 2, 179–223.18.Coville J., Dupaigne L.: Propagation speed of travelling fronts in nonlocal reaction diffusion equations. Nonl. Anal. 60, 797–819 (2005)MathSciNetCrossRefMATH19.Coville J., Dupaigne L.: On a nonlocal equation arising in population dynamics. Proc. Roy. Soc. Edinburgh 137, 1–29 (2007)MathSciNetCrossRefMATH20.E. B. Davies, “Linear Operators and their Spectra”. Cambridge studies in advanced mathematics 106, Cambridge University Press, Cambridge, 2007.21.P. Fife, Some nonclassical trends in parabolic and parabolic-like evolutions, in “Trends in nonlinear analysis”, pp. 153–191, Springer-Verlag, Berlin, 2003.22.Hutson V., Martínez S., Mischaikow K., Vickers G. T.: The evolution of dispersal. J. Math. Biol. 47, 483–517 (2003)MathSciNetCrossRefMATH23.Ignat L. I., Rossi J. D.: Refined asymptotic expansions for nonlocal evolution equations. J. Evol. Eqns. 8, 617–629 (2008)MathSciNetCrossRefMATH24.M. G. Krein, R. Rutman, Linear operators leaving invariant a cone in a Banach space. Amer. Math. Soc. Translation 1950 (1950) no. 26, 128 pp.25.L. Kong and W. Shen. Positive stationary solutions and spreading speeds of KPP equations in locally spatially inhomogeneous media. Methods Appl. Anal. 18 (2011), no. 4, 427–456.26.W. T. Li, J. W. Sun, and F. Y. Yang, Approximate the Fokker-Planck equation by a class of nonlocal dispersal problems. Nonlinear Anal. 74 (2011), no. 11, 3501–3509.27.Schumacher K.: Travelling-front solutions for integro-differential equations I. J. Reine Angew. Math. 316, 54–70 (1980)MathSciNetMATH28.W. Shen and A. Zhang. Spreading speeds for monostable equations with nonlocal dispersal in space periodic habitats. J. Differential Equations 249 (2010), no. 4, 747–795.29.W. Shen and A. Zhang. Stationary solutions and spreading speeds of nonlocal monostable equations in space periodic habitats. Proc. Amer. Math. Soc. 140 (2012), no. 5, 1681–1696.30.Zhang L.: Existence, uniqueness and exponential stability of traveling wave solutions of some integral differential equations arising from neural networks. J. Diff. Eqns. 197, 162–196 (2004)CrossRefMATH About this Article Title An inhomogeneous nonlocal diffusion problem with unbounded steps Journal Journal of Evolution Equations Volume 16, Issue 1 , pp 209-232 Cover Date2016-03 DOI 10.1007/s00028-015-0299-x Print ISSN 1424-3199 Online ISSN 1424-3202 Publisher Springer International Publishing Additional Links Register for Journal Updates Editorial Board About This Journal Manuscript Submission Topics Analysis Authors Carmen Cortázar (1) Manuel Elgueta (1) Jorge García-Melián (2) (3) Salomé Martínez (4) (5) Author Affiliations 1. Departamento de Matemáticas, Facultad de Matemáticas, Pontificia, Universidad Católica de Chile, Casilla 306, Correo 22, Santiago, Chile 2. Departamento de Análisis Matemático, Universidad de La Laguna, C/. Astrofísico Francisco Sánchez s/n, 38271, La Laguna, Spain 3. Instituto Universitario de Estudios Avanzados, (IUdEA) en Física Atómica, Molecular y Fotónica, Universidad de La Laguna, C/. Astrofísico Francisco Sánchez s/n, 38203, La Laguna, Spain 4. Departamento de Ingeniería Matemática, Universidad de Chile, Blanco Encalada 2120, 5° piso, Santiago, Chile 5. Centro de Modelamiento Matemático, UMI 2807 CNRS-UChile, Universidad de Chile, Casilla 170, Correo 3, Santiago, Chile Continue reading... To view the rest of this content please follow the download PDF link above.