On a simple criterion for the existence of a principal eigenfunction of some nonlocal operators. (English) Zbl 1218.45002
The paper deals with integral operators of the form
\[ Mu=\int_\Omega k(x,y)u(y) dy -b(x)u\,, \]
where \(\Omega\subset\mathbb{R}^n\) is an open connected set,
\[ k(x,y)=J\left(\frac{x-y}{g(y)}\right)\frac{1}{g^n(y)}\,, \]
\(J\) is a continuous nonnegative compactly supported function with \(J(0)>0\), \(g\in L^\infty(\Omega)\) (\(0<\alpha\leq g\leq\beta\)), and \(b\in C(\bar\Omega)\cap L^\infty(\Omega)\). The author studies the existence and certain properties of principal eigenpairs for \(M\). As an application the asymptotic behavior of solutions to the associated evolution problem \(u_t=Mu+f(x,u)\) is described.
\[ Mu=\int_\Omega k(x,y)u(y) dy -b(x)u\,, \]
where \(\Omega\subset\mathbb{R}^n\) is an open connected set,
\[ k(x,y)=J\left(\frac{x-y}{g(y)}\right)\frac{1}{g^n(y)}\,, \]
\(J\) is a continuous nonnegative compactly supported function with \(J(0)>0\), \(g\in L^\infty(\Omega)\) (\(0<\alpha\leq g\leq\beta\)), and \(b\in C(\bar\Omega)\cap L^\infty(\Omega)\). The author studies the existence and certain properties of principal eigenpairs for \(M\). As an application the asymptotic behavior of solutions to the associated evolution problem \(u_t=Mu+f(x,u)\) is described.
Reviewer: Aleksander Pankov (Baltimore)
MSC:
| 45C05 | Eigenvalue problems for integral equations |
| 35B50 | Maximum principles in context of PDEs |
| 47G20 | Integro-differential operators |
| 35R09 | Integro-partial differential equations |
| 35B40 | Asymptotic behavior of solutions to PDEs |
| 45K05 | Integro-partial differential equations |
Keywords:
nonlocal diffusion operator; principal eigenvalue; asymptotic behavior; principal eigenfunction; integral operators; principal eigenpairs; evolution problemReferences:
| [1] | Alberti, Giovanni; Bellettini, Giovanni, A nonlocal anisotropic model for phase transitions. I. The optimal profile problem, Math. Ann., 310, 3, 527-560 (1998) · Zbl 0891.49021 |
| [2] | Bates, Peter W.; Chmaj, Adam, An integrodifferential model for phase transitions: stationary solutions in higher space dimensions, J. Stat. Phys., 95, 5-6, 1119-1139 (1999) · Zbl 0958.82015 |
| [3] | Bates, Peter W.; Fife, Paul C.; Ren, Xiaofeng; Wang, Xuefeng, Traveling waves in a convolution model for phase transitions, Arch. Ration. Mech. Anal., 138, 2, 105-136 (1997) · Zbl 0889.45012 |
| [4] | Bates, Peter W.; Zhao, Guangyu, Existence, uniqueness and stability of the stationary solution to a nonlocal evolution equation arising in population dispersal, J. Math. Anal. Appl., 332, 1, 428-440 (2007) · Zbl 1114.35017 |
| [5] | Berestycki, H.; Nirenberg, L.; Varadhan, S. R.S., The principal eigenvalue and maximum principle for second-order elliptic operators in general domains, Comm. Pure Appl. Math., 47, 1, 47-92 (1994) · Zbl 0806.35129 |
| [6] | Berestycki, Henri; Hamel, François; Roques, Lionel, Analysis of the periodically fragmented environment model. I. Species persistence, J. Math. Biol., 51, 1, 75-113 (2005) · Zbl 1066.92047 |
| [7] | Berestycki, Henri; Hamel, François; Rossi, Luca, Liouville-type results for semilinear elliptic equations in unbounded domains, Ann. Mat. Pura Appl. (4), 186, 3, 469-507 (2007) · Zbl 1223.35022 |
| [8] | Browder, Felix E., On the spectral theory of elliptic differential operators. I, Math. Ann., 142, 22-130 (1960/1961) · Zbl 0104.07502 |
| [9] | Cain, Michael L.; Milligan, Brook G.; Strand, Allan E., Long-distance seed dispersal in plant populations, Am. J. Bot., 87, 9, 1217-1227 (2000) |
| [10] | Chasseigne, Emmanuel; Chaves, Manuela; Rossi, Julio D., Asymptotic behavior for nonlocal diffusion equations, J. Math. Pures Appl. (9), 86, 3, 271-291 (2006) · Zbl 1126.35081 |
| [11] | Chen, Xinfu, Existence, uniqueness, and asymptotic stability of traveling waves in nonlocal evolution equations, Adv. Differential Equations, 2, 1, 125-160 (1997) · Zbl 1023.35513 |
| [12] | Chmaj, Adam J. J.; Ren, Xiaofeng, The nonlocal bistable equation: stationary solutions on a bounded interval, Electron. J. Differential Equations (2002), No. 02, 12 pp. (electronic) · Zbl 0992.45002 |
| [13] | Clark, James S., Why trees migrate so fast: Confronting theory with dispersal biology and the paleorecord, Amer. Natural., 152, 2, 204-224 (1998) |
| [14] | Cortázar, C.; Coville, J.; Elgueta, M.; Martínez, S., A nonlocal inhomogeneous dispersal process, J. Differential Equations, 241, 2, 332-358 (2007) · Zbl 1127.45003 |
| [15] | Cortazar, Carmen; Elgueta, Manuel; Rossi, Julio D., A nonlocal diffusion equation whose solutions develop a free boundary, Ann. Henri Poincaré, 6, 2, 269-281 (2005) · Zbl 1067.35136 |
| [16] | Coville, Jérôme, On uniqueness and monotonicity of solutions of non-local reaction diffusion equation, Ann. Mat. Pura Appl. (4), 185, 3, 461-485 (2006) · Zbl 1232.35084 |
| [17] | Coville, Jérôme, Remarks on the strong maximum principle for nonlocal operators, Electron. J. Differential Equations (2008), No. 66, 10 pp · Zbl 1173.35390 |
| [18] | Jérôme Coville, Harnack’s inequality for some nonlocal equations and application, preprint du MPI, February 2008.; Jérôme Coville, Harnack’s inequality for some nonlocal equations and application, preprint du MPI, February 2008. |
| [19] | Jérôme Coville, Travelling fronts in asymmetric nonlocal reaction diffusion equation: The bistable and ignition case, preprint du CMM, July 2006.; Jérôme Coville, Travelling fronts in asymmetric nonlocal reaction diffusion equation: The bistable and ignition case, preprint du CMM, July 2006. |
| [20] | Coville, Jérôme; Dávila, Juan; Martínez, Salomé, Existence and uniqueness of solutions to a nonlocal equation with monostable nonlinearity, SIAM J. Math. Anal., 39, 5, 1693-1709 (2008) · Zbl 1161.45003 |
| [21] | Coville, Jérôme; Dávila, Juan; Martínez, Salomé, Nonlocal anisotropic dispersal with monostable nonlinearity, J. Differential Equations, 244, 12, 3080-3118 (2008) · Zbl 1148.45011 |
| [22] | Coville, Jérôme; Dupaigne, Louis, Propagation speed of travelling fronts in non local reaction-diffusion equations, Nonlinear Anal., 60, 5, 797-819 (2005) · Zbl 1069.45008 |
| [23] | Coville, Jerome; Dupaigne, Louis, On a non-local equation arising in population dynamics, Proc. Roy. Soc. Edinburgh Sect. A, 137, 4, 727-755 (2007) · Zbl 1133.35056 |
| [24] | De Masi, A.; Gobron, T.; Presutti, E., Travelling fronts in non-local evolution equations, Arch. Ration. Mech. Anal., 132, 2, 143-205 (1995) · Zbl 0847.45008 |
| [25] | E. Deveaux, C. Klein, Estimation de la dispersion de pollen à longue distance à l’echelle d’un paysage agicole : une approche expérimentale, Publication du Laboratoire Ecologie, Systématique et Evolution, 2004.; E. Deveaux, C. Klein, Estimation de la dispersion de pollen à longue distance à l’echelle d’un paysage agicole : une approche expérimentale, Publication du Laboratoire Ecologie, Systématique et Evolution, 2004. |
| [26] | Donsker, Monroe D.; Varadhan, S. R.S., On a variational formula for the principal eigenvalue for operators with maximum principle, Proc. Natl. Acad. Sci. USA, 72, 780-783 (1975) · Zbl 0353.49039 |
| [27] | Edmunds, D. E.; Potter, A. J.B.; Stuart, C. A., Non-compact positive operators, Proc. R. Soc. London Ser. A, 328, 1572, 67-81 (1972) · Zbl 0232.47035 |
| [28] | Evans, Lawrence C., Partial Differential Equations, Grad. Stud. Math., vol. 19 (1998), American Mathematical Society: American Mathematical Society Providence, RI · Zbl 0902.35002 |
| [29] | Fife, Paul C., Mathematical Aspects of Reacting and Diffusing Systems, Lecture Notes in Biomath., vol. 28 (1979), Springer-Verlag: Springer-Verlag Berlin · Zbl 0403.92004 |
| [30] | Fife, Paul C., An integrodifferential analog of semilinear parabolic PDEs, (Partial Differential Equations and Applications. Partial Differential Equations and Applications, Lect. Notes Pure Appl. Math., vol. 177 (1996), Dekker: Dekker New York), 137-145 · Zbl 0846.45004 |
| [31] | García-Melián, Jorge; Rossi, Julio D., On the principal eigenvalue of some nonlocal diffusion problems, J. Differential Equations, 246, 1, 21-38 (2009) · Zbl 1162.35055 |
| [32] | Hutson, V.; Martinez, S.; Mischaikow, K.; Vickers, G. T., The evolution of dispersal, J. Math. Biol., 47, 6, 483-517 (2003) · Zbl 1052.92042 |
| [33] | Medlock, Jan; Kot, Mark, Spreading disease: integro-differential equations old and new, Math. Biosci., 184, 2, 201-222 (2003) · Zbl 1036.92030 |
| [34] | Murray, J. D., Mathematical Biology, Biomathematics, vol. 19 (1993), Springer-Verlag: Springer-Verlag Berlin · Zbl 0779.92001 |
| [35] | Nussbaum, Roger D., The radius of the essential spectrum, Duke Math. J., 37, 473-478 (1970) · Zbl 0216.41602 |
| [36] | Nussbaum, Roger D.; Pinchover, Yehuda, On variational principles for the generalized principal eigenvalue of second order elliptic operators and some applications, J. Anal. Math., 59, 161-177 (1992), Festschrift on the occasion of the 70th birthday of Shmuel Agmon · Zbl 0816.35095 |
| [37] | Protter, Murray H.; Weinberger, Hans F., Maximum Principles in Differential Equations (1967), Prentice Hall Inc.: Prentice Hall Inc. Englewood Cliffs, NJ · Zbl 0549.35002 |
| [38] | Pucci, Carlo, Maximum and minimum first eigenvalues for a class of elliptic operators, Proc. Amer. Math. Soc., 17, 788-795 (1966) · Zbl 0149.07601 |
| [39] | Schurr, Frank M.; Steinitz, Ofer; Nathan, Ran, Plant fecundity and seed dispersal in spatially heterogeneous environments: models, mechanisms and estimation, J. Ecol., 96, 4, 628-641 (2008) |
| [40] | Zeidler, Eberhard, Nonlinear Functional Analysis and Its Applications. I: Fixed-Point Theorems (1986), Springer-Verlag: Springer-Verlag New York, translated from the German by Peter R. Wadsack · Zbl 0583.47050 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
