American Institute of Mathematical Sciences

February  2012, 32(2): 619-641. doi: 10.3934/dcds.2012.32.619

The periodic-parabolic logistic equation on $\mathbb{R}^N$

 1 Department of Mathematics, Xuzhou Normal University, Xuzhou, 221116, China 2 Hebei University of Engineering, Handan City, Hebei Province, 056038, China

Received  August 2010 Revised  June 2011 Published  September 2011

In this article, we investigate the periodic-parabolic logistic equation on the entire space $\mathbb{R}^N\ (N\geq1)$: $$$$\left\{\begin{array}{ll} \partial_t u-\Delta u=a(x,t)u-b(x,t)u^p\ \ \ \ & {\rm in}\ \mathbb{R}^N\times(0,T),\\ u(x,0)=u(x,T) \ & {\rm in}\ \mathbb{R}^N, \end{array} \right.$$$$ where the constants $T>0$ and $p>1$, and the functions $a,\ b$ with $b>0$ are smooth in $\mathbb{R}^N\times\mathbb{R}$ and $T$-periodic in time. Under the assumptions that $a(x,t)/{|x|^\gamma}$ and $b(x,t)/{|x|^\tau}$ are bounded away from $0$ and infinity for all large $|x|$, where the constants $\gamma>-2$ and $\tau\in\mathbb{R}$, we study the existence and uniqueness of positive $T$-periodic solutions. In particular, we determine the asymptotic behavior of the unique positive $T$-periodic solution as $|x|\to\infty$, which turns out to depend on the sign of $\gamma$. Our investigation considerably generalizes the existing results.
Citation: Rui Peng, Dong Wei. The periodic-parabolic logistic equation on $\mathbb{R}^N$. Discrete and Continuous Dynamical Systems, 2012, 32 (2) : 619-641. doi: 10.3934/dcds.2012.32.619
References:
 [1] G. Afrouzi and K. J. Brown, On a diffusive logistic equation, J. Math. Anal. Appl., 225 (1998), 326-339. doi: 10.1006/jmaa.1998.6044. [2] S. Alama and G. Tarantello, On semilinear elliptic equations with indefinite nonlinearities, Calc. Var. Partial Differential Equations, 1 (1993), 439-475. [3] S. Alama and G. Tarantello, Elliptic problems with nonlinearities indefinite in sign, J. Funct. Anal., 141 (1996), 159-215. doi: 10.1006/jfan.1996.0125. [4] H. Amann and J. López-Gómez, A priori bounds and multiple solutions for superlinear indefinite elliptic problems, J. Differential Equations, 146 (1998), 336-374. doi: 10.1006/jdeq.1998.3440. [5] H. Berestycki, I. Capuzzo-Dolcetta and L. Nirenberg, Superlinear indefinite elliptic problems and nonlinear Liouville theorems, Topol. Methods Nonl. Anal., 4 (1994), 59-78. [6] H. Berestycki, I. Capuzzo-Dolcetta and L. Nirenberg, Variational methods for indefinite superlinear homogeneous elliptic problems, Nonlinear Differential Equations Appl., 2 (1995), 553-572. doi: 10.1007/BF01210623. [7] H. Berestycki, F. Hamel and G. Nadin, Asymptotic spreading in heterogeneous diffusive excitable media, J. Funct. Anal., 255 (2008), 2146-2189. doi: 10.1016/j.jfa.2008.06.030. [8] H. Berestycki, F. Hamel and L. Roques, Analysis of the periodically fragmented environment model. I. Species persistence, J. Math. Biol., 51 (2005), 75-113. doi: 10.1007/s00285-004-0313-3. [9] H. Berestycki, F. Hamel and L. Rossi, Liouville-type results for semilinear elliptic equations in unbounded domains, Ann. Mat. Pura Appl., 186 (2007), 469-507. doi: 10.1007/s10231-006-0015-0. [10] D. Costa and H. Tehrani, Existence of positive solutions for a class of indefinite elliptic problems in $\mathbbR^N$, Calc. Var. Partial Differential Equations, 13 (2001), 159-189. [11] W. Dong, Positive solutions for logistic type quasilinear elliptic equations on $\mathbbR^N$, J. Math. Anal. Appl., 290 (2004), 469-480. doi: 10.1016/j.jmaa.2003.10.034. [12] Y. Du, Multiplicity of positive solutions for an indefinite superlinear elliptic problem on $\mathbbR^N$, Ann. Inst. H. Poincaré Anal. Non Linéaire, 21 (2004), 657-672. [13] Y. Du, Order Structure and Topological Methods in Nonlinear Partial Differential Equations. Vol. 1. Maximum Principles and Applications "Order Structure and Topological Methods in Nonlinear Partial Differential Equations. Vol. 1. Maximum Principles and Applications," Series in Partial Differential Equations and Applications, 2, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2006. [14] Y. Du and Z. Guo, Liouville type results and eventual flatness of positive solutions for $p$-Laplacian equations, Adv. Differential Equations, 7 (2002), 1479-1512. [15] Y. Du and Q. Huang, Blow-up solutions for a class of semilinear elliptic and parabolic equations, SIAM J. Math. Anal., 31 (1999), 1-18. doi: 10.1137/S0036141099352844. [16] Y. Du and L. Liu, Remarks on the uniqueness problem for the logistic equation on the entire space, Bull. Austral. Math. Soc., 73 (2006), 129-137. doi: 10.1017/S0004972700038685. [17] Y. Du and L. Ma, Logistic type equations on $\mathbbR^N$ by a squeezing method involving boundary blow-up solutions, J. London Math. Soc., 64 (2001), 107-124. doi: 10.1017/S0024610701002289. [18] Y. Du and L. Ma, Positive solutions of an elliptic partial differential equation on $\mathbbR^N$, J. Math. Anal. Appl., 271 (2002), 409-425. doi: 10.1016/S0022-247X(02)00124-5. [19] Y. Du and R. Peng, The periodic logistic equation with spatial and temporal degeneracies,, Trans. Amer. Math. Soc., (). [20] Y. Du and Y. Yamada, On the long-time limit of positive solutions to the degenerate logistic equation, Disc. Cont. Dyna. Syst. A, 25 (2009), 123-132. doi: 10.3934/dcds.2009.25.123. [21] J. Fraile, P. Koch Medina, J. López-Gómez and S. Merino, Elliptic eigenvalue problems and unbounded continua of positive solutions of a semilinear elliptic equation, J. Differential Equations, 127 (1996), 295-319. doi: 10.1006/jdeq.1996.0071. [22] A. Friedman, "Partial Differential Equations of Parabolic Type,'' Printice-Hall, Englewood Cliffs, N.J., 1964. [23] J. García-Melián, R. Gómez-Reñasco, J. López-Gómez and J. Sabina de Lis, Pointwise growth and uniqueness of positive solutions for a class of sublinear elliptic problems where bifurcation from infinity occurs, Arch. Ration. Mech. Anal., 145 (1998), 261-289. [24] P. Hess, "Periodic-Parabolic Boundary Value Problems and Positivity,'' Pitman Res., Notes in Mathematics, 247, Longman Sci. Tech., Harlow, copublished in the United States with John Wiley & Sons, Inc., New York, 1991. [25] M. Hieber, P. Koch Medina and S. Merino, Diffusive logistic growth on $\mathbbR^N$, Nonlinear Analysis, 27 (1996), 879-894. doi: 10.1016/0362-546X(95)00035-T. [26] J. Kazdan and F. Warner, Scalar curvature and conformal deformation of Riemannian structure, J. Differential Geometry, 10 (1975), 113-134. [27] P. Koch Medina and G. Schätti, Long-time behaviour of reaction-diffusion equations on $\mathbbR^N$, Nonlinear Analysis, 25 (1995), 831-870. [28] A. N. Kolmogorov, I. G. Petrovskiĭ and N. S. Piskunov, A study of the diffusion equation with increase in the amount of substance, and its application to a biological problem, Bull. Univ. Moscow Ser. Internat, Sec. A, 1 (1937), 1-26. [29] O. Ladyzenskaja, V. Solonnikov and N. Ural'ceva, "Linear and Quasi-Linear Equations of Parabolic Type,'' Translations of Mathematical Monographs, 23, AMS, Providence, RI, 1967. [30] G. Lieberman, "Second Order Parabolic Differential Equations,'' World Scientific Publ. Co., Inc., River Edge, NJ, 1996. [31] L. Ma and X. W. Xu, Positive solutions of a logistic equation on unbounded intervals, Proc. Amer. Math. Soc., 130 (2002), 2947-2958. doi: 10.1090/S0002-9939-02-06405-5. [32] M. Marcus and L. Véron, Uniqueness and asymptotic behavior of solutions with boundary blow-up for a class of nonlinear elliptic equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 14 (1997), 237-274. [33] M. Marcus and L. Véron, The boundary trace of positive solutions of semilinear elliptic equations: The subcritical case, Arch. Ration. Mech. Anal., 144 (1998), 201-231. doi: 10.1007/s002050050116. [34] T. Ouyang, On the positive solutions of semilinear equations $\Delta u+\lambda u+hu^p=0$ on compact manifolds. II, Indiana Univ. Math. J., 40 (1991), 1083-1141. doi: 10.1512/iumj.1991.40.40049. [35] T. Ouyang, On the positive solutions of semilinear equations $\Delta u+\lambda u-hu^p=0$ on the compact manifolds, Trans. Amer. Math. Soc., 331 (1992), 503-527. doi: 10.2307/2154124. [36] S. Pigola, M. Rigoli and A. Setti, Existence and non-existence results for a logistic-type equation on manifolds, Trans. Amer. Math. Soc., 362 (2010), 1907-1936. doi: 10.1090/S0002-9947-09-04752-7. [37] L. Rossi, Liouville type results for periodic and almost periodic linear operators, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 2481-2502. [38] J. Shi and R. Shivaji, Semilinear elliptic equations with generalized cubic nonlinearities,, Disc. Cont. Dyna. Syst., 2005 (): 798. [39] H. Tehrani, On indefinite superlinear elliptic equations, Calc. Var. Partial Differential Equations, 4 (1996), 139-153.

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References:
 [1] G. Afrouzi and K. J. Brown, On a diffusive logistic equation, J. Math. Anal. Appl., 225 (1998), 326-339. doi: 10.1006/jmaa.1998.6044. [2] S. Alama and G. Tarantello, On semilinear elliptic equations with indefinite nonlinearities, Calc. Var. Partial Differential Equations, 1 (1993), 439-475. [3] S. Alama and G. Tarantello, Elliptic problems with nonlinearities indefinite in sign, J. Funct. Anal., 141 (1996), 159-215. doi: 10.1006/jfan.1996.0125. [4] H. Amann and J. López-Gómez, A priori bounds and multiple solutions for superlinear indefinite elliptic problems, J. Differential Equations, 146 (1998), 336-374. doi: 10.1006/jdeq.1998.3440. [5] H. Berestycki, I. Capuzzo-Dolcetta and L. Nirenberg, Superlinear indefinite elliptic problems and nonlinear Liouville theorems, Topol. Methods Nonl. Anal., 4 (1994), 59-78. [6] H. Berestycki, I. Capuzzo-Dolcetta and L. Nirenberg, Variational methods for indefinite superlinear homogeneous elliptic problems, Nonlinear Differential Equations Appl., 2 (1995), 553-572. doi: 10.1007/BF01210623. [7] H. Berestycki, F. Hamel and G. Nadin, Asymptotic spreading in heterogeneous diffusive excitable media, J. Funct. Anal., 255 (2008), 2146-2189. doi: 10.1016/j.jfa.2008.06.030. [8] H. Berestycki, F. Hamel and L. Roques, Analysis of the periodically fragmented environment model. I. Species persistence, J. Math. Biol., 51 (2005), 75-113. doi: 10.1007/s00285-004-0313-3. [9] H. Berestycki, F. Hamel and L. Rossi, Liouville-type results for semilinear elliptic equations in unbounded domains, Ann. Mat. Pura Appl., 186 (2007), 469-507. doi: 10.1007/s10231-006-0015-0. [10] D. Costa and H. Tehrani, Existence of positive solutions for a class of indefinite elliptic problems in $\mathbbR^N$, Calc. Var. Partial Differential Equations, 13 (2001), 159-189. [11] W. Dong, Positive solutions for logistic type quasilinear elliptic equations on $\mathbbR^N$, J. Math. Anal. Appl., 290 (2004), 469-480. doi: 10.1016/j.jmaa.2003.10.034. [12] Y. Du, Multiplicity of positive solutions for an indefinite superlinear elliptic problem on $\mathbbR^N$, Ann. Inst. H. Poincaré Anal. Non Linéaire, 21 (2004), 657-672. [13] Y. Du, Order Structure and Topological Methods in Nonlinear Partial Differential Equations. Vol. 1. Maximum Principles and Applications "Order Structure and Topological Methods in Nonlinear Partial Differential Equations. Vol. 1. Maximum Principles and Applications," Series in Partial Differential Equations and Applications, 2, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2006. [14] Y. Du and Z. Guo, Liouville type results and eventual flatness of positive solutions for $p$-Laplacian equations, Adv. Differential Equations, 7 (2002), 1479-1512. [15] Y. Du and Q. Huang, Blow-up solutions for a class of semilinear elliptic and parabolic equations, SIAM J. Math. Anal., 31 (1999), 1-18. doi: 10.1137/S0036141099352844. [16] Y. Du and L. Liu, Remarks on the uniqueness problem for the logistic equation on the entire space, Bull. Austral. Math. Soc., 73 (2006), 129-137. doi: 10.1017/S0004972700038685. [17] Y. Du and L. Ma, Logistic type equations on $\mathbbR^N$ by a squeezing method involving boundary blow-up solutions, J. London Math. Soc., 64 (2001), 107-124. doi: 10.1017/S0024610701002289. [18] Y. Du and L. Ma, Positive solutions of an elliptic partial differential equation on $\mathbbR^N$, J. Math. Anal. Appl., 271 (2002), 409-425. doi: 10.1016/S0022-247X(02)00124-5. [19] Y. Du and R. Peng, The periodic logistic equation with spatial and temporal degeneracies,, Trans. Amer. Math. Soc., (). [20] Y. Du and Y. Yamada, On the long-time limit of positive solutions to the degenerate logistic equation, Disc. Cont. Dyna. Syst. A, 25 (2009), 123-132. doi: 10.3934/dcds.2009.25.123. [21] J. Fraile, P. Koch Medina, J. López-Gómez and S. Merino, Elliptic eigenvalue problems and unbounded continua of positive solutions of a semilinear elliptic equation, J. Differential Equations, 127 (1996), 295-319. doi: 10.1006/jdeq.1996.0071. [22] A. Friedman, "Partial Differential Equations of Parabolic Type,'' Printice-Hall, Englewood Cliffs, N.J., 1964. [23] J. García-Melián, R. Gómez-Reñasco, J. López-Gómez and J. Sabina de Lis, Pointwise growth and uniqueness of positive solutions for a class of sublinear elliptic problems where bifurcation from infinity occurs, Arch. Ration. Mech. Anal., 145 (1998), 261-289. [24] P. Hess, "Periodic-Parabolic Boundary Value Problems and Positivity,'' Pitman Res., Notes in Mathematics, 247, Longman Sci. Tech., Harlow, copublished in the United States with John Wiley & Sons, Inc., New York, 1991. [25] M. Hieber, P. Koch Medina and S. Merino, Diffusive logistic growth on $\mathbbR^N$, Nonlinear Analysis, 27 (1996), 879-894. doi: 10.1016/0362-546X(95)00035-T. [26] J. Kazdan and F. Warner, Scalar curvature and conformal deformation of Riemannian structure, J. Differential Geometry, 10 (1975), 113-134. [27] P. Koch Medina and G. Schätti, Long-time behaviour of reaction-diffusion equations on $\mathbbR^N$, Nonlinear Analysis, 25 (1995), 831-870. [28] A. N. Kolmogorov, I. G. Petrovskiĭ and N. S. Piskunov, A study of the diffusion equation with increase in the amount of substance, and its application to a biological problem, Bull. Univ. Moscow Ser. Internat, Sec. A, 1 (1937), 1-26. [29] O. Ladyzenskaja, V. Solonnikov and N. Ural'ceva, "Linear and Quasi-Linear Equations of Parabolic Type,'' Translations of Mathematical Monographs, 23, AMS, Providence, RI, 1967. [30] G. Lieberman, "Second Order Parabolic Differential Equations,'' World Scientific Publ. Co., Inc., River Edge, NJ, 1996. [31] L. Ma and X. W. Xu, Positive solutions of a logistic equation on unbounded intervals, Proc. Amer. Math. Soc., 130 (2002), 2947-2958. doi: 10.1090/S0002-9939-02-06405-5. [32] M. Marcus and L. Véron, Uniqueness and asymptotic behavior of solutions with boundary blow-up for a class of nonlinear elliptic equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 14 (1997), 237-274. [33] M. Marcus and L. Véron, The boundary trace of positive solutions of semilinear elliptic equations: The subcritical case, Arch. Ration. Mech. Anal., 144 (1998), 201-231. doi: 10.1007/s002050050116. [34] T. Ouyang, On the positive solutions of semilinear equations $\Delta u+\lambda u+hu^p=0$ on compact manifolds. II, Indiana Univ. Math. J., 40 (1991), 1083-1141. doi: 10.1512/iumj.1991.40.40049. [35] T. Ouyang, On the positive solutions of semilinear equations $\Delta u+\lambda u-hu^p=0$ on the compact manifolds, Trans. Amer. Math. Soc., 331 (1992), 503-527. doi: 10.2307/2154124. [36] S. Pigola, M. Rigoli and A. Setti, Existence and non-existence results for a logistic-type equation on manifolds, Trans. Amer. Math. Soc., 362 (2010), 1907-1936. doi: 10.1090/S0002-9947-09-04752-7. [37] L. Rossi, Liouville type results for periodic and almost periodic linear operators, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 2481-2502. [38] J. Shi and R. Shivaji, Semilinear elliptic equations with generalized cubic nonlinearities,, Disc. Cont. Dyna. Syst., 2005 (): 798. [39] H. Tehrani, On indefinite superlinear elliptic equations, Calc. Var. Partial Differential Equations, 4 (1996), 139-153.
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