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 $\mathbb{R}^N\ (N\geq1)$: $$ \begin{equation} \left\{\begin{array}{ll} \partial_t u-\Delta u=a(x,t)u-b(x,t)u^p\ \ \ \ & {\rm in}\ \mathbb{R}^N\times(0,T),\\ u(x,0)=u(x,T) \ & {\rm in}\ \mathbb{R}^N, \end{array} \right. \end{equation} $$ 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 & Continuous Dynamical Systems - A, 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.  doi: 10.1006/jmaa.1998.6044.  Google Scholar

[2]

S. Alama and G. Tarantello, On semilinear elliptic equations with indefinite nonlinearities,, Calc. Var. Partial Differential Equations, 1 (1993), 439.   Google Scholar

[3]

S. Alama and G. Tarantello, Elliptic problems with nonlinearities indefinite in sign,, J. Funct. Anal., 141 (1996), 159.  doi: 10.1006/jfan.1996.0125.  Google Scholar

[4]

H. Amann and J. López-Gómez, A priori bounds and multiple solutions for superlinear indefinite elliptic problems,, J. Differential Equations, 146 (1998), 336.  doi: 10.1006/jdeq.1998.3440.  Google Scholar

[5]

H. Berestycki, I. Capuzzo-Dolcetta and L. Nirenberg, Superlinear indefinite elliptic problems and nonlinear Liouville theorems,, Topol. Methods Nonl. Anal., 4 (1994), 59.   Google Scholar

[6]

H. Berestycki, I. Capuzzo-Dolcetta and L. Nirenberg, Variational methods for indefinite superlinear homogeneous elliptic problems,, Nonlinear Differential Equations Appl., 2 (1995), 553.  doi: 10.1007/BF01210623.  Google Scholar

[7]

H. Berestycki, F. Hamel and G. Nadin, Asymptotic spreading in heterogeneous diffusive excitable media,, J. Funct. Anal., 255 (2008), 2146.  doi: 10.1016/j.jfa.2008.06.030.  Google Scholar

[8]

H. Berestycki, F. Hamel and L. Roques, Analysis of the periodically fragmented environment model. I. Species persistence,, J. Math. Biol., 51 (2005), 75.  doi: 10.1007/s00285-004-0313-3.  Google Scholar

[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.  doi: 10.1007/s10231-006-0015-0.  Google Scholar

[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.   Google Scholar

[11]

W. Dong, Positive solutions for logistic type quasilinear elliptic equations on $\mathbbR^N$,, J. Math. Anal. Appl., 290 (2004), 469.  doi: 10.1016/j.jmaa.2003.10.034.  Google Scholar

[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.   Google Scholar

[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, 2 (2006).   Google Scholar

[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.   Google Scholar

[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.  doi: 10.1137/S0036141099352844.  Google Scholar

[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.  doi: 10.1017/S0004972700038685.  Google Scholar

[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.  doi: 10.1017/S0024610701002289.  Google Scholar

[18]

Y. Du and L. Ma, Positive solutions of an elliptic partial differential equation on $\mathbbR^N$,, J. Math. Anal. Appl., 271 (2002), 409.  doi: 10.1016/S0022-247X(02)00124-5.  Google Scholar

[19]

Y. Du and R. Peng, The periodic logistic equation with spatial and temporal degeneracies,, Trans. Amer. Math. Soc., ().   Google Scholar

[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.  doi: 10.3934/dcds.2009.25.123.  Google Scholar

[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.  doi: 10.1006/jdeq.1996.0071.  Google Scholar

[22]

A. Friedman, "Partial Differential Equations of Parabolic Type,'', Printice-Hall, (1964).   Google Scholar

[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.   Google Scholar

[24]

P. Hess, "Periodic-Parabolic Boundary Value Problems and Positivity,'', Pitman Res., 247 (1991).   Google Scholar

[25]

M. Hieber, P. Koch Medina and S. Merino, Diffusive logistic growth on $\mathbbR^N$,, Nonlinear Analysis, 27 (1996), 879.  doi: 10.1016/0362-546X(95)00035-T.  Google Scholar

[26]

J. Kazdan and F. Warner, Scalar curvature and conformal deformation of Riemannian structure,, J. Differential Geometry, 10 (1975), 113.   Google Scholar

[27]

P. Koch Medina and G. Schätti, Long-time behaviour of reaction-diffusion equations on $\mathbbR^N$,, Nonlinear Analysis, 25 (1995), 831.   Google Scholar

[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, 1 (1937), 1.   Google Scholar

[29]

O. Ladyzenskaja, V. Solonnikov and N. Ural'ceva, "Linear and Quasi-Linear Equations of Parabolic Type,'', Translations of Mathematical Monographs, 23 (1967).   Google Scholar

[30]

G. Lieberman, "Second Order Parabolic Differential Equations,'', World Scientific Publ. Co., (1996).   Google Scholar

[31]

L. Ma and X. W. Xu, Positive solutions of a logistic equation on unbounded intervals,, Proc. Amer. Math. Soc., 130 (2002), 2947.  doi: 10.1090/S0002-9939-02-06405-5.  Google Scholar

[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.   Google Scholar

[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.  doi: 10.1007/s002050050116.  Google Scholar

[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.  doi: 10.1512/iumj.1991.40.40049.  Google Scholar

[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.  doi: 10.2307/2154124.  Google Scholar

[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.  doi: 10.1090/S0002-9947-09-04752-7.  Google Scholar

[37]

L. Rossi, Liouville type results for periodic and almost periodic linear operators,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 2481.   Google Scholar

[38]

J. Shi and R. Shivaji, Semilinear elliptic equations with generalized cubic nonlinearities,, Disc. Cont. Dyna. Syst., 2005 (): 798.   Google Scholar

[39]

H. Tehrani, On indefinite superlinear elliptic equations,, Calc. Var. Partial Differential Equations, 4 (1996), 139.   Google Scholar

show all references

References:
[1]

G. Afrouzi and K. J. Brown, On a diffusive logistic equation,, J. Math. Anal. Appl., 225 (1998), 326.  doi: 10.1006/jmaa.1998.6044.  Google Scholar

[2]

S. Alama and G. Tarantello, On semilinear elliptic equations with indefinite nonlinearities,, Calc. Var. Partial Differential Equations, 1 (1993), 439.   Google Scholar

[3]

S. Alama and G. Tarantello, Elliptic problems with nonlinearities indefinite in sign,, J. Funct. Anal., 141 (1996), 159.  doi: 10.1006/jfan.1996.0125.  Google Scholar

[4]

H. Amann and J. López-Gómez, A priori bounds and multiple solutions for superlinear indefinite elliptic problems,, J. Differential Equations, 146 (1998), 336.  doi: 10.1006/jdeq.1998.3440.  Google Scholar

[5]

H. Berestycki, I. Capuzzo-Dolcetta and L. Nirenberg, Superlinear indefinite elliptic problems and nonlinear Liouville theorems,, Topol. Methods Nonl. Anal., 4 (1994), 59.   Google Scholar

[6]

H. Berestycki, I. Capuzzo-Dolcetta and L. Nirenberg, Variational methods for indefinite superlinear homogeneous elliptic problems,, Nonlinear Differential Equations Appl., 2 (1995), 553.  doi: 10.1007/BF01210623.  Google Scholar

[7]

H. Berestycki, F. Hamel and G. Nadin, Asymptotic spreading in heterogeneous diffusive excitable media,, J. Funct. Anal., 255 (2008), 2146.  doi: 10.1016/j.jfa.2008.06.030.  Google Scholar

[8]

H. Berestycki, F. Hamel and L. Roques, Analysis of the periodically fragmented environment model. I. Species persistence,, J. Math. Biol., 51 (2005), 75.  doi: 10.1007/s00285-004-0313-3.  Google Scholar

[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.  doi: 10.1007/s10231-006-0015-0.  Google Scholar

[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.   Google Scholar

[11]

W. Dong, Positive solutions for logistic type quasilinear elliptic equations on $\mathbbR^N$,, J. Math. Anal. Appl., 290 (2004), 469.  doi: 10.1016/j.jmaa.2003.10.034.  Google Scholar

[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.   Google Scholar

[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, 2 (2006).   Google Scholar

[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.   Google Scholar

[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.  doi: 10.1137/S0036141099352844.  Google Scholar

[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.  doi: 10.1017/S0004972700038685.  Google Scholar

[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.  doi: 10.1017/S0024610701002289.  Google Scholar

[18]

Y. Du and L. Ma, Positive solutions of an elliptic partial differential equation on $\mathbbR^N$,, J. Math. Anal. Appl., 271 (2002), 409.  doi: 10.1016/S0022-247X(02)00124-5.  Google Scholar

[19]

Y. Du and R. Peng, The periodic logistic equation with spatial and temporal degeneracies,, Trans. Amer. Math. Soc., ().   Google Scholar

[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.  doi: 10.3934/dcds.2009.25.123.  Google Scholar

[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.  doi: 10.1006/jdeq.1996.0071.  Google Scholar

[22]

A. Friedman, "Partial Differential Equations of Parabolic Type,'', Printice-Hall, (1964).   Google Scholar

[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.   Google Scholar

[24]

P. Hess, "Periodic-Parabolic Boundary Value Problems and Positivity,'', Pitman Res., 247 (1991).   Google Scholar

[25]

M. Hieber, P. Koch Medina and S. Merino, Diffusive logistic growth on $\mathbbR^N$,, Nonlinear Analysis, 27 (1996), 879.  doi: 10.1016/0362-546X(95)00035-T.  Google Scholar

[26]

J. Kazdan and F. Warner, Scalar curvature and conformal deformation of Riemannian structure,, J. Differential Geometry, 10 (1975), 113.   Google Scholar

[27]

P. Koch Medina and G. Schätti, Long-time behaviour of reaction-diffusion equations on $\mathbbR^N$,, Nonlinear Analysis, 25 (1995), 831.   Google Scholar

[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, 1 (1937), 1.   Google Scholar

[29]

O. Ladyzenskaja, V. Solonnikov and N. Ural'ceva, "Linear and Quasi-Linear Equations of Parabolic Type,'', Translations of Mathematical Monographs, 23 (1967).   Google Scholar

[30]

G. Lieberman, "Second Order Parabolic Differential Equations,'', World Scientific Publ. Co., (1996).   Google Scholar

[31]

L. Ma and X. W. Xu, Positive solutions of a logistic equation on unbounded intervals,, Proc. Amer. Math. Soc., 130 (2002), 2947.  doi: 10.1090/S0002-9939-02-06405-5.  Google Scholar

[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.   Google Scholar

[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.  doi: 10.1007/s002050050116.  Google Scholar

[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.  doi: 10.1512/iumj.1991.40.40049.  Google Scholar

[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.  doi: 10.2307/2154124.  Google Scholar

[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.  doi: 10.1090/S0002-9947-09-04752-7.  Google Scholar

[37]

L. Rossi, Liouville type results for periodic and almost periodic linear operators,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 2481.   Google Scholar

[38]

J. Shi and R. Shivaji, Semilinear elliptic equations with generalized cubic nonlinearities,, Disc. Cont. Dyna. Syst., 2005 (): 798.   Google Scholar

[39]

H. Tehrani, On indefinite superlinear elliptic equations,, Calc. Var. Partial Differential Equations, 4 (1996), 139.   Google Scholar

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