Applied Mathematics and Computation 148 (2004) 453–460 www.elsevier.com/locate/amc A sufficient condition for the existence of periodic solution for a reaction diffusion equation with infinite delay q Yanbin Tang *, Li Zhou Department of Mathematics, Huazhong University of Science and Technology, Wuhan, Hubei 430074, PR China Abstract For a periodic reaction diffusion equation with infinite delay, a sufficient condition of existence and uniqueness of periodic solution is given By using the periodic monotone method, the asymptotic behavior of the time-dependent solutions is investigated, that is, the time-dependent solutions converge to the corresponding periodic solution in the time variable Ó 2002 Elsevier Inc All rights reserved Keywords: Reaction diffusion equation; Periodic solution; Existence; Asymptotic behavior; Delay Introduction In [1], Redlinger considered a single species population model with diffusion and infinite delay Luðt; xÞ ¼ uðt; xÞ a À buðt; xÞ À Z kðsÞuðt À s; xÞ ds ; ðt; xÞ Rþ  X; ð1:1Þ q The project supported by National Natural Science Foundation of China Corresponding author E-mail address: tangyb@public.wh.hb.cn (Y Tang) * 0096-3003/$ - see front matter Ó 2002 Elsevier Inc All rights reserved doi:10.1016/S0096-3003(02)00860-3 454 Y Tang, L Zhou / Appl Math Comput 148 (2004) 453–460 ou ðt; xÞ ¼ 0; on ðt; xÞ Rþ  oX; uðt; xÞ ¼ /ðt; xÞ; ð1:2Þ ðt; xÞ RÀ  X; ð1:3Þ where L ¼ ðo=otÞ À M, a and b are positive constants, X is a bounded domain in Rn with its boundary oX being a C ––manifold, / CðRÀ  XÞ is a bounded nonnegative function and /ð0; ÁÞ C ðXÞ Redlinger obtained the following result: 1.1 Suppose that k CðRþ Þ \ L1 ðRþ Þ which k 6¼ and b > RTheorem jkðsÞj ds, then the initial boundary value problem (1.1)–(1.3) has a unique bounded, nonnegative and regular solution uðt; xÞ for each positive, continuous and bounded function / on RÀ  X Moreover, if /ð0;R ÁÞ 6¼ 0, then uðt; xÞ > for all ðt; xÞ ð0; þ1Þ Â X and limt!1 uðt; xÞ ¼ a=ðb þ kðsÞ dsÞ In [2], Shi and Chen extended Theorem 1.1 to the case of a Volterra reaction diffusion equation with variable coefficients as following: Z Luðt; xÞ ¼ uðt; xÞ aðt; xÞ À bðt; xÞuðt; xÞ À cðt; xÞ uðt À s; xÞ dlðsÞ ; ð1:4Þ where a, b and c are sufficiently smooth functions and < a1 aðt; xÞ a2 , < b1 bðt; xÞ b2 , < c1 cðt; xÞ c2 , lðÁÞ is of bounded variation with lð0Þ ¼ Let MðtÞ denote the total variation of lðÁÞ on ½0; t and M Æ ðtÞ ¼ ½MðtÞ Æ lðtÞ=2 for all t Rþ , then MðtÞ and M Æ ðtÞ are nonnegative and nondecreasing on Rþ Denote M0 ¼ limt!þ1 MðtÞ, M0Æ ¼ limt!þ1 M Æ ðtÞ, l0 ¼ limt!þ1 lðtÞ Shi and Chen obtained the following asymptotic behavior: Theorem 1.2 Suppose that b1 > c1 M0À , a1 ðb1 À c1 M0À Þ > c2 a2 M0þ and uðt; xÞ is a solution of (1.2)–(1.4) Then there exist positive constants a, bð0 < a bÞ such that a lim inf uðt; xÞ lim sup max uðt; xÞ b: t!þ1 x2X t!þ1 x2X In [3], Zhou and Fu considered a periodic logistic delay equation with discrete delay effect as following: Luðt; xÞ ¼ uðt; xÞ½1 þ ðt; xÞ À buðt; xÞ À cuðt À s; xÞ; ðt; xÞ Rþ  X; ð1:5Þ where ðt; xÞ is T -periodic in t and there is a constant ^ with < ^ ( such that À^ ðt; xÞ ^ on X  ½0; T , b and c are positive constants They obtained the following conclusion Y Tang, L Zhou / Appl Math Comput 148 (2004) 453–460 455 Theorem 1.3 Let b > c and ^ be sufficiently small ThenEthere is a unique T D ^ à 1 ^ ; bþc þ bÀc periodic solution u in the sector J ¼ bþc À bÀc Moreover, for any initial function /ðt; xÞ J , the corresponding solution uðt; xÞ of (1.5), (1.2), (1.3) satisfies uðt; xÞ ! uà ðt; xÞ as t ! þ1 uniformly in x X In this paper we consider the following periodic reaction diffusion equation with infinite delay: Z Luðt; xÞ ¼ uðt; xÞ aðt; xÞ À bðt; xÞuðt; xÞ À cðt; xÞ uðt À s; xÞ dlðsÞ ; ð1:6Þ ou ðt; xÞ ¼ 0; on ðt; xÞ Rþ  oX; uðt; xÞ ¼ /ðt; xÞ; ð1:7Þ ðt; xÞ RÀ  X: ð1:8Þ where a, b and c are sufficiently smooth functions, and strictly positive and periodic in the time variable t with period T > Given a function g ¼ gðt; xÞ which is continuous, strictly positive and T -periodic in t on R  X Let gM and gL denote the maximum and minimum of g on R  X respectively In this paper we consider the existence and uniqueness of the periodic solution to the periodic boundary value problem (1.6) and (1.7), and the asymptotic behavior of the solution to the initial boundary value problem (1.6)– (1.8) Main result Let a and b be given by the following system of linear equations: aM À bL b À cL aM0þ þ cM bM0À ¼ 0; ð2:1Þ aL À bM a À cM bM0þ þ cL aM0À ¼ 0: We assume that aL bL > cM ðaL M0À þ aM M0þ Þ: ð2:2Þ Then it is easy to solve (2.1) and a unique solution is given by a¼ b¼ aL ðbL À cM M0À Þ À aM cM M0þ ðbL À cM M0À ÞðbM À cL M0À Þ À cL cM ðM0þ Þ aM ðbM À cL M0À Þ À aL cL M0þ ðbL À cM M0À ÞðbM À cL M0À Þ À cL cM ðM0þ Þ ; ð2:3Þ : 456 Y Tang, L Zhou / Appl Math Comput 148 (2004) 453–460 Eq (2.2) implies that < a b We obtain the following theorem by making use of the method of T -upper and lower solutions and the bootstrap technique of periodic monotone iteration developed in [3–6] Theorem 2.1 Assume that (2.2) holds true, and also aM À 2bL a À cL aM0þ þ cM bM0þ þ 2cM bM0À < 0; ð2:4Þ then problem (1.6) and (1.7) has a unique T -periodic solution uà ðt; xÞ in the sector ha; bi, and this solution is an attractor of system (1.6)–(1.8) That is, for any initial function /ðt; xÞ ha; bi, the corresponding solution uðt; xÞ of (1.6)–(1.8) satisfies uðt; xÞ ! uà ðt; xÞ; t ! 1ðx XÞ: Proof We know that the problem (1.6)–(1.8) is equivalent to the following: Z Z þ À Luðt; xÞ ¼ u a À bu À c uðt À s; xÞ dM ðsÞ þ c uðt À s; xÞ dM ðsÞ ; 0 ð2:5Þ ou ðt; xÞ ¼ 0; on ðt; xÞ Rþ  oX; uðt; xÞ ¼ /ðt; xÞ; ð2:6Þ ðt; xÞ RÀ  X: ð2:7Þ From (2.1), it is easy to check that b and a are the constant coupled upper and lower solutions of (2.5) and (2.6) According to the bootstrap technique developed in [4] and the existence theorem of quasi-solutions given in [3], Theorem 3.2, there is a pair of T -periodic quasi-solutions hðt; xÞ and hðt; xÞ with ð2:8Þ a hðt; xÞ hðt; xÞ b such that Z Z Lhðt; xÞ ¼ h a À bh À c hðt À s; xÞ dM þ ðsÞ þ c hðt À s; xÞ dM À ðsÞ ; 0 Z Z þ hðt À s; xÞ dM ðsÞ þ c hðt À s; xÞ dM À ðsÞ ; Lhðt; xÞ ¼ h a À bh À c 0 þ ðt; xÞ R  X; oh oh ðt; xÞ ¼ ðt; xÞ ¼ 0; on on ðt; xÞ Rþ  oX: ð2:9Þ Moreover, for any initial function /ðt; xÞ satisfying a / b in RÀ  X, the corresponding solution uðt; xÞ of (2.5)–(2.7) satisfies Y Tang, L Zhou / Appl Math Comput 148 (2004) 453–460 457 t ! 1ðx XÞ: hðt; xÞ uðt; xÞ hðt; xÞ; ð2:10Þ According to the equivalence of problem (2.5)–(2.7) and problem (1.6)–(1.8), the solution of (1.6)–(1.8) also possesses the property (2.10) From (2.9), we know that the T -periodic quasi-solution hðt; xÞ and hðt; xÞ of (2.5) and (2.6) are not the solutions of the problem in general It is obvious that hðt; xÞ and hðt; xÞ is a pair of coupled T -periodic upper and lower solutions of (1.6) and (1.7), namely, Z Lhðt; xÞ P h a À bh À c hðt À s; xÞ dlðsÞ ; Z Lhðt; xÞ h a À bh À c hðt À s; xÞ dlðsÞ ; ðt; xÞ Rþ  X; ð2:11Þ oh oh ðt; xÞ ¼ ðt; xÞ ¼ 0; on on ðt; xÞ Rþ  oX: Therefore, if hðt; xÞ ¼ hðt; xÞ, then (2.11) implies that the quasi-solution hðt; xÞ (or hðt; xÞ) is exactly the T -periodic solution of (1.6) and (1.7) We are going to show that hðt; xÞ ¼ hðt; xÞ, provided that (2.4) holds true Due to the periodicity of hðt; xÞ and hðt; xÞ, (2.9) implies that 0¼ Z Z Z T Z o ðh À hÞ dx ¼ dt ðh À hÞ ot X X 0 Z Z Z hs dM þ ðsÞ þ c hs dM À ðsÞ dx À  Mh þ h a À bh À c T dt ðh À hÞ Â Z ðh À hÞ Mh þ h a À bh À c Z ¼À Z Z T À jrðh À hÞj2 dx þ dt Z Z T hs dM þ þ h Z ðaM À 2bL aÞ Z dt X Z T Z Z T hs dM À þ h dx X Z hs dM þ À h Z cðh À hÞ dt X cðh À hÞ2 hs dM À ðsÞ aðh À hÞ2 dx ðh À hÞ dx dt X Z T Z Z 0 X Z  Àh hs dM þ ðsÞ þ c T bðh þ hÞðh À hÞ2 dx þ dt Z dt X À Z dt 0 X T hs dM þ dx Z hs dM À dx 458 Y Tang, L Zhou / Appl Math Comput 148 (2004) 453–460 þ Z Z T dt þ Z Z T dt Z chðh À hÞ Z dt hs dM Z À dx ðhs À hs Þ dM X T Z X þ cðh À hÞ chðh À hÞ Z À dx ðhs À hs Þ dM þ dx X Z T Z dt ðh À hÞ2 dx ðaM À 2bL a À cL aM0þ þ cM bM0À Þ X Z T Z Z þ cM b dt ðh À hÞ ðhs À hs Þ dM À dx X 0 Z T Z Z þ dt ðh À hÞ ðhs À hs Þ dM dx: þ cM b 0 X First, we consider the second term in the above: Z Z T Z À dt ðh À hÞ ðhs À hs Þ dM dx ¼ Z Z X dM À ðsÞ Â Z À dM ðsÞ Z Z T dt 1=2 ðhðt; xÞ À hðt; xÞÞ dx X 1=2 ðhðt À s; xÞ À hðt À s; xÞÞ dx dt X Z Z T ½ðhðt; xÞ À hðt; xÞÞðhðt À s; xÞ À hðt À s; xÞÞ dx dt 0 T Z : X Using the periodicity of hðt; xÞ and hðt; xÞ, we have Z T Z dt ðhðt À s; xÞ À hðt À s; xÞÞ2 dx X Z Z T Z Z T Às ¼ dx ðhðt À s; xÞ À hðt À s; xÞÞ2 dt ¼ dx ðhðs; xÞ À hðs; xÞÞ2 ds X X Às Z Z T Z T Z ¼ dx ðhðs; xÞ À hðs; xÞÞ ds ¼ dt ðhðt; xÞ À hðt; xÞÞ2 dx: X Then we obtain Z T Z Z dt ðh À hÞ X T ðhs À hs ÞdM À dx Z dM À ðsÞ dt ðhðt; xÞ À hðt; xÞÞ dx X 0 Z T Z À ¼ M0 dt ðh À hÞ dx: Z Z X X Y Tang, L Zhou / Appl Math Comput 148 (2004) 453–460 459 Similarly, we can get the estimate of the third term: Z T Z Z þ dt ðh À hÞ ðhs À hs ÞdM dx X Z þ Z T Z dM ðsÞ dt ðhðt; xÞ À hðt; xÞÞ dx X Z T Z dt ðh À hÞ dx: ¼ M0þ 0 X Therefore, we have Z T Z o 0¼ dt ðh À hÞ ðh À hÞ dx ot X Z T Z dt ðh À hÞ dx ½aM À 2bL a À cL aM0þ þ cM bM0À þ cM bðM0þ þ M0À Þ X Z T Z ¼ ½aM À 2bL a À cL aM0þ þ cM bM0þ þ 2cM bM0À dt ðh À hÞ dx 0: X ð2:12Þ The last inequality follows from the assumption (2.4) Then (2.12) implies that hðt; xÞ ¼ hðt; xÞ for ðt; xÞ Rþ  X This means that hðt; xÞ ¼ hðt; xÞ ¼ uà ðt; xÞ is a T -periodic solution of (1.6) and (1.7) From (2.10), for any initial function /ðt; xÞ satisfying a / b in RÀ  X, the solution uðt; xÞ of (1.6)–(1.8) satisfies uðt; xÞ ! uà ðt; xÞ as t ! 1ðx XÞ This completes the proof à Discussion The method of T -upper and lower solutions and the bootstrap technique of periodic monotone iteration are very useful in the research of the periodic systems But in general, we just obtain the quasi-solutions of the periodic boundary value problems, they are not the solutions of the problems However, for a pair of quasi-solutions hðt; xÞ and hðt; xÞ, the sector hhðt; xÞ; hðt; xÞi is an attractor of the associated initial boundary value problem Furthermore, if hðt; xÞ ¼ hðt; xÞ, then the quasi-solution hðt; xÞ (or hðt; xÞ) is exactly the solution, and the asymptotic behavior of the solutions to the associated initial boundary value problem is also obtained Therefore, the sufficient conditions of hðt; xÞ ¼ hðt; xÞ can help us to get more detailed information about periodic systems 460 Y Tang, L Zhou / Appl Math Comput 148 (2004) 453–460 In our main result Theorem 2.1, if we take l0 ðsÞ ¼ dðs À rÞ, aðt; xÞ ¼ þ eðt; xÞ, Àbe e be , bðt; xÞ b, cðt; xÞ c, where r, b, c are positive constants, be > is sufficiently small, then we have M0þ ¼ 1; M0À ¼ 0; a¼ be À ; bþc bÀc b¼ be þ bþc bÀc and the conditions (2.2) and (2.4) become b>c þ be ; À be À bÀc bþc be < 0: þ be þ bþc bÀc For sufficiently small be > 0, it is obvious that Theorem 1.3 is special case of Theorem 2.1 References [1] R Redlinger, On VolterraÕs population equation with diffusion, SIAM J Math Anal 16 (1) (1985) 135–142 [2] B Shi, Y Chen, A prior bounds and stability of solutions for a Volterra reaction diffusion equation with infinite delay, Nonlinear Anal., TMA 44 (2001) 97–121 [3] L Zhou, Y Fu, Existence and stability of periodic quasisolutions in nonlinear parabolic systems with discrete delays, J Math Anal Appl 250 (2000) 139–161 [4] S Ahmad, A.C Lazer, Asymptotic behavior of solutions of periodic competition diffusion system, Nonlinear Anal., TMA 13 (1989) 263–284 [5] C.V Pao, Quasisolutions and global attractor of reaction diffusion system, Nonlinear Anal., TMA 26 (1996) 1889–1903 [6] C.V Pao, Periodic solutions of parabolic system with nonlinear boundary conditions, J Math Anal Appl 234 (1999) 695–716 ... diffusion equation with infinite delay, Nonlinear Anal., TMA 44 (2001) 97–121 [3] L Zhou, Y Fu, Existence and stability of periodic quasisolutions in nonlinear parabolic systems with discrete delays,... (1.2)–(1.4) Then there exist positive constants a, bð0 < a bÞ such that a lim inf uðt; xÞ lim sup max uðt; xÞ b: t!þ1 x2X t!þ1 x2X In [3], Zhou and Fu considered a periodic logistic delay equation with... M0þ ðbL À cM M0À ÞðbM À cL M0À Þ À cL cM ðM0þ Þ ; ð2:3Þ : 456 Y Tang, L Zhou / Appl Math Comput 148 (2004) 453–460 Eq (2.2) implies that < a b We obtain the following theorem by making use of