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Propagation and blocking in periodically hostile environments. (English) Zbl 1263.35036
The main purpose of this paper is to study the persistence and propagation phenomena for one species in a periodically hostile environment. This is modeled by the following reaction-diffusion equation $\begin{cases} u_t-\nabla\cdot (A(x)\nabla u)=F(x,u), & \;x\in \overline\Omega \\ u(t,x)=0, & \;x \in \partial \Omega \\ \end{cases}$ where $$\Omega$$ is an unbounded domain in $$\mathbb{R}^N$$, $$A(x)=(A_{ij}(x))$$ is a symmetric matrix and is uniformly positive definite, $$F(x,u)$$ is assumed to satisfy the condition that for all $$x\in \overline\Omega$$, $$F(x,0)=0$$ and $$F(x,M)\leq 0$$ for some $$M>0$$. In addition, the domain $$\Omega$$, the functions $$A_{ij}(\cdot)$$, and the function $$F(\cdot,u)$$ are all periodic in the sense that they are invariant under the translation $$x\rightarrow x+k$$ for $$k\in L_1\mathbb{Z}\times \cdots \times L_N\mathbb{Z}$$, where $$L_1, \cdots, L_N$$ are fixed positive numbers. A result on the existence of a minimal nonnegative stationary solution and the asymptotic behavior of the solution of the initial boundary value problem is presented.

##### MSC:
 35B40 Asymptotic behavior of solutions to PDEs 35K57 Reaction-diffusion equations 35K58 Semilinear parabolic equations 35K20 Initial-boundary value problems for second-order parabolic equations
##### Keywords:
minimal nonnegative stationary solution
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##### References:
 [1] Aronson D.G., Weinberger H.F.: Multidimensional nonlinear diffusions arising in population genetics. Adv. Math. 30, 33–76 (1978) · Zbl 0407.92014 [2] Berestycki H.: Le nombre de solutions de certains problèmes semi-linéaires elliptiques. J. Funct. Anal. 40, 1–29 (1981) · Zbl 0452.35038 [3] Berestycki H., Hamel F.: Front propagation in periodic excitable media. Commun. Pure Appl. Math. 55, 949–1032 (2002) · Zbl 1024.37054 [4] Berestycki H., Hamel F., Nadin G.: Asymptotic spreading in heterogeneous diffusive media. J. Funct. Anal. 255, 2146–2189 (2008) · Zbl 1234.35033 [5] Berestycki H., Hamel F., Roques L.: Analysis of the periodically fragmented environment model: I. Species persistence. J. Math. Biol. 51, 75–113 (2005) · Zbl 1066.92047 [6] Berestycki H., Hamel F., Roques L.: Analysis of the periodically fragmented environment model: II. Biological invasions and pulsating travelling fronts. J. Math. Pures Appl. 84, 1101–1146 (2005) · Zbl 1083.92036 [7] Berestycki H., Hamel F., Rossi L.: Liouville type results for semilinear elliptic equations in unbounded domains. Ann. Mat. Pura Appl. 186, 469–507 (2007) · Zbl 1223.35022 [8] Berestycki H., Nirenberg L.: Travelling fronts in cylinders. Ann. Inst. H. Poincaré Analyse Non Linéaire 9, 497–572 (1992) · Zbl 0799.35073 [9] Berestycki H., Nirenberg L., Varadhan S.R.S.: The principal eigenvalue and maximum principle for second order elliptic operators in general domains. Commun. Pure Appl. Math. 47, 47–92 (1994) · Zbl 0806.35129 [10] Cantrell R.S., Cosner C.: The effects of spatial heterogeneity in population dynamics. J. Math. Biol. 29, 315–338 (1991) · Zbl 0722.92018 [11] Cantrell R.S., Cosner C.: On the effects of spatial heterogeneity on the persistence of interacting species. J. Math. Biol. 37, 103–145 (1998) · Zbl 0948.92021 [12] Du Y., Matano H.: Convergence and sharp thresholds for propagation in nonlinear diffusion problems. J. Eur. Math. Soc. 12, 279–312 (2010) · Zbl 1207.35061 [13] El Smaily M.: Pulsating travelling fronts: Asymptotics and homogenization regimes. Eur. J. Appl. Math. 19, 393–434 (2008) · Zbl 1162.35064 [14] Fisher R.A.: The advance of advantageous genes. Ann. Eugenics 7, 335–369 (1937) · JFM 63.1111.04 [15] Freidlin M., Gärtner J.: On the propagation of concentration waves in periodic and random media. Sov. Math. Dokl. 20, 1282–1286 (1979) · Zbl 0447.60060 [16] Hamel F.: Qualitative properties of monostable pulsating fronts: exponential decay and monotonicity. J. Math. Pures Appl. 89, 355–399 (2008) · Zbl 1171.35061 [17] Hamel F., Roques L.: Uniqueness and stability properties of monostable pulsating fronts. J. Eur. Math. Soc. 13, 345–390 (2011) · Zbl 1219.35035 [18] Heinze, L.: Large convection limits for KPP fronts. Preprint [19] Kinezaki N., Kawasaki K., Shigesada N.: Spatial dynamics of invasion in sinusoidally varying environments. Popul. Ecol. 48, 263–270 (2006) [20] Kolmogorov A.N., Petrovsky I.G., Piskunov N.S.: Étude de l’équation de la diffusion avec croissance de la quantité de matière et son application à à un problème biologique. Bull. Univ. Etat Moscou, Sér. Intern. A 1, 1–26 (1937) [21] Liang X., Lin X., Matano H.: A variational problem associated with the minimal speed of travelling waves for spatially periodic reaction-diffusion equations. Trans. Am. Math. Soc. 362, 5605–5633 (2010) · Zbl 1206.35061 [22] Liang X., Zhao X.Q.: Asymptotic speeds of spread and traveling waves for monostable semiflows with applications. Commun. Pure Appl. Math. 60, 1–40 (2007) · Zbl 1106.76008 [23] Liang X., Zhao X.Q.: Spreading speeds and traveling waves for abstract monostable evolution systems. J. Funct. Anal. 259, 857–903 (2010) · Zbl 1201.35068 [24] Muratov C.B., Novaga M.: Front propagation in infinite cylinders, I. A variational approach. Commun. Math. Sci. 6, 799–826 (2008) · Zbl 1173.35537 [25] Murray J.D.: Mathematical Biology. Springer, Berlin (2003) · Zbl 1006.92002 [26] Nadin G.: Travelling fronts in space-time periodic media. J. Math. Pures Appl. 92, 232–262 (2009) · Zbl 1182.35074 [27] Nadin G.: Some dependence results between the spreading speed and the coefficients of the space-time periodic Fisher-KPP equation. Eur. J. Appl. Math. 22, 169–185 (2011) · Zbl 1225.35022 [28] Nolen J., Rudd M., Xin J.: Existence of KPP fronts in spatially-temporally periodic advection and variational principle for propagation speeds. Dyn. Partial Differ. Equ. 2, 1–24 (2005) · Zbl 1179.35166 [29] Nolen J., Xin J.: Existence of KPP type fronts in space-time periodic shear flows and a study of minimal speeds based on variational principle. Disc. Contin. Dyn. Syst. A 13, 1217–1234 (2005) · Zbl 1097.35072 [30] Polacik P.: Threshold solutions and sharp transitions for nonautonomous parabolic equations on $${$$\backslash$$mathbb{R}\^N}$$ . Arch. Rational Mech. Anal. 199, 69–97 (2011) · Zbl 1262.35130 [31] Roquejoffre J.-M.: Eventual monotonicity and convergence to travelling fronts for the solutions of parabolic equations in cylinders. Ann. Inst. H. Poincaré, Analyse Non Linéaire 14, 499–552 (1997) · Zbl 0884.35013 [32] Ryzhik L., Zlatoš A.: KPP pulsating front speed-up by flows. Commun. Math. Sci. 5, 575–593 (2007) · Zbl 1152.35055 [33] Shigesada, N., Kawasaki, K.: Biological Invasions: Theory and Practice. Oxford Series in Ecology and Evolution. Oxford University Press, Oxford, 1997 [34] Vega J.M.: Travelling waves fronts of reaction-diffusion equations in cylindrical domains. Commun. Partial Differ. Equ. 18, 505–531 (1993) · Zbl 0816.35058 [35] Weinberger H.F.: On spreading speeds and traveling waves for growth and migration in periodic habitat. J. Math. Biol. 45, 511–548 (2002) · Zbl 1058.92036 [36] Xin X.: Existence of planar flame fronts in convective-diffusive periodic media. Arch. Rational Mech. Anal. 121, 205–233 (1992) · Zbl 0764.76074 [37] Xin J.X.: Analysis and modeling of front propagation in heterogeneous media. SIAM Rev. 42, 161–230 (2000) · Zbl 0951.35060 [38] Zlatoš A.: Sharp transition between extinction and propagation of reaction. J. Am. Math. Soc. 19, 251–263 (2006) · Zbl 1081.35011 [39] Zlatoš A.: Sharp asymptotics for KPP pulsating front speed-up and diffusion enhancement by flows. Arch. Rational Mech. Anal. 195, 441–453 (2010) · Zbl 1185.35205
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