Tineo, Antonio Existence of global coexistence state for periodic competition diffusion systems. (English) Zbl 0779.35058 Nonlinear Anal., Theory Methods Appl. 19, No. 4, 335-344 (1992). Consider the following system of parabolic equations: \[ u_ t= P(u)+ u[a(t,x)-b(t,x)u-c(t,x)v], \qquad v_ t= Q(u)+ v[d(t,x)- e(t,x)u- f(t,x)v] \tag{1} \] where \(a,\dots,f: \mathbb{R}\times \overline{\Omega} \to \mathbb{R}\) are positive continuous functions, which are periodic in the time \(t\) with period \(T>0\); \(\Omega\) is a bounded domain in \(\mathbb{R}^ n\) whose boundary \(\partial\Omega\) belongs to \(C^ 2\) and \(P\), \(Q\) are uniformly elliptic operators in \(\mathbb{R} \times \overline {\Omega}\), whose coefficients are \(T\)-periodic in the first variable.When \(a,\dots,f\) and \(\partial\Omega\) are sufficiently smooth, the author studies the existence and uniqueness of a positive solution \((u,v)\) \((u>0\), \(v>0)\) of (1) such that (2) \(u(t+T,x)=u(t,x)\), \(v(t+T,x)=v(t,x)\), (3) \(\partial u/\partial \nu=\partial v/\partial \nu=0\) in \(\mathbb{R}\times \partial \Omega\), where \(\partial/\partial\nu\) denotes differentiation in the direction of the outer normal to \(\partial\Omega\).The author also studies the asymptotic behavior of positive solutions \((u,v)\) of (1), which satisfy the following initial boundary value problem: \[ (4) \quad \partial u/\partial\nu= \partial v/\partial\nu=0 \text{ in } (0,\infty)\times \partial\Omega, \qquad (5) \quad u(0,x)=\varphi(x),\;v(0,x)=\Psi(x) \text{ in } \Omega \] for sufficiently smooth nonnegative functions \(\varphi\), \(\Psi: \overline{\Omega}\to \mathbb{R}\) such that \(\partial \varphi/\partial\nu= \partial\Psi/\partial\nu=0\), in \(\partial\Omega\). To be precise, let us define from now on: \(B=b/a\), \(C=c/a\), \(E=e/d\), \(F=f/d\), and for all bounded functions \(g: X\to\mathbb{R}\) (some nonempty set \(X\)); \(g_ L=\inf \{g(x)\): \(x\in X\}\) and \(g_ M=\sup\{g(x)\): \(x\in X\}\). In the “stable” case (6) \(B_ L>E_ M\) and \(F_ L>C_ M\), the author proves the existence of a positive solution to (1)–(3) and he obtains a priori bounds for the positive solutions to this problem, whose components belong to \(C^{2,1}(\mathbb{R}\times \Omega)\cap C^{1,0}(\mathbb{R}\times \overline{\Omega})\).Further, if (6) and (7) \(C_ M E_ M(F_ M-C_ L)(B_ M- E_ L)< B_ LF_ L (F_ L- C_ M)(B_ L-E_ M)\) hold, then the problem (1)- -(3) has exactly one positive solution. Moreover, this solution is globally asymptotically stable. Reviewer: Y.Ebihara (Fukuoka) Cited in 15 Documents MSC: 35K57 Reaction-diffusion equations 35B40 Asymptotic behavior of solutions to PDEs 35B35 Stability in context of PDEs 35D05 Existence of generalized solutions of PDE (MSC2000) Keywords:existence; uniqueness; positive solution; asymptotic behavior; initial boundary value problem; a priori bounds PDFBibTeX XMLCite \textit{A. Tineo}, Nonlinear Anal., Theory Methods Appl. 19, No. 4, 335--344 (1992; Zbl 0779.35058) Full Text: DOI References: [1] Ahmad, S.; Lazer, A., Asymptotic behaviour of solutions of periodic competition diffusion systems, Nonlinear Analysis, 13, 263-284 (1989) · Zbl 0686.35060 [2] Alvarez, C.; Tineo, A., Asymptotically stable solutions of Lotka-Volterra equations, Rad. Mat., 4, 309-319 (1988) · Zbl 0662.34052 [3] Pao, C. V., Positive solutions of nonlinear boundary value problems of parabolic type, J. diff. Eqns, 22, 145-163 (1976) · Zbl 0326.35041 [4] Pao, C. V., Coexistence and stability of a competition-diffusion system in population dynamics, J. math. Analysis Applic., 83, 54-76 (1981) · Zbl 0479.92013 [5] De Mottoni, P.; Schiaffino, A., Competition systems with periodic coefficients: a geometric approach, J. Math. Biol., 11, 319-335 (1981) · Zbl 0474.92015 [6] Smoller, J., Shock Waves and Reaction-Diffusion Equations (1983), Springer: Springer New York · Zbl 0508.35002 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.