×

Existence and stability of periodic quasisolutions in nonlinear parabolic systems with discrete delays. (English) Zbl 0970.35004

There are studied nonlinear systems of parabolic equations with delays of the form: \[ \partial u_i/\partial t- L_iu_i= f_i(x, t,u,u_\sigma)\quad\text{on } \Omega\times (0,\infty), \]
\[ B_i[u_i](x,t)= h_i(x,t)\quad\text{on }\partial\Omega\times (0,\infty), \] \(u_i(x,t)= \mu_i(x,t)\) on \(\Omega\times [-\sigma_i, 0]\), where \(u(x,t)= (u_1(x, t),\dots, u_n(x, t))\), \(u_\sigma(x,t)= (u_1(x, t-\sigma_1),\dots, u_n(x, t-\sigma_n))\); \(\Omega\) is a bounded domain in \(\mathbb{R}^N\) with the boundary \(\partial\Omega\); \(\sigma_1,\dots,\sigma_n\) are positive constants \[ L_iu_i\equiv \sum^N_{j,k= 1} a^i_{jk}(x, t)\partial^2 u_i/\partial x_j\partial x_k+ \sum^N_{j=1} b^i_j(x, t)\partial u_i/\partial x_j, \] (for each \(t\in \mathbb{R}\), \(L_i\) is a uniformly elliptic operator); \(B_iu_i\equiv \alpha_i\partial u_i/\partial n+ \beta_i(x, t)u_i\) (\(\alpha_i=0\), \(\beta_i(x,t)= 1\) or \(\alpha_i(x,t)= 1\), \(\beta_i(x, t)\geq 0\) and it is allowed to be of different type for different \(i\)); the functions \(a^i_{jk}\), \(h_i\), \(\beta_i\) belong, respectively, to some spaces of \(T\)-periodic, in \(t\), functions; \(f_i\), are \(T\)-periodic in \(t\); \(\eta_i\) are Hölder continuous.
The authors obtain the existence of periodic quasisolutions and some asymptotic properties; some applications to some models in ecology are given to illustrate the obtained results.

MSC:

35B10 Periodic solutions to PDEs
35K50 Systems of parabolic equations, boundary value problems (MSC2000)
35R10 Partial functional-differential equations
35K55 Nonlinear parabolic equations
92D40 Ecology
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Pao, C. V., Coupled nonlinear parabolic systems with time delay, J. Math. Anal. Appl., 196, 237-265 (1995) · Zbl 0854.35122
[2] Pao, C. V., Dynamics of nonlinear parabolic systems with time delays, J. Math. Anal. Appl., 198, 751-779 (1996) · Zbl 0860.35138
[3] Hess, P., Periodic-Parabolic Boundary Value Problems and Positivity. Periodic-Parabolic Boundary Value Problems and Positivity, Pitman Research Notes in Mathematics Series, 247 (1991), Longman: Longman Harlow · Zbl 0731.35050
[4] Martin, R. H.; Smith, H. L., Abstract functional differential equation and reaction-diffusion systems, Trans. Amer. Math. Soc., 321, 1-44 (1990) · Zbl 0722.35046
[5] Amann, H., Periodic solutions of semilinear parabolic equations, (Cesari; Kannan; Weinberger, Nonlinear Analysis (1978), Academic Press: Academic Press New York) · Zbl 0464.35050
[6] Ahmad, S.; Lazer, A. C., Asymptotic behavior of solutions of periodic competition diffusion system, Nonlinear Anal., 13, 263-284 (1989) · Zbl 0686.35060
[7] Tineo, A., Asymptotic behavior of solutions of a periodic reaction-diffusion system of a competitor-competitor-mutualist model, J. Differential Equations, 108, 326-341 (1994) · Zbl 0806.35095
[8] Fu, S.; Ma, R., Existence of a global coexistence state for periodic competition diffusion systems, Nonlinear Anal., 28, 1265-1271 (1997) · Zbl 0871.35051
[9] Zhao, X.-Q., Global asymptotic behavior in a periodic competitor-competitor-mutualist parabolic system, Nonlinear Anal., 29, 551-568 (1997) · Zbl 0876.35058
[10] Pao, C. V., Systems of parabolic equations with continuous and discrete delays, J. Math. Anal. Appl., 205, 157-185 (1997) · Zbl 0880.35126
[11] Feng, W.; Lu, X., Asymptotic periodicity in diffusive logistic equations with discrete delays, Nonlinear Anal., 26, 171-178 (1996) · Zbl 0842.35129
[12] Lu, X.; Feng, W., Periodic solution and oscillation in a competition model with diffusion and distributed delay effects, Nonlinear Anal., 27, 699-709 (1996) · Zbl 0862.35134
[13] Beltramo, A.; Hess, P., On the principal eigenvalue of a periodic-parabolic operator, Comm. Partial Differential Equations, 9, 919-941 (1984) · Zbl 0563.35033
[14] Pao, C. V., Quasisolutons and global attractor of reaction-diffusion systems, Nonlinear Anal., 26, 1889-1903 (1996) · Zbl 0853.35056
[15] Pao, C. V., Periodic solutions of parabolic systems with nonlinear boundary conditions, J. Math. Anal. Appl., 234, 695-716 (1999) · Zbl 0932.35111
[16] Leung, A., System of Nonlinear Partial Differential Equations. System of Nonlinear Partial Differential Equations, Applications to Biology and Engineering, Math. and Its Appl. (1989), Kluwer Academic: Kluwer Academic Dordrecht
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.