Advances in Differential Equations

Positive periodic solutions for semilinear reaction diffusion systems on $\bf {R}^N$

Sandro Merino

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The existence and nonexistence of positive time-periodic solutions of semilinear reaction diffusion systems of the type $$ \begin{cases} \partial_{t}u +\mathcal{A}_{1}u = au-bg_{1}(u)u - h_{1}(u,v)u\\ \partial_{t}v +\mathcal{A}_{2}v = dv-fg_{2}(v)v + h_{2}(u)v \end{cases} \quad \text{ in } \ \mathbb{r}^{N}\times (0,\infty ), $$ is discussed. Here the coefficients $a,b,d,f,$ and the coupling terms $g_{1},g_{2},h_{1},h_{2}$ depend on space and time and satisfy suitable conditions. The differential operators $\mathcal{A}_{1}$ and $\mathcal{A}_{2}$ are elliptic with space- and time-dependent coefficients. In particular, the time dependence is always assumed to be periodic with a fixed common period. A topological method, based on the Leray-Schauder degree, is used to obtain the existence of positive time-periodic solutions. Finally, we apply our results to various population models.

Article information

Adv. Differential Equations, Volume 1, Number 4 (1996), 579-609.

First available in Project Euclid: 25 April 2013

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35K57: Reaction-diffusion equations
Secondary: 35B10: Periodic solutions 92D25: Population dynamics (general)


Merino, Sandro. Positive periodic solutions for semilinear reaction diffusion systems on $\bf {R}^N$. Adv. Differential Equations 1 (1996), no. 4, 579--609.

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