Classes of reaction diffusion equations where a parameter influences the equation as well as the boundary condition

We study positive solutions to steady-state reaction–diffusion models of the form $$ \textstyle\begin{cases} -\Delta u=\lambda f(v);\quad\Omega, \\ -\Delta v=\lambda g(u);\quad\Omega, \\ \frac{\partial u}{\partial \eta }+\sqrt{\lambda } u=0;\quad \partial \Omega, \\ \frac{\partial v}{\partial \eta }+\sqrt{\lambda }v=0; \quad\partial \Omega, \end{cases} $$ { − Δ u = λ f ( v ) ; Ω , − Δ v = λ g ( u ) ; Ω , ∂ u ∂ η + λ u = 0 ; ∂ Ω , ∂ v ∂ η + λ v = 0 ; ∂ Ω , where $\lambda >0$ λ > 0 is a positive parameter, Ω is a bounded domain in $\mathbb{R}^{N}$ R N $(N > 1)$ ( N > 1 ) with smooth boundary ∂Ω, or $\Omega =(0,1)$ Ω = ( 0 , 1 ) , $\frac{\partial z}{\partial \eta }$ ∂ z ∂ η is the outward normal derivative of z. We assume that f and g are continuous increasing functions such that $f(0) = 0 = g(0)$ f ( 0 ) = 0 = g ( 0 ) and $\lim_{s \rightarrow \infty } \frac{f(Mg(s))}{s} = 0$ lim s → ∞ f ( M g ( s ) ) s = 0 for all $M>0$ M > 0 . In particular, we extend the results for the single equation case discussed in (Fonseka et al. in J. Math. Anal. Appl. 476(2):480-494, 2019) to the above system.

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“…In [7] the authors analyzed and established several results for positive solutions for reaction-diffusion models of the form…”

Section: Introductionmentioning

confidence: 99%

Analysis of reaction–diffusion systems where a parameter influences both the reaction terms as well as the boundary

Acharya

Fonseka²,

Shivaji

2021

Bound Value Probl

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“…is such that f (s) ≤ M s for s ∈ [0, ∞). But σ λ λ −→ ∞ as λ −→ 0 (see [6]) and hence L > 0 for λ ≈ 0, which is a contradiction. Thus (3) has no positive solutions for λ ≈ 0.…”

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confidence: 96%

A diffusive weak Allee effect model with U-shaped emigration and matrix hostility

Fonseka¹,

Goddard²,

Shivaji³

et al. 2021

DCDS-B

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“…Introduction. Recently, there has been growing interest on the study of the reaction diffusion models where a parameter influences the equation as well as the boundary condition (see [1], [2], [6], [8], [7], and [9]). In this paper, we study reaction diffusion systems of the form:…”

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confidence: 99%

$ \Sigma $-shaped bifurcation curves for classes of elliptic systems

Acharya

Shivaji

Fonseka³

2022

DCDS-S

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We study positive solutions to classes of steady state reaction diffusion systems of the form:<disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} \left\lbrace \begin{matrix}-\Delta u = \lambda f(v) ;\; \Omega\\ -\Delta v = \lambda g(u) ;\; \Omega\\ \frac{\partial u}{\partial \eta}+\sqrt{\lambda} u = 0; \; \partial \Omega\\ \frac{\partial v}{\partial \eta}+\sqrt{\lambda}v = 0; \; \partial \Omega\ \end{matrix} \right. \end{equation*} $\end{document} </tex-math></disp-formula>where <inline-formula><tex-math id="M2">\begin{document}$ \lambda>0 $\end{document}</tex-math></inline-formula> is a positive parameter, <inline-formula><tex-math id="M3">\begin{document}$ \Omega $\end{document}</tex-math></inline-formula> is a bounded domain in <inline-formula><tex-math id="M4">\begin{document}$ \mathbb{R}^N $\end{document}</tex-math></inline-formula>; <inline-formula><tex-math id="M5">\begin{document}$ N > 1 $\end{document}</tex-math></inline-formula> with smooth boundary <inline-formula><tex-math id="M6">\begin{document}$ \partial \Omega $\end{document}</tex-math></inline-formula> or <inline-formula><tex-math id="M7">\begin{document}$ \Omega = (0, 1) $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M8">\begin{document}$ \frac{\partial z}{\partial \eta} $\end{document}</tex-math></inline-formula> is the outward normal derivative of <inline-formula><tex-math id="M9">\begin{document}$ z $\end{document}</tex-math></inline-formula>. Here <inline-formula><tex-math id="M10">\begin{document}$ f, g \in C^2[0, r) \cap C[0, \infty) $\end{document}</tex-math></inline-formula> for some <inline-formula><tex-math id="M11">\begin{document}$ r>0 $\end{document}</tex-math></inline-formula>. Further, we assume that <inline-formula><tex-math id="M12">\begin{document}$ f $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M13">\begin{document}$ g $\end{document}</tex-math></inline-formula> are increasing functions such that <inline-formula><tex-math id="M14">\begin{document}$ f(0) = 0 = g(0) $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M15">\begin{document}$ f'(0) = g'(0) = 1 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M16">\begin{document}$ f''(0)>0, g''(0)>0 $\end{document}</tex-math></inline-formula>, and <inline-formula><tex-math id="M17">\begin{document}$ \lim\limits_{s \rightarrow \infty} \frac{f(Mg(s))}{s} = 0 $\end{document}</tex-math></inline-formula> for all <inline-formula><tex-math id="M18">\begin{document}$ M>0 $\end{document}</tex-math></inline-formula>. Under certain additional assumptions on <inline-formula><tex-math id="M19">\begin{document}$ f $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M20">\begin{document}$ g $\end{document}</tex-math></inline-formula> we prove that the bifurcation diagram for positive solutions of this system is at least <inline-formula><tex-math id="M21">\begin{document}$ \Sigma- $\end{document}</tex-math></inline-formula>shaped. We also discuss an example where <inline-formula><tex-math id="M22">\begin{document}$ f $\end{document}</tex-math></inline-formula> is sublinear at <inline-formula><tex-math id="M23">\begin{document}$ \infty $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M24">\begin{document}$ g $\end{document}</tex-math></inline-formula> is superlinear at <inline-formula><tex-math id="M25">\begin{document}$ \infty $\end{document}</tex-math></inline-formula> which satisfy our hypotheses.

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Classes of reaction diffusion equations where a parameter influences the equation as well as the boundary condition

Cited by 8 publications

References 11 publications

Analysis of reaction–diffusion systems where a parameter influences both the reaction terms as well as the boundary

Analysis of reaction–diffusion systems where a parameter influences both the reaction terms as well as the boundary

A diffusive weak Allee effect model with U-shaped emigration and matrix hostility

$ \Sigma $-shaped bifurcation curves for classes of elliptic systems

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