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A predator-prey model with combined death and competition terms

Joon Hyuk Kang, Jungho Lee (2010)

Czechoslovak Mathematical Journal

The existence of a positive solution for the generalized predator-prey model for two species Δ u + u ( a + g ( u , v ) ) = 0 in Ω , Δ v + v ( d + h ( u , v ) ) = 0 in Ω , u = v = 0 on Ω , are investigated. The techniques used in the paper are the elliptic theory, upper-lower solutions, maximum principles and spectrum estimates. The arguments also rely on some detailed properties of the solution of logistic equations.

A variational approach to bifurcation in reaction-diffusion systems with Signorini type boundary conditions

Jamol I. Baltaev, Milan Kučera, Martin Väth (2012)

Applications of Mathematics

We consider a simple reaction-diffusion system exhibiting Turing's diffusion driven instability if supplemented with classical homogeneous mixed boundary conditions. We consider the case when the Neumann boundary condition is replaced by a unilateral condition of Signorini type on a part of the boundary and show the existence and location of bifurcation of stationary spatially non-homogeneous solutions. The nonsymmetric problem is reformulated as a single variational inequality with a potential...

A variational approach to bifurcation points of a reaction-diffusion system with obstacles and Neumann boundary conditions

Jan Eisner, Milan Kučera, Martin Väth (2016)

Applications of Mathematics

Given a reaction-diffusion system which exhibits Turing's diffusion-driven instability, the influence of unilateral obstacles of opposite sign (source and sink) on bifurcation and critical points is studied. In particular, in some cases it is shown that spatially nonhomogeneous stationary solutions (spatial patterns) bifurcate from a basic spatially homogeneous steady state for an arbitrarily small ratio of diffusions of inhibitor and activator, while a sufficiently large ratio is necessary in the...

Boundary blow-up solutions for a cooperative system involving the p-Laplacian

Li Chen, Yujuan Chen, Dang Luo (2013)

Annales Polonici Mathematici

We study necessary and sufficient conditions for the existence of nonnegative boundary blow-up solutions to the cooperative system Δ p u = g ( u - α v ) , Δ p v = f ( v - β u ) in a smooth bounded domain of N , where Δ p is the p-Laplacian operator defined by Δ p u = d i v ( | u | p - 2 u ) with p > 1, f and g are nondecreasing, nonnegative C¹ functions, and α and β are two positive parameters. The asymptotic behavior of solutions near the boundary is obtained and we get a uniqueness result for p = 2.

Convergence of minimax structures and continuation of critical points for singularly perturbed systems

Benedetta Noris, Hugo Tavares, Susanna Terracini, Gianmaria Verzini (2012)

Journal of the European Mathematical Society

In the recent literature, the phenomenon of phase separation for binary mixtures of Bose–Einstein condensates can be understood, from a mathematical point of view, as governed by the asymptotic limit of the stationary Gross–Pitaevskii system - Δ u + u 3 + β u v 2 = λ u , - Δ v + v 3 + β u 2 v = μ v , u , v H 0 1 ( Ω ) , u , v > 0 , as the interspecies scattering length β goes to + . For this system we consider the associated energy functionals J β , β ( 0 , + ) , with L 2 -mass constraints, which limit J (as β + ) is strongly irregular. For such functionals, we construct multiple critical points via a common...

Existence and asymptotic behavior of positive solutions for elliptic systems with nonstandard growth conditions

Honghui Yin, Zuodong Yang (2012)

Annales Polonici Mathematici

Our main purpose is to establish the existence of a positive solution of the system ⎧ - p ( x ) u = F ( x , u , v ) , x ∈ Ω, ⎨ - q ( x ) v = H ( x , u , v ) , x ∈ Ω, ⎩u = v = 0, x ∈ ∂Ω, where Ω N is a bounded domain with C² boundary, F ( x , u , v ) = λ p ( x ) [ g ( x ) a ( u ) + f ( v ) ] , H ( x , u , v ) = λ q ( x ) [ g ( x ) b ( v ) + h ( u ) ] , λ > 0 is a parameter, p(x),q(x) are functions which satisfy some conditions, and - p ( x ) u = - d i v ( | u | p ( x ) - 2 u ) is called the p(x)-Laplacian. We give existence results and consider the asymptotic behavior of solutions near the boundary. We do not assume any symmetry conditions on the system.

Existence of solutions to the (rot,div)-system in L p -weighted spaces

Wojciech M. Zajączkowski (2010)

Applicationes Mathematicae

The existence of solutions to the elliptic problem rot v = w, div v = 0 in a bounded domain Ω ⊂ ℝ³, v · n ̅ | S = 0 , S = ∂Ω in weighted L p -Sobolev spaces is proved. It is assumed that an axis L crosses Ω and the weight is a negative power function of the distance to the axis. The main part of the proof is devoted to examining solutions of the problem in a neighbourhood of L. The existence in Ω follows from the technique of regularization.

Generalized RBSDEs with Random Terminal Time and Applications to PDEs

Katarzyna Jańczak-Borkowska (2011)

Bulletin of the Polish Academy of Sciences. Mathematics

Generalized reflected backward stochastic differential equations have been considered so far only in the case of a deterministic interval. In this paper the existence and uniqueness of solution for generalized reflected backward stochastic differential equations in a convex domain with random terminal time is studied. Applications to the obstacle problem with Neumann boundary conditions for partial differential equations of elliptic type are given.

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