A reaction-diffusion equation on a net-shaped thin domain
Thomas Elsken (2004)
Studia Mathematica
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Thomas Elsken (2004)
Studia Mathematica
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Liliana Klimczak (2015)
Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica
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We consider a nonlinear Neumann problem with a nonhomogeneous elliptic differential operator. With some natural conditions for its structure and some general assumptions on the growth of the reaction term we prove that the problem has two nontrivial solutions of constant sign. In the proof we use variational methods with truncation and minimization techniques.
Jan Eisner, Jan Žilavý (2023)
Archivum Mathematicum
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We show the location of so called critical points, i.e., couples of diffusion coefficients for which a non-trivial solution of a linear reaction-diffusion system of activator-inhibitor type on an interval with Neumann boundary conditions and with additional non-linear unilateral condition at one or two points on the boundary and/or in the interior exists. Simultaneously, we show the profile of such solutions.
Salah Badraoui (1999)
Applicationes Mathematicae
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We are concerned with the boundedness and large time behaviour of the solution for a system of reaction-diffusion equations modelling complex consecutive reactions on a bounded domain under homogeneous Neumann boundary conditions. Using the techniques of E. Conway, D. Hoff and J. Smoller [3] we also show that the bounded solution converges to a constant function as t → ∞. Finally, we investigate the rate of this convergence.
Jan Chabrowski, Kyril Tintarev (2005)
Bulletin of the Polish Academy of Sciences. Mathematics
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We establish the existence of a solution to the Neumann problem in the half-space with a subcritical nonlinearity on the boundary. Solutions are obtained through the constrained minimization or minimax. The existence of solutions depends on the shape of a boundary coefficient.
Wei-Ming Ni (2004)
Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni
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We discuss the stability and instability properties of steady state solutions to single equations, shadow systems, as well as systems. Our basic observation is that the more complicated the pattern are, the more unstable they tend to be.
Kouachi, Said (2002)
Electronic Journal of Qualitative Theory of Differential Equations [electronic only]
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Jifeng Chu, Daqing Jiang, Donal O'Regan, R. P. Agarwal (2005)
Applicationes Mathematicae
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We study the existence of positive solutions to second order nonlinear differential equations with Neumann boundary conditions. The proof relies on a fixed point theorem in cones, and the positivity of Green's function plays a crucial role in our study.
Martin Stynes, R. Bruce Kellogg (2002)
Mathematica Bohemica
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Kolmogorov -widths are an approximation theory concept that, for a given problem, yields information about the optimal rate of convergence attainable by any numerical method applied to that problem. We survey sharp bounds recently obtained for the -widths of certain singularly perturbed convection-diffusion and reaction-diffusion boundary value problems.
Benchérif-Madani, Abdelatif (2009)
Serdica Mathematical Journal
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2000 Mathematics Subject Classification: Primary 60J45, 60J50, 35Cxx; Secondary 31Cxx. We give a probabilistic formula for the solution of a non-homogeneous Neumann problem for a symmetric nondegenerate operator of second order in a bounded domain. We begin with a g-Hölder matrix and a C^1,g domain, g > 0, and then consider extensions. The solutions are expressed as a double layer potential instead of a single layer potential; in particular a new boundary function is discovered...