A reaction-diffusion system exhibiting Turing's diffusion driven instability is considered. The equation for an activator is supplemented by unilateral terms of the type s − (x)u − , s + (x)u + describing sources and sinks active only if the concentration decreases below and increases above, respectively, the value of the basic spatially constant solution which is shifted to zero. We show that the domain of diffusion parameters in which spatially non-homogeneous stationary solutions can bifurcate from that constant solution is smaller than in the classical case without unilateral terms. It is a dual information to previous results stating that analogous terms in the equation for an inhibitor imply the existence of bifurcation points even in diffusion parameters for which bifurcation is excluded without unilateral sources. The case of mixed (Dirichlet-Neumann) boundary conditions as well as that of pure Neumann conditions is described.

This paper analyzes the structure of the set of nodal solutions, i.e., solutions changing sign, of a class of one-dimensional superlinear indefinite boundary values problems with an indefinite weight functions in front of the spectral parameter. Quite surprisingly, the associated high order eigenvalues may not be concave as is the case for the lowest one. As a consequence, in many circumstances, the nodal solutions can bifurcate from three or even four bifurcation points from the trivial solution. This paper combines analytical and numerical tools. The analysis carried out is a paradigm of how mathematical analysis aids the numerical study of a problem, whereas simultaneously the numerical study confirms and illuminates the analysis. 2020 MSC: 34B15, 34B08, 34L16.

We consider a general reaction-diusion system exhibiting Turing's diusion-driven instability. In the rst part of the paper, we supplement the activator equation by unilateral integral sources and sinks of the type K u(x) |K| dK − and K u(x) |K| dK +. These terms measure an average of the concentration over the set K and are active only when this average decreases bellow or increases above the value of the reference spatially homogeneous steady state, which is shifted to the origin. We show that the set of diusion parameters in which spatially heterogeneous stationary solutions can bifurcate from the reference state is smaller than in the classical case without any unilateral integral terms. This problem is studied for the case of mixed, pure Neumann and periodic boundary conditions. In the second part of the paper, we investigate the eect of both unilateral terms of the type u − , u + and unilateral integral terms on the pattern formation using numerical experiments on the system with well-known Schnakenberg kinetics.

This paper analyzes the structure of the set of positive solutions of a class of one-dimensional superlinear indefinite bvp’s. It is a paradigm of how mathematical analysis aids the numerical study of a problem, whereas simultaneously its numerical study confirms and illuminates the analysis. On the analytical side, we establish the fast decay of the positive solutions as λ ↓ −∞ in the region where a(x) < 0 (see (1.1)), as well as the decay of the solutions of the parabolic counterpart of the model (see (1.2)) as λ ↓ −∞ on any subinterval of [0, 1] where u 0 = 0, provided u 0 is a subsolution of (1.1). This result provides us with a proof of a conjecture of [26] under an additional condition of a dynamical nature. On the numerical side, this paper ascertains the global structure of the set of positive solutions on some paradigmatic prototypes whose intricate behavior is far from predictable from existing analytical results.