Diffusion-induced blowup solutions for the shadow limit model of a singular Gierer–Meinhardt system

In this paper, we provide a thorough investigation of the blowing up behavior induced via diffusion of the solution of the following non-local problem: [Formula: see text] where [Formula: see text] is a bounded domain in [Formula: see text] with smooth boundary [Formula: see text] such problem is derived as the shadow limit of a singular Gierer–Meinhardt system, Kavallaris and Suzuki [On the dynamics of a non-local parabolic equation arising from the Gierer–Meinhardt system, Nonlinearity (2017) 1734–1761; Non-Local Partial Differential Equations for Engineering and Biology: Mathematical Modeling and Analysis, Mathematics for Industry, Vol. 31 (Springer, 2018)]. Under the Turing type condition [Formula: see text] we construct a solution which blows up in finite time and only at an interior point [Formula: see text] of [Formula: see text] i.e. [Formula: see text] where [Formula: see text] More precisely, we also give a description on the final asymptotic profile at the blowup point [Formula: see text] and thus we unveil the form of the Turing patterns occurring in that case due to driven-diffusion instability. The applied technique for the construction of the preceding blowing up solution mainly relies on the approach developed in [F. Merle and H. Zaag, Reconnection of vortex with the boundary and finite time quenching, Nonlinearity 10 (1997) 1497–1550] and [G. K. Duong and H. Zaag, Profile of a touch-down solution to a nonlocal MEMS model, Math. Models Methods Appl. Sci. 29 (2019) 1279–1348].

EFFECT OF VISCOSITY ON THE DIFFUSION MIXING PROCESS IN ISOTHERMAL MULTICOMPONENT GAS MIXTURES

In this article, the influence of viscosity on the stability of the diffusion process in isothermal multicomponent gas mixtures is experimentally studied. For gas systems 0,4300 C3H8 + 0,5700 He - 0,4300 C3H8 + 0,5700 CH4 and 0,4300 С3Н8 + 0,5700 Не - 0,4200 C3H8 + 0,5800 Ne, the results of calculating the number of partial losses of components at different pressures are given. The dependence of the amount of diffused components (partial expenses) on the pressure for this initial composition was studied. Then, the characteristic features of mass transfer for systems with different viscosities were compared over the same time intervals. Studies have shown a qualitative change in the nature of the mixing process. An increase in the viscosity of the mixture led to a stabilization of the diffusion process with an increase in the pressure in one case and the time of the process in the other. Analysis of the results showed that the viscosity affects the nature of the diffusion instability of the gas mixing process.

Small-Scale Effect on Longitudinal Wave Propagation in Laser-Excited Plates

Longitudinal wave propagation in an elastic isotopic laser-excited solid plate with atomic defect (vacancies, interstitials) generation is studied by the nonlocal continuum model. The nonlocal differential constitutive equations of Eringen are used in the formulations. The coupled governing equations for the dynamic of elastic displacement and atomic defect concentration fields are obtained. The frequency equations for the symmetrical and antisymmetrical motions of the plate are found and discussed. Explicit expressions for different characteristics of waves like phase velocity and attenuation (amplification) coefficients are derived. It is shown that coupling between the displacement and defect concentration fields affects the wave dispersion characteristics in the nonlocal elasticity. The dispersion curves of the elastic-diffusion instability are investigated for different pump parameters and larger wave numbers.