In the present paper, we study a conservation law with periodic boundary conditions and an impulsive source term, which contains an approximation φ _ε (t) of the Dirac delta function δ _(t=0) . Along with suppφ _ε =[0,ε ] , the scaling mapping t=t/ε:[0,ε ]↦ [0,1] leads to a new problem on the infinitesimal initial layer. In the limit as ε→ 0 , this new problem enables us to obtain new initial data.

We study the two-dimensional Cauchy-Dirichlet problems for the quasilinear pseudoparabolic integro-differential Volterra equation and for the quasilinear pseudoparabolic integro-differential Fredholm equation. In each of these two problems, the integral term depends on a positive integer parameter and, as , converges weakly to the expression incorporating the Dirac delta-function, which models an instantaneous impulsive impact. For fixed we prove the well-posedness of each of the two problems in the class of regular weak solutions. Further, for each of the two problems, we establish that the initial shock layer, associated with the Dirac delta-function, is formed as , and that the family of regular weak solutions of the original problem converges to the strong solution of a limit two-scale microscopic-macroscopic model.

We study the multi-dimensional initial-boundary value problem for the semilinear pseudoparabolic equation with a regular nonlinear minor term, which, in general, may be superlinear. This term models a non-instantaneous but a very rapid absorption with q(x)-growth. The minor term depends on a positive integer parameter n and, as n→+∞, converges weakly⋆ to the expression incorporating the Dirac delta function, which, in turn, models an instant absorption at the initial moment. We prove that an infinitesimal initial layer, associated with the Dirac delta function, is formed as n→+∞, and that the family of regular weak solutions to the original problem converges to the strong solution of a two-scale microscopic-macroscopic model. This model consists of two equations and the set of initial, boundary, and matching conditions, so that the 'outer' macroscopic solution beyond the initial layer is governed by the linear homogeneous pseudoparabolic equation at the macroscopic ('slow') timescale, while the initial layer solution is defined at the microscopic level and obeys the semilinear pseudoparabolic equation at the microscopic ('fast') timescale. The latter equation inherits the full information about the profile of the original non-instantaneous absorption. In general, the research is devoted to pseudoparabolic equations with measure data depending on an unknown solution.

We study the multi-dimensional Cauchy-Dirichlet problem for the p(x, t)-parabolic equation with a regular nonlinear minor term, which models a non-instantaneous but very rapid absorption with the q(x, t)-growth. The minor term depends on a positive integer parameter n and, as n -> +infinity, converges weakly star to the expression incorporating the Dirac delta function, which, in turn, models an instant absorption at the initial moment. We prove that an infinitesimal initial layer, associated with the Dirac delta function, is formed as n -> +infinity, and that the family of regular weak solutions of the original problem converges to the so-called 'strong-weak' solution of a two-scale microscopic-macroscopic model. Furthermore, the equation of the microstructure can be integrated explicitly, which leads in a number of cases to the purely macroscopic formulation for the p(x, t)-parabolic equation provided with the corrected initial data.

Steady state stability features of a diode with electrons and positrons entering from opposite boundaries and moving without collisions in plasma are numerically studied. The most complex regime when charged particles are reflected from potential barriers is considered. This problem arises, in particular, when modeling pulsar diodes. A small perturbation evolution is studied. It has been established that at the initial stage of the process the perturbation amplitude changes in time according to an exponential law. It is shown that stationary solutions with a potential barrier for electrons located near the electron-emitting electrode and a potential barrier for positrons located near the opposite electrode are stable when the inter-electrode distance is below a certain threshold. As the inter-electrode distance increases, the solutions become unstable. Solutions of another type when barriers reflecting particles are located in the opposite to the emitting electrode parts of the gap are also studied. However, these solutions turned out to be unstable.

We study the Dirichlet problem for the pseudo-parabolic equationut=div(|∇u|p(x,t)−2∇u)+Δut+f(x,t,u,∇u) in the cylinder (x,t)∈QT=Ω×(0,T), Ω⊂Rd, d≥2. It is shown that under appropriate conditions on the regularity of the data and the growth of the source f with respect to the second and third arguments, the problem has a global in time solution with the propertiesu∈L∞(0,T;H02(Ω)),ut,|∇ut|∈L2(QT),|∇u|∈L∞(0,T;Lp(⋅)(Ω))∩Lp(⋅,⋅)+δ(QT) with some δ>0. For special choices of the source f, sufficient conditions of uniqueness are derived, stability of solutions with respect to perturbations of the nonlinear structure of the equation is proven, and the rate of vanishing of ‖u‖W1,2(Ω) is found.

This paper continues studying stability features of steady states of a diode with counter-streaming electron and ion flows. In our recent paper, an integral-differential equa-tion for the potential perturbation amplitude in the mode without potential barriers reflecting charged particles within the plasma was derived. Its exact solution was found for homogeneous steady-state field distribution. In this paper, we propose a semi-analytical method to solve the integral-differential equation for potential perturbation amplitude in the case of inhomogene-ous steady-state solutions. It is based on the use of the piecewise linear approximation of the integral operator kernel and the variable coefficient as well as the potential perturbation distri-bution. A dispersion equation is obtained and five first dispersion branches are constructed. As a result, we have proved that all steady state potential distributions with the values of dimen-sionless inter-electrode gap up to 10 & pi;/-42 are unstable. Numerical calculations of the potential perturbation development confirm analytical results.

Stability of steady states of a planar geometry diode with counter flows of electrons and positrons is studied. The study is related to the elucidation of pulsar RF radiation nature. The equation for the electric field perturbation is derived. Its exact solution is obtained for the case of a homogeneous steady-state field. The study of the dispersion equation obtained has shown that there is a threshold for the inter-electrode gap value, above which steady-state solutions are unstable. The instability threshold turned out to be & RADIC;2 times higher than the known Pierce threshold.

We study the initial-boundary value problem for the one-dimensional Oskolkov pseudoparabolic equation of viscoelasticity with a nonlinear convective term and a linear absorption term. The absorption term depends on a positive integer parameter n and, as n → + ∞ , converges weakly * to the expression incorporating the Dirac deltafunction, which models an instant absorption at the initial moment of time. We prove that the infinitesimal initial layer, associated with the Dirac delta function, is formed as n → + ∞ , and that the family of regular weak solutions of the original problem converges to the strong solution of a two-scale microscopic-macroscopic model. The main novelty of the article consists of taking into account of the effect of convection. In the final section, some possible generalizations and applications are briefly discussed, in particular with regard to active fluids.

We study the Dirichlet problem for the pseudo-parabolic equation ut−diva(x,t)|∇u|p(x,t)−2∇u−Δut=b(x,t)|u|q(x,t)−2uin the cylinder QT=Ω×(0,T), where Ω⊂Rd is a sufficiently smooth domain. The positive coefficients a, b and the exponents p≥2, q>2 are given Lipschitz-continuous functions. The functions a, p are monotone decreasing, and b, q are monotone increasing in t. It is shown that there exists a positive constant M=M(|Ω|,sup(x,t)∈QTp(x,t),sup(x,t)∈QTq(x,t)), such if the initial energy is negative, E(0)=∫Ωa(x,0)p(x,0)|∇u0(x)|p(x,0)−b(x,0)q(x,0)|u0(x)|q(x,0)dx<−M,then the problem admits a local in time solution with negative energy E(t). If p and q are independent of t, then M=0. For the solutions from this class, sufficient conditions for the finite time blow-up are derived.

We study the multi-dimensional initial–boundary value problem for the quasilinear pseudoparabolic equation with a regular nonlinear minor term, which models a non-instantaneous impulsive impact. The minor term depends on a small parameter ɛ>0 and, as ɛ→0, converges weakly⋆ to the expression incorporating the Dirac delta function, which, in turn, models an instantaneous impulsive impact. We prove that the infinitesimal transition layer, associated with the Dirac delta function, is formed as ɛ→0, and that the family of weak solutions of the original problem converges to the weak solution of a two-scale microscopic–macroscopic model. This model consists of two equations and the set of initial, boundary, and matching conditions, so that the ‘outer’ macroscopic solution beyond the transition layer is governed by the quasilinear homogeneous pseudoparabolic equation at the macroscopic (‘slow’) timescale, while the transition layer solution is defined at the microscopic level and obeys the semilinear pseudoparabolic equation at the microscopic (‘fast’) timescale. The latter equation inherits the full information about the profile of the original non-instantaneous impulsive impact.

Instability features of steady states of the plasma diode with electron and positron counter flows are studied. There are several types of such states for each value of the inter-electrode distance. The case when charged particles moving in the diode plasma are not reflected from potential extrema is considered. We have solved an equation for the amplitude of the electric field perturbation for steady states with an inhomogeneous field distribution. Studying the dispersion equation has shown that all considered solutions are unstable. We have also confirmed this result when simulating small perturbation evolution of a steady-state solution.

The possibility of creating an alternating current source based on a thermionic en-ergy converter is because, under certain conditions, an electron instability can develop in such a diode in the collisionless mode, leading to a sharp current cut-off. To implement this effect, it is enough to short electrodes through inductance. To select the optimal operation mode of the generator, it is necessary to study external inductance influence on the development of the instability. This problem is theoretically studied in the proposed work, and both over-neu-tralized and under-neutralized modes are considered. Dispersion equations are obtained. It is shown that when external inductance is included an instability threshold can be moved below the Pierce one. Besides, this type of instability can develop only for inductance values from a limited range.

We study the two-dimensional Cauchy problem for the quasilinear pseudoparabolic equation with a regular nonlinear minor term endowed with periodic initial data and periodicity conditions. The minor term depends on a small parameter ɛ>0 and, as ɛ→0, converges weakly⋆ to the expression incorporating the Dirac delta function, which models an instantaneous impulsive impact. We establish that the transition (shock) layer, associated with the Dirac delta function, is formed as ɛ→0, and that the family of strong solutions of the original problem converges to the strong solution of a two-scale microscopic–macroscopic model. This model consists of two equations and the set of initial and matching conditions, so that the ‘outer’ macroscopic solution beyond the transition layer is governed by the quasilinear homogeneous pseudoparabolic equation at the macroscopic (‘slow’) timescale, while the transition layer solution is defined at the microscopic level and obeys the semilinear pseudoparabolic equation at the microscopic (‘fast’) timescale. The latter is derived based on the microstructure of the transition layer profile.

We study the Cauchy problem with periodic initial data for the heat equation, which includes a nonlocal in time integral term. This term models a fading memory effect and has the form of convolution of a nonlinear function depending on solution with a smooth relaxation kernel. The kernel contains a small parameter ε >0 and, as ε→ 0 , collapses to the Dirac delta function supported at some given moment of time t=τ . The Dirac delta function, in turn, models a shock (impulsive) loading at the moment t=τ . We establish that the transition (shock) layer, associated with the Dirac delta function, is formed as ε→ 0 , and the family of weak solutions of the considered problem converges to a solution of the two-scale microscopic-macroscopic model. This model consists of two equations and the set of initial and matching conditions, so that the ‘outer’ macroscopic solution beyond the transition layer is governed by the classical homogeneous heat equation, while the transition layer solution is defined on the microscopic level and obeys the Volterra integro-differential equation derived from the microstructure of the relaxation profile.

This article describes an experimental study of the electrical and kinetic parameters of a Knudsen-mode thermionic converter (TIC) equipped both with a multicavity and a smooth emitter that converts heat into electrical energy using a combination of cesium and barium vapors. In the study, we recorded and analyzed data on how the maximum electron current density and the specific power are affected by the values of cesium and barium vapors pressure, the width of the interelectrode gap, and such parameters of the emitters as temperature and surface geometry. By analyzing the influence of surface geometry, it was revealed that the values of the electron current density and the electric power are four times higher in a TIC with a multicavity emitter compared to a TIC with a smooth emitter. TIC efficiency was analyzed taking into account the effective emissivity of the multicavity emitter. It is shown that the efficiency of such a TIC exceeds that of a TIC with a smooth emitter despite a slight increase in heat transfer from the emitter to the collector due to radiation.

We measured the critical temperatures and critical switching and retrapping currents of wide and narrow thinfilm quasi-one-dimensional superconducting aluminum structures of the same thickness in zero magnetic field. For the first time, we found that the narrower the structure, the lower the critical temperature and critical current density in the structure. Probably, the influence of depairing centers that are on dirty longitudinal boundaries of the structure, is the stronger than the narrower the structure. It is found for the first time that, in most cases, the temperature-dependent switching critical current in both structures is approximated by two functions. At temperatures below the temperature corresponding to the bottom of the resistive N-S transition of structures, the switching critical current is described by the Kupriyanov-Lukichev theory. At temperatures close to the top of the N-S transition, the switching current is linear with temperature and coincides with the critical Josephson current. At these temperatures, Josephson SNS junctions are formed in structures.

In the present article, we study singular limits of weak energy solutions to non-instantaneous single-and multi-impulsive advection-diffusion-reaction equation as impulsive source terms collapse to time-dependent the Dirac delta-functions, i.e., to instant impulses. We establish that the limiting functions are the solutions of the instantaneous impulsive advection-diffusion-reaction equations. Besides, in the multi-impulsive case we find the effect of transition to an equilibrium as a number of impulses grows infinitely. The results can be applied to further modeling of such processes as frost-quakes in glaciology.

We measured the rectification of an ac voltage in a structure of superconducting circularly-asymmetric aluminum rings in series, permeated with a magnetic flux and biased with a low-frequency alternating current (without a dc component). This rectification is due to the shift of the maxima of the critical currents of different polarity relative to the zero flux in opposite directions along the flux axis in the asymmetric ring. For the first time, we propose a model for a temperature-dependent phase shift equal to difference between dimensionless kinetic inductances of wide and narrow semirings having the same length and thickness. The shift is not zero in the case of different critical currents densities in both semirings. This is possible only in a situation of different critical temperatures of both semirings. The model describes well the temperature-dependent shift of the maxima of the critical currents, answers the long-standing mysterious challenge of the shift and removes extremely strange contradiction between the results of different measurements, previously found in circularly-asymmetric aluminum structures.