Self-similar solutions of semilinear wave equations with a focusing nonlinearity

We prove that in three space dimensions a nonlinear wave equation $u_{tt}-\Delta u = u^p$ with $p\geq 7$ being an odd integer has a countable family of regular spherically symmetric self-similar solutions.Comment: 12 pages, 3 figures, minor corrections to match the published versio

Singular and regular solutions of a non-linear parabolic system

We study a dissipative nonlinear equation modelling certain features of the Navier-Stokes equations. We prove that the evolution of radially symmetric compactly supported initial data does not lead to singularities in dimensions $n\leq 4$. For dimensions $n>4$ we present strong numerical evidence supporting existence of blow-up solutions. Moreover, using the same techniques we numerically confirm a conjecture of Lepin regarding existence of self-similar singular solutions to a semi-linear heat equation.Comment: 16 page

Shrinkers, expanders, and the unique continuation beyond generic blowup in the heat flow for harmonic maps between spheres

Using mixed analytical and numerical methods we investigate the development of singularities in the heat flow for corotational harmonic maps from the $d$-dimensional sphere to itself for $3\leq d\leq 6$. By gluing together shrinking and expanding asymptotically self-similar solutions we construct global weak solutions which are smooth everywhere except for a sequence of times $T_1<T_2<...<T_k<\infty$ at which there occurs the type I blow-up at one of the poles of the sphere. We show that in the generic case the continuation beyond blow-up is unique, the topological degree of the map changes by one at each blow-up time $T_i$, and eventually the solution comes to rest at the zero energy constant map.Comment: 24 pages, 8 figures, minor corrections, matches published versio

Use of Onion-Like Carbon to Reinforce Carbon Composites

Onion-like carbon reinforced carbon-carbon composite was fabricated, and the influence of onion-like carbon (OLC) on the microstructure and mechanical and friction properties was investigated by porosity analysis, scanning electron microscopy, three-point bending test, nanoindentation test and ring-on-ring friction test. The results show that the sample containing OLC has a higher flexural strength (by 7.3%) and compressive strength (by 29.3%), hardness (by 2.1 times) and apparent density (by 1.1%) and smaller open porosity (7.9% vs 9.8%) and mesoporevolume, which is confirmed by porosity analysis and is attributed to improved fiber/matrix interface performance. The presence of OLC results in higher hardness and elastic modulus of carbon matrix under nanoindentation testing, which leads to modification of friction mechanism and a decrease in the wear rate under friction (by 3.3 times). Besides, OLC particles form self-lubricating film and show a graphitic carbon solid lubricant properties

Five types of blow-up in a semilinear fourth-order reaction-diffusion equation: an analytic-numerical approach

Five types of blow-up patterns that can occur for the 4th-order semilinear parabolic equation of reaction-diffusion type u_t= -\Delta^2 u + |u|^{p-1} u \quad {in} \quad \ren \times (0,T), p>1, \quad \lim_{t \to T^-}\sup_{x \in \ren} |u(x,t)|= +\iy, are discussed. For the semilinear heat equation $u_t= \Delta u+ u^p$, various blow-up patterns were under scrutiny since 1980s, while the case of higher-order diffusion was studied much less, regardless a wide range of its application.Comment: 41 pages, 27 figure

Propargylated Novolac Resins for Solvent-Free Technology for High-Performance Composites

Propargyl substituted novolac phenolic resin diluted with unsaturated bisphenol-A ethers was used for glass fiber solvent-free impregnation for the formation of high-performance composites. The addition of 20% mass of diallyl (DAEBA) or dipropargyl (DPEBA) bisphenol-A to propargyl substituted novolac phenolic resin resulted in viscosity drop from 2000 mPa∙s to 400‒500 mPa∙s at 140 °C. This proved to be enough to achieve complete impregnation of the twisted glass fibers, as illustrated by SEM analysis. This  improvement in impregnation was shown to result in increasing both flexural strength and modulus of the unidirectional glass fiber composite material approximately with a factor of two compared to the composite impregnated with resin without bisphenol-A ethers. DPEBA was shown to be more suitable for high-temperate applications since its addition does not seem to result in a decrease of the heat deflection temperature (HDT)

Математическое моделирование тепловых режимов космических аппаратов негерметичного исполнения

A description of the software complex for modeling the thermal conditions of non-hermetic spacecrafts in circular and elliptical orbits is given. The software complex structure, mathematical models of thermal processes in the presence of thermal control system are presented. For thermal processes calculation the method of lumped parameters is used. The results of thermal stabilization for model of orbital optical device are given. The software complex can be used in the design and development of thermal control systems of nonhermetic spacecraft, and in the determination of the causes of emergency situations in these systems in orbit as well.Предлагается описание программного комплекса для моделирования тепловых режимов космических аппаратов негерметичного исполнения в условиях полета по круговым и эллиптическим орбитам. Приводятся состав и структура программного комплекса, математические модели тепловых процессов при наличии системы обеспечения тепловых режимов космических аппаратов. Для расчета тепловых процессов используется метод сосредоточенных параметров. Даются результаты вычислительных экспериментов по термостабилизации макета орбитального оптического прибора. Программный комплекс может быть использован при проектировании и создании систем обеспечения тепловых режимов космических аппаратов негерметичного исполнения, а также при определении причин нештатных ситуаций в данных системах на орбите