This paper is concerned with the blowup phenomena for initial–boundary value problem ...(0.1)τ∂t2u(x,t)−Δu(x,t)+a(x)∂tu(x,t)=λ|u(x,t)|p,(x,t)∈CΣ×(0,T),u(x,t)=0,(x,t)∈∂CΣ×(0,T),u(x,0)=εf(x),x∈CΣ,τ∂tu(x,0)=τεg(x),x∈CΣ,where CΣ is a cone-like domain in RN (N≥2) defined as CΣ=intrω∈RN;r≥0,ω∈Σ with a connected open set Σ in SN−1 with smooth boundary ∂Σ. If N=1, then we only consider two cases CΣ=(0,∞) and CΣ=R. Here a(x) is a non-zero coefficient of ∂tu which could be complex-valued and space-dependent, λ∈ℂ is a fixed constant, and ε>0 is a small parameter. The constants τ=0,1 switch the parabolicity and hyperbolicity of the problem (0.1). The result proposes an argument for sharp upper lifespan estimates of solutions to (0.1) by a test function method based on Mitidieri and Pokhozhaev (2001). The crucial idea is to introduce an ordinary differential inequality with a parameter as a variable.
We consider the Cauchy problem of the system of nonlinear Schrödinger equations with derivative nonlinearlity. This system was introduced by Colin and Colin (Differ Int Equ 17:297–330, 2004) as a ...model of laser-plasma interactions. We study existence of ground state solutions and the global well-posedness of this system by using the variational methods. We also consider the stability of traveling waves for this system. These problems are proposed by Colin–Colin as the open problems. We give a subset of the ground-states set which satisfies the condition of stability. In particular, we prove the stability of the set of traveling waves with small speed for 1-dimension.
We consider the large time behavior of the solutions to the Cauchy problem for the BBM–Burgers equation. We prove that the solution to this problem goes to the self-similar solution to the Burgers ...equation called the nonlinear diffusion wave. Moreover, we construct the appropriate second asymptotic profiles of the solutions depending on the initial data. Based on that discussion, we investigate the effect of the initial data on the large time behavior of the solution, and derive the optimal asymptotic rate to the nonlinear diffusion wave. Especially, the important point of this study is that the second asymptotic profiles of the solutions with slowly decaying data, whose case has not been studied, are obtained.
•A link of the glass-forming ability (GFA) to the Arrhenius crossover (AC) is explored.•The thermodynamic liquid model of Kauzmann is related to our BSCNF model.•The BSCNF model provides a ...microscopic picture to understand the AC and the GFA.•It is suggested that fragile glass-forming liquids exhibiting the AC have a high GFA.
We studied on a link between the Arrhenius crossover (AC) of structural relaxation and the glass-forming ability (GFA) in fragile liquids based on the bond strength–coordination number fluctuation (BSCNF) model proposed by the authors. According to the BSCNF model, an Arrhenius-like relaxation behavior appears at high temperature due to the presence of large fluctuations with strong mean binding energy, which could lead to an enhancement of GFA. We show that this behavior is consistent with the experimental observation in fragile glass-formers and with the thermodynamic liquid model which enables to assess the free energy barrier between amorphous and crystalline phases. In the present study, the implication of the AC-like behavior of fragile systems is discussed in terms of the GFA.
The theory of the vacancy formation in crystals has been extended by generalising the vacancy formation energy that depends on the number of vacancies resulting from thermal activation. This ...extension enables to describe the freezing of defect concentrations at lower temperature, while the derived vacancy concentration reduces to the Boltzmann-Arrhenius law at higher temperature. The implications of the low-temperature patterns and its relevance to non-equilibrium process in the defect concentrations have been also touched upon. In addition, based on the approach explored, we further construct a model for the ionic conductivity in the intrinsic and extrinsic regions, which is formulated by focusing on the relation between the number concentration and the mobility of ionic charge carriers. A theoretical background to the continuity (knee) across which the both ionic conductivity patterns are smoothly connected is corroborated by the present work.
Herein, the relation between the Arrhenius crossover (AC) behaviors in the structural relaxation and the electrical conductivity relaxation is explored in terms of the Bond Strength–Coordination ...Number Fluctuation (BSCNF) model developed by the authors. For the ionic glass‐forming liquids and solution examined, Ca2K3(NO3)7, Ca2Rb3(NO3)7, ZBLAN20 and (LiCl)0.16(H2O)0.84, the electric conductivity at high temperature resembles the behavior of the high‐temperature structural relaxation. This mutual relationship suggests a similar origin of the AC behaviors for both, structural and electrical relaxation. The difference between them is reflected in the different temperatures that demarcate the AC behaviors. The implication of this difference to the occurrence of the high ionic conduction via the decoupling mechanism is discussed in terms of the BSCNF model and the fractional Stokes–Einstein law.
The Bond Strength–Coordination Number Fluctuation (BSCNF) model suggests that the Arrhenius crossover behavior in the structural relaxation has a similar origin to that of the electrical conductivity relaxation. The present study provides a relation between these relaxation behaviors, which could be important in the study of the decoupling phenomenon observed in solid electrolytes.
In this paper we consider the following initial-boundary value problem with the power type nonlinearity |u|p with 1<p≤2 in a two-dimensional exterior ...domain(0.1){τ∂t2u(x,t)−Δu(x,t)+eiζ∂tu(x,t)=λ|u(x,t)|p,(x,t)∈Ω×(0,T),u(x,t)=0,(x,t)∈∂Ω×(0,T),u(x,0)=εf(x),x∈Ω,∂tu(x,0)=εg(x),x∈Ω, where Ω is given by Ω={x∈R2;|x|>1}, ζ∈−π2,π2, λ∈C and τ∈{0,1} switches the parabolicity, dispersivity and hyperbolicity. Remark that 2=1+2/N is well-known as the Fujita exponent. If p>2, then there exists a small global-in-time solution of (0.1) for some initial data small enough (see Ikehata 11), and if p<2, then global-in-time solutions cannot exist for any positive initial data (see Ogawa–Takeda 22 and Lai–Yin 14). The result is that for given initial data (f,τg)∈H01(Ω)×L2(Ω) satisfying (f+τg)log|x|∈L1(Ω) with some requirement, the solution blows up at finite time, and moreover, the upper bound for lifespan of solutions to (0.1) is given as the following double exponential type when p=2:LifeSpan(u)≤expexp(Cε−1). The crucial idea is to use test functions which approximates the harmonic function log|x| satisfying Dirichlet boundary condition and the technique modified from 9.
Metallic glass-forming liquids (MGFLs) exhibit a quite unusual pattern in the temperature dependence of the viscosity when plotted in a wide range of temperature. Among the fragile liquids, the MGFLs ...have the propensity that at high temperature the viscosity exhibits a very low activation energy, while at low temperature a straight-line viscosity behavior appears in the Arrhenius representation. This unusual pattern hinders a theoretical description over a wide temperature range and the understanding of its microscopic mechanism. In the present study, we show that the modified BSCNF model describes well the temperature dependence of the viscosity of MGFLs. The relevance of the result to the fragile-to-strong transition that can occur in the supercooled regime is discussed briefly.
In this paper, we consider the initial-boundary value problems with several fundamental boundary conditions (the Dirichlet/Neumann/Robin boundary condition) for the multi-component system of ...semi-linear classical damped wave equations outside a ball. By applying a test function approach with a judicious choice of test functions, which approximates the harmonic functions being subject to these boundary conditions on
∂
Ω
, simultaneously we have succeeded in proving the blow-up result in a finite time as well as in catching the upper bound of lifespan estimates for small solutions in all spatial dimensions. Moreover, such kind of these results, which become sharp in the subcritical cases for one-dimensional case, will be discussed at the end of this paper.
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