150 research outputs found
Non-uniqueness for a critical heat equation in two dimensions with singular data
No full textNonlinear heat equations in two dimensions with singular initial data are studied. In recent works nonlinearities with exponential growth of Trudinger-Moser type have been shown to manifest critical behavior: well-posedness in the subcritical case and non-existence for certain supercritical data. In this article we propose a specific model nonlinearity with Trudinger-Moser growth for which we obtain surprisingly complete results: a) for initial data strictly below a certain singular threshold function u ̃ the problem is well-posed, b) for initial data above this threshold function u ̃, there exists no solution, c) for the singular initial datum u ̃ there is non-uniqueness. The function u ̃ is a weak stationary singular solution of the problem, and we show that there exists also a regularizing classical solution with the same initial datum u ̃
Existence, non-existence, and uniqueness for a heat equation with exponential nonlinearity in R^2
No full textWe consider a semilinear heat equation with exponential nonlinearity in R2. We prove that local solutions do not exist for certain data in the Orlicz space exp L2(R2), even though a small data global existence result holds in the same space exp L2(R2). Moreover, some suitable subclass of exp L2(R2) for local existence and uniqueness is proposed
Non-radial maximizers for functionals with exponential non-linearity in R-2
No full textWe consider the functional F:H-0(1)(B(0,1))-> R
F(u)=integral(B(0,1)) vertical bar x vertical bar(alpha)(e(p vertical bar u vertical bar gamma)-1-p vertical bar u vertical bar(gamma))dx
where alpha>0, p>0, 1<= 2, and B(0,1) is the unit ball in R-2. We prove that for any p>0, 1<2 and 0<4 pi, gamma=2 no maximizer of F(u) on the unit ball in H-0(1) is radially symmetric provided that alpha is large enough. This extends a result of Smets, Su and Willem concerning the existence of non-radial ground state solutions for the Rayleigh quotient related to the Henon equation with Dirichlet boundary conditions
Asymptotics for a parabolic equation with critical exponential nonlinearity
No full textWe consider the Cauchy problem: {∂tu=Δu-u+λf(u)in(0,T)×R2,u(0,x)=u0(x)inR2,where λ> 0 , f(u):=2α0ueα0u2,for someα0>0,with initial data u∈ H1(R2). The nonlinear term f has a critical growth at infinity in the energy space H1(R2) in view of the Trudinger-Moser embedding. Our goal is to investigate from the initial data u∈ H1(R2) whether the solution blows up in finite time or the solution is global in time. For 0<12α0, we prove that for initial data with energies below or equal to the ground state level, the dichotomy between finite time blow-up and global existence can be determined by means of a potential well argument
Distribution of resonance widths in localized tight-binding models
No full textWe numerically analyze the distribution of scattering resonance
widths in one- and quasi-one dimensional tight binding models, in the
localized regime. We detect and discuss an algebraic decay of the
distribution, similar, though not identical, to recent theoretical predictions
Blow-up and global solutions for subcritical and critical parabolic equations in
No full textWe study the local well-posedness in the framework of the Sobolev space (Formula presented), for a semilinear parabolic equation with asymptotically polynomial nonlinearity up to the critical Sobolev growth. Then we establish the dichotomy between blow-up and global existence for solutions with small energy by means of variational methods and the so-called potential well argument
An application of the multilevel regression technique to validate a social stratification scale
No full textThis chapter applies the multilevel regression technique in order to validate the ranking of occupational categories in a reputational scale of social desirability. Such attention to the differences inherent in work roles resumes the topic of the unequal distribution of material and symbolic rewards within societies. A tool widely used by sociologists to grasp the distributive inequalities associated with jobs is the occupational stratification scale or the hierarchical ordering of occupations. The aim of the model presented in this chapter is to validate an occupational stratification scale constructed in 2007 on the basis of the scale developed by de Lillo-Schizzerotto in 1985. The scale consists of 110 occupational categories constructed as the aggregate of 676 occupations (described in detail) which 2000 interviewees were asked to evaluate in terms of their social desirability. The ordering of the scale is validated through decomposition of the heterogeneity of the evaluations. The multilevel model shows that the 110 categories explain large part of this heterogeneity, also with the socio-demographic characteristics of the interviewees remaining equal
Comparative response of brain to chronic hypoxia and hyperoxia
No full textTwo antithetic terms, hypoxia and hyperoxia, i.e., insufficient and excess oxygen availability with respect to needs, are thought to trigger opposite responses in cells and tissues. This review aims at summarizing the molecular and cellular mechanisms underlying hypoxia and hyperoxia in brain and cerebral tissue, a context that may prove to be useful for characterizing not only several clinically relevant aspects, but also aspects related to the evolution of oxygen transport and use by the tissues. While the response to acute hypoxia/hyperoxia presumably recruits only a minor portion of the potentially involved cell machinery, focusing into chronic conditions, instead, enables to take into consideration a wider range of potential responses to oxygen-linked stress, spanning from metabolic to genic. We will examine how various brain subsystems, including energetic metabolism, oxygen sensing, recruitment of pro-survival pathways as protein kinase B (Akt), mitogen-activated protein kinases (MAPK), neurotrophins (BDNF), erythropoietin (Epo) and its receptors (EpoR), neuroglobin (Ngb), nitric oxide (NO), carbon monoxide (CO), deal with chronic hypoxia and hyperoxia to end-up with the final outcomes, oxidative stress and brain damage. A more complex than expected pattern results, which emphasizes the delicate balance between the severity of the stress imposed by hypoxia and hyperoxia and the recruitment of molecular and cellular defense patterns. While for certain functions the expectation that hypoxia and hyperoxia should cause opposite responses is actually met, for others it is not, and both emerge as dangerous treatments
Asymptotic behavior and decay estimates of the solutions for a nonlinear parabolic equation with exponential nonlinearity
No full textWe consider a nonlinear parabolic equation with an exponential nonlinearity which is critical with respect to the growth of the nonlinearity and the regularity of the initial data. After showing the equivalence of the notions of weak and mild solutions, we derive decay estimates and the asymptotic behavior for small global-in-time solutions
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