444 research outputs found
Energy solutions to one-dimensional singular parabolic problems with data are viscosity solutions
We study one-dimensional very singular parabolic equations with periodic
boundary conditions and initial data in , which is the energy space. We
show existence of solutions in this energy space and then we prove that they
are viscosity solutions in the sense of Giga-Giga.Comment: 15 page
Universality in Blow-Up for Nonlinear Heat Equations
We consider the classical problem of the blowing-up of solutions of the
nonlinear heat equation. We show that there exist infinitely many profiles
around the blow-up point, and for each integer , we construct a set of
codimension in the space of initial data giving rise to solutions that
blow-up according to the given profile.Comment: 38 page
Very Singular Diffusion Equations-Second and Fourth Order Problems
This paper studies singular diffusion equations whose diffusion effect is so strong that the speed of evolution becomes a nonlocal quantity. Typical examples include the total variation flow as well as crystalline flow which are formally of second order. This paper includes fourth order models which are less studied compared with second order models. A typical example of this model is an H−1 gradient flow of total variation. It turns out that such a flow is quite different from the second order total variation flow. For example, we prove that the solution may instantaneously develop jump discontinuity for the fourth order total variation flow by giving an explicit example
Continuity of the blow-up profile with respect to initial data and to the blow-up point for a semilinear heat equation
International audienceWe consider blow-up solutions for semilinear heat equations with Sobolev subcritical power nonlinearity. Given a blow-up point , we have from earlier literature, the asymptotic behavior in similarity variables. Our aim is to discuss the stability of that behavior, with respect to perturbations in the blow-up point and in initial data. Introducing the notion of ``profile order", we show that it is upper semicontinuous, and continuous only at points where it is a local minimum
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Leadership and approaches to the management of workplace bullying
Leadership behaviour has been identified as an important antecedent of workplace bullying, since managers may prevent, permit or engage in the mistreatment of others. However, the issue of how managers respond when bullying occurs has received limited attention. With this in mind, the aim of this study was to explore how managers behave when bullying occurs in their work group, and to elucidate the contextual issues that underlie this behaviour. This was achieved through analysis of in-depth interviews with individuals involved in cases of bullying. The findings revealed a typology of four types of management behaviour in cases of bullying, each underpinned by contextual factors at the individual, group and organizational level. The study shows that the role of leadership in workplace bullying is more complex than previously thought, and suggests several ways in which managers and organizations could deal with bullying behaviour
Numerical computations of facetted pattern formation in snow crystal growth
Facetted growth of snow crystals leads to a rich diversity of forms, and
exhibits a remarkable sixfold symmetry. Snow crystal structures result from
diffusion limited crystal growth in the presence of anisotropic surface energy
and anisotropic attachment kinetics. It is by now well understood that the
morphological stability of ice crystals strongly depends on supersaturation,
crystal size and temperature. Until very recently it was very difficult to
perform numerical simulations of this highly anisotropic crystal growth. In
particular, obtaining facet growth in combination with dendritic branching is a
challenging task. We present numerical simulations of snow crystal growth in
two and three space dimensions using a new computational method recently
introduced by the authors. We present both qualitative and quantitative
computations. In particular, a linear relationship between tip velocity and
supersaturation is observed. The computations also suggest that surface energy
effects, although small, have a larger effect on crystal growth than previously
expected. We compute solid plates, solid prisms, hollow columns, needles,
dendrites, capped columns and scrolls on plates. Although all these forms
appear in nature, most of these forms are computed here for the first time in
numerical simulations for a continuum model.Comment: 12 pages, 28 figure
Renormalizing Partial Differential Equations
In this review paper, we explain how to apply Renormalization Group ideas to
the analysis of the long-time asymptotics of solutions of partial differential
equations. We illustrate the method on several examples of nonlinear parabolic
equations. We discuss many applications, including the stability of profiles
and fronts in the Ginzburg-Landau equation, anomalous scaling laws in
reaction-diffusion equations, and the shape of a solution near a blow-up point.Comment: 34 pages, Latex; [email protected]; [email protected]
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