1,895 research outputs found
Positive Stationary Solutions and Spreading Speeds of KPP Equations in Locally Spatially Inhomogeneous Media
The current paper is concerned with positive stationary solutions and spatial
spreading speeds of KPP type evolution equations with random or nonlocal or
discrete dispersal in locally spatially inhomogeneous media. It is shown that
such an equation has a unique globally stable positive stationary solution and
has a spreading speed in every direction. Moreover, it is shown that the
localized spatial inhomogeneity of the medium neither slows down nor speeds up
the spatial spreading in all the directions
Local and Global Well-Posedness for Aggregation Equations and Patlak-Keller-Segel Models with Degenerate Diffusion
Recently, there has been a wide interest in the study of aggregation
equations and Patlak-Keller-Segel (PKS) models for chemotaxis with degenerate
diffusion. The focus of this paper is the unification and generalization of the
well-posedness theory of these models. We prove local well-posedness on bounded
domains for dimensions and in all of space for , the
uniqueness being a result previously not known for PKS with degenerate
diffusion. We generalize the notion of criticality for PKS and show that
subcritical problems are globally well-posed. For a fairly general class of
problems, we prove the existence of a critical mass which sharply divides the
possibility of finite time blow up and global existence. Moreover, we compute
the critical mass for fully general problems and show that solutions with
smaller mass exists globally. For a class of supercritical problems we prove
finite time blow up is possible for initial data of arbitrary mass.Comment: 31 page
Exact analytical solutions to the master equation of quantum Brownian motion for a general environment
We revisit the model of a quantum Brownian oscillator linearly coupled to an
environment of quantum oscillators at finite temperature. By introducing a
compact and particularly well-suited formulation, we give a rather quick and
direct derivation of the master equation and its solutions for general spectral
functions and arbitrary temperatures. The flexibility of our approach allows
for an immediate generalization to cases with an external force and with an
arbitrary number of Brownian oscillators. More importantly, we point out an
important mathematical subtlety concerning boundary-value problems for
integro-differential equations which led to incorrect master equation
coefficients and impacts on the description of nonlocal dissipation effects in
all earlier derivations. Furthermore, we provide explicit, exact analytical
results for the master equation coefficients and its solutions in a wide
variety of cases, including ohmic, sub-ohmic and supra-ohmic environments with
a finite cut-off.Comment: 37 pages (26 + appendices), 14 figures; this paper is an evolution of
arXiv:0705.2766v1, but contains far more general and significant results; v2
minor changes, double column, improved Appendix
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