957 research outputs found
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 , 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
Ill-posedness of degenerate dispersive equations
In this article we provide numerical and analytical evidence that some
degenerate dispersive partial differential equations are ill-posed.
Specifically we study the K(2,2) equation and
the "degenerate Airy" equation . For K(2,2) our results are
computational in nature: we conduct a series of numerical simulations which
demonstrate that data which is very small in can be of unit size at a
fixed time which is independent of the data's size. For the degenerate Airy
equation, our results are fully rigorous: we prove the existence of a compactly
supported self-similar solution which, when combined with certain scaling
invariances, implies ill-posedness (also in )
A particle system with explosions: law of large numbers for the density of particles and the blow-up time
Consider a system of independent random walks in the discrete torus with
creation-annihilation of particles and possible explosion of the total number
of particles in finite time. Rescaling space and rates for
diffusion/creation/annihilation of particles, we obtain a stong law of large
numbers for the density of particles in the supremum norm. The limiting object
is a classical solution to the semilinear heat equation u_t =u_{xx} + f(u). If
f(u)=u^p, 1<p \le 3, we also obtain a law of large numbers for the explosion
time
New variable separation approach: application to nonlinear diffusion equations
The concept of the derivative-dependent functional separable solution, as a
generalization to the functional separable solution, is proposed. As an
application, it is used to discuss the generalized nonlinear diffusion
equations based on the generalized conditional symmetry approach. As a
consequence, a complete list of canonical forms for such equations which admit
the derivative-dependent functional separable solutions is obtained and some
exact solutions to the resulting equations are described.Comment: 19 pages, 2 fig
Interaction effects on magnetooscillations in a two-dimensional electron gas
Motivated by recent experiments, we study the interaction corrections to the
damping of magnetooscillations in a two-dimensional electron gas (2DEG). We
identify leading contributions to the interaction-induced damping which are
induced by corrections to the effective mass and quantum scattering time. The
damping factor is calculated for Coulomb and short-range interaction in the
whole range of temperatures, from the ballistic to the diffusive regime. It is
shown that the dominant effect is that of the renormalization of the effective
electron mass due to the interplay of the interaction and impurity scattering.
The results are relevant to the analysis of experiments on magnetooscillations
(in particular, for extracting the value of the effective mass) and are
expected to be useful for understanding the physics of a high-mobility 2DEG
near the apparent metal-insulator transition.Comment: 24 pages; subsection adde
Overscreening Diamagnetism in Cylindrical Superconductor-Normal Metal-Heterostructures
We study the linear diamagnetic response of a superconducting cylinder coated
by a normal-metal layer due to the proximity effect using the clean limit
quasiclassical Eilenberger equations. We compare the results for the
susceptibility with those for a planar geometry. Interestingly, for
the cylinder exhibits a stronger overscreening of the magnetic field, i.e., at
the interface to the superconductor it can be less than (-1/2) of the applied
field. Even for , the diamagnetism can be increased as compared to the
planar case, viz. the magnetic susceptibility becomes smaller than
-3/4. This behaviour can be explained by an intriguing spatial oscillation of
the magnetic field in the normal layer
Electron transport through interacting quantum dots
We present a detailed theoretical investigation of the effect of Coulomb
interactions on electron transport through quantum dots and double barrier
structures connected to a voltage source via an arbitrary linear impedance.
Combining real time path integral techniques with the scattering matrix
approach we derive the effective action and evaluate the current-voltage
characteristics of quantum dots at sufficiently large conductances. Our
analysis reveals a reach variety of different regimes which we specify in
details for the case of chaotic quantum dots. At sufficiently low energies the
interaction correction to the current depends logarithmically on temperature
and voltage. We identify two different logarithmic regimes with the crossover
between them occurring at energies of order of the inverse dwell time of
electrons in the dot. We also analyze the frequency-dependent shot noise in
chaotic quantum dots and elucidate its direct relation to interaction effects
in mesoscopic electron transport.Comment: 21 pages, 4 figures. References added, discussion slightly extende
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