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Register a new userThe uniform spreading speed in cooperative systems with non-uniform initial data
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This paper considers the spreading speed of cooperative nonlocal dispersal system with irreducible reaction functions and non-uniform initial data. Here the non-uniformity means that all components of initial data decay exponentially but their decay rates are different. It is well-known that in a monostable reaction-diffusion or nonlocal dispersal equation, different decay rates of initial data yield different spreading speeds. In this paper, we show that due to the cooperation and irreducibility of reaction functions, all components of the solution with non-uniform initial data will possess a uniform spreading speed which non-increasingly depends only on the smallest decay rate of initial data. The nonincreasing property of the uniform spreading speed further implies that the component with the smallest decay rate can accelerate the spatial propagation of other components. In addition, all the methods in this paper can be carried over to the cooperative system with classical diffusion (i.e. random diffusion).
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In this paper, we first establish the local well-posedness (existence, uniqueness and continuous dependence) for the Fornberg-Whitham equation in both supercritical Besov spaces $B^s_{p,r}, s>1+frac{1}{p}, 1leq p,rleq+infty$ and critical Besov spaces $B^{1+frac{1}{p}}_{p,1}, 1leq p<+infty$, which improves the previous work cite{y2,ho,ht}. Then, we prove the solution is not uniformly continuous dependence on the initial data in supercritical Besov spaces $B^s_{p,r}, s>1+frac{1}{p}, 1leq pleq+infty, 1leq r<+infty$ and critical Besov spaces $B^{1+frac{1}{p}}_{p,1}, 1leq p<+infty$. At last, we show that the solution is ill-posed in $B^{sigma}_{p,infty}$ with $sigma>3+frac{1}{p}, 1leq pleq+infty$.
We consider a class of cooperative reaction-diffusion systems with free boundaries in one space dimension, where the diffusion terms are nonlocal, given by integral operators involving suitable kernel functions, and they are allowed not to appear in some of the equations in the system. The problem is monostable in nature, resembling the well known Fisher-KPP equation. Such a system covers various models arising from mathematical biology, with the Fisher-KPP equation as the simplest special case, where a spreading-vanishing dichotomy is known to govern the long time dynamical behaviour. The question of spreading speed is widely open for such systems except for the scalar case. In this paper, we develop a systematic approach to determine the spreading profile of the system, and obtain threshold conditions on the kernel functions which decide exactly when the spreading has finite speed, or infinite speed (accelerated spreading). This relies on a rather complete understanding of both the associated semi-waves and traveling waves. When the spreading speed is finite, we show that the speed is determined by a particular semi-wave, and obtain sharp estimates of the semi-wave profile and the spreading speed. For kernel functions that behave like $|x|^{-gamma}$ near infinity, we are able to obtain better estimates of the spreading speed for both the finite speed case, and the infinite speed case, which appear to be the first for this kind of free boundary problems, even for the special Fisher-KPP equation.
The aim of this paper is to study, in dimensions 2 and 3, the pure-power non-linear Schrodinger equation with an external uniform magnetic field included. In particular, we derive a general criteria on the initial data and the power of the non-linearity so that the corresponding solution blows up in finite time, and we show that the time for blow up to occur decreases as the strength of the magnetic field increases. In addition, we also discuss some observations about Strichartz estimates in 2 dimensions for the Mehler kernel, as well as similar blow-up results for the non-linear Pauli equation.
We give a comprehensive study of strong uniform attractors of non-autonomous dissipative systems for the case where the external forces are not translation compact. We introduce several new classes of external forces which are not translation compact, but nevertheless allow to verify the attraction in a strong topology of the phase space and discuss in a more detailed way the class of so-called normal external forces introduced before. We also develop a unified approach to verify the asymptotic compactness for such systems based on the energy method and apply it to a number of equations of mathematical physics including the Navier-Stokes equations, damped wave equations and reaction-diffusing equations in unbounded domains.
Many results in modern astrophysics rest on the notion that the Initial Mass Function (IMF) is universal. Our observations of HI selected galaxies in the light of H-alpha and the far-ultraviolet (FUV) challenge this notion. The flux ratio H-alpha/FUV from these two star formation tracers shows strong correlations with the surface-brightness in H-alpha and the R band: Low Surface Brightness (LSB) galaxies have lower ratios compared to High Surface Brightness galaxies and to expectations from equilibrium star formation models using commonly favored IMF parameters. Weaker but significant correlations of H-alpha/FUV with luminosity, rotational velocity and dynamical mass are found as well as a systematic trend with morphology. The correlated variations of H-alpha/FUV with other global parameters are thus part of the larger family of galaxy scaling relations. The H-alpha/FUV correlations can not be due to dust correction errors, while systematic variations in the star formation history can not explain the trends with both H-alpha and R surface brightness. LSB galaxies are unlikely to have a higher escape fraction of ionizing photons considering their high gas fraction, and color-magnitude diagrams. The most plausible explanation for the correlations are systematic variations of the upper mass limit and/or slope of the IMF at the upper end. We outline a scenario of pressure driving the correlations by setting the efficiency of the formation of the dense star clusters where the highest mass stars form. Our results imply that the star formation rate measured in a galaxy is highly sensitive to the tracer used in the measurement. A non-universal IMF also calls into question the interpretation of metal abundance patterns in dwarf galaxies and star formation histories derived from color magnitude diagrams. Abridged.
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