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Register a new userExistence of traveling waves for a nonlocal monostable equation: an abstract approach
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Authors
Hiroki Yagisita
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We consider a nonlocal analogue of the Fisher-KPP equation. We do not assume that the Borel-measure for the convolution is absolutely continuous. In order to show the main result, we modify a recursive method for abstract monotone discrete dynamical systems by Weinberger. We note that the monotone semiflow generated by the equation does not have compactness with respect to the compact-open topology. At the end, we propose a discrete model that describes the measurement process.
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We consider the nonlocal analogue of the Fisher-KPP equation. We do not assume that the Borel-measure is absolutely continuous with respect to the Lebesgue measure. We gives a sufficient condition for existence of traveling waves, and a necessary condition for existence of periodic traveling waves.
We consider traveling fronts to the nonlocal bistable equation. We do not assume that the Borel-measure is absolutely continuous with respect to the Lebesgue measure. We show that there is a traveling wave solution with monotone profile. In order to prove this result, we would develop a recursive method for abstract monotone dynamical systems and apply it to the equation.
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.
We study traveling wave solutions of the nonlinear variational wave equation. In particular, we show how to obtain global, bounded, weak traveling wave solutions from local, classical ones. The resulting waves consist of monotone and constant segments, glued together at points where at least one one-sided derivative is unbounded. Applying the method of proof to the Camassa--Holm equation, we recover some well-known results on its traveling wave solutions.
Let $Omegasubsetmathbb{R}^n$ be a $C^2$ bounded domain and $chi>0$ be a constant. We will prove the existence of constants $lambda_Ngelambda_N^{ast}gelambda^{ast}(1+chiint_{Omega}frac{dx}{1-w_{ast}})^2$ for the nonlocal MEMS equation $-Delta v=lam/(1-v)^2(1+chiint_{Omega}1/(1-v)dx)^2$ in $Omega$, $v=0$ on $1Omega$, such that a solution exists for any $0lelambda<lambda_N^{ast}$ and no solution exists for any $lambda>lambda_N$ where $lambda^{ast}$ is the pull-in voltage and $w_{ast}$ is the limit of the minimal solution of $-Delta v=lam/(1-v)^2$ in $Omega$ with $v=0$ on $1Omega$ as $lambda earrow lambda^{ast}$. We will prove the existence, uniqueness and asymptotic behaviour of the global solution of the corresponding parabolic nonlocal MEMS equation under various boundedness conditions on $lambda$. We also obtain the quenching behaviour of the solution of the parabolic nonlocal MEMS equation when $lambda$ is large.
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