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Register a new userExistence of traveling wave solutions for a nonlocal bistable equation: an abstract approach
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Authors
Hiroki Yagisita
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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.
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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.
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 an epidemic model with direct transmission given by a system of nonlinear partial differential equations and study the existence of traveling wave solutions. When the basic reproductive number of the considered model is less than one, we show that there is no nontrivial traveling wave solution. On the other hand, when the basic reproductive number is greater than one, we prove that there is a minimum wave speed $c^*$ such that the system has a traveling wave solution with speed $c$ connecting both equilibrium points for any $cge c^*$. Moreover, under suitable assumption on the diffusion rates, we show that there is no traveling wave solution with speed less than $c^*$. We conclude with numerical simulations to illustrate our findings. The numerical experiments supports the validity of our theoretical results.
Several stains of the intracellular parasitic bacterium Wolbachia limit severely the competence of the mosquitoes Aedes aegypti as a vector of dengue fever and possibly other arboviroses. For this reason, the release of mosquitoes infected by this bacterium in natural populations is presently considered a promising tool in the control of these diseases. Following works by M. Turelli [4] and subsequently M. Strugarek et al. [21, 22], we consider a simple scalar reaction-diffusion model describing the evolution of the proportion of infected mosquitoes, sufficient to reveal the bistable nature of the Wolbachia dynamics. A simple distributed feedback law is proposed, whose application on a compact domain during finite time is shown to be sufficient to invade the whole space. The corresponding stabilization result is established for any space dimension.
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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