DCDS
Spreading speeds and traveling waves of a parabolic-elliptic chemotaxis system with logistic source on $\mathbb{R}^N$
Rachidi B. Salako Wenxian Shen
The current paper is devoted to the study of spreading speeds and traveling wave solutions of the following parabolic-elliptic chemotaxis system,
$\label{IntroEq0-2}\begin{cases}u_{t}=Δ{u}-χ\nabla·(u\nabla{v})+u(1-u),{x}∈\mathbb{R}^N,\\{0}=Δ{v}-v+u,{x}∈\mathbb{R}^N,\end{cases}$
where $u(x, t)$ represents the population density of a mobile species and $v(x, t)$ represents the population density of a chemoattractant, and $χ$ represents the chemotaxis sensitivity. We first give a detailed study in the case $N=1$. In this case, it has been shown in an earlier work by the authors of the current paper that, when $0 < χ < 1$, for every nonnegative uniformly continuous and bounded function $u_0(x)$, the system has a unique globally bounded classical solution $(u(x, t;u_0), v(x, t;u_0))$ with initial condition $u(x, 0;u_0)=u_0(x)$. Furthermore, it was shown that, if $0 < χ < \frac{1}{2}$, then the constant steady-state solution $(1, 1)$ is asymptotically stable with respect to strictly positive perturbations. In the current paper, we show that if $0 < χ < 1$, then there are nonnegative constants $c_{-}^*(χ)≤q c_+^*(χ)$ such that for every nonnegative initial function $u_0(·)$ with non-empty and compact support ${\rm{supp}}(u_0)$,
$\mathop {\lim }\limits_{t \to \infty } \mathop {\sup }\limits_{|x| \le ct} [|u(x,t;{u_0}) - 1| + |v(x,t;{u_0}) - 1|] = 0\quad \forall {\mkern 1mu} {\mkern 1mu} 0 < c < c_ - ^*(\chi )$
and
$\mathop {\lim }\limits_{t \to \infty } \mathop {\sup }\limits_{|x| \le ct} [u(x,t;{u_0}) + v(x,t;{u_0})] = 0\quad \forall {\mkern 1mu} {\mkern 1mu} c > c_ + ^*(\chi ).$
We also show that if $0 < χ < \frac{1}{2}$, there is a positive constant $c^*(χ)$ such that for every $c≥q{c}^*(χ)$, the system has a traveling wave solution $(u(x, t), v(x, t))$ with speed $c$ and connecting $(1, 1)$ and $(0, 0)$, that is, $(u(x, t), v(x, t))=(U(x-ct), V(x-ct))$ for some functions $U(·)$ and $V(·)$ satisfying $(U(-∞), V(-∞))=(1, 1)$ and $(U(∞), V(∞))=(0, 0)$. Moreover, we show that
$\mathop {\lim }\limits_{\chi \to 0} {c^*}(\chi ) = \mathop {\lim }\limits_{\chi \to 0} c_ + ^*(\chi ) = \mathop {\lim }\limits_{\chi \to 0} c_ - ^*(\chi ) = 2.$
We then consider the extensions of the results in the case $N=1$ to the case $N≥q2$.
keywords: Parabolic-elliptic chemotaxis system logistic source comparison principles spreading speed traveling wave solution
DCDS
Two species competition with an inhibitor involved
Georg Hetzer Wenxian Shen
The dynamics of the solution flow of a two-species Lotka-Volterra competition model with an extra equation for simple inhibitor dynamics is investigated. The model fits into the abstract framework of two-species competition systems (or $K$-monotone systems), but the equilibrium representing the extinction of both species is not a repeller. This feature distinguishes our problem from the case of classical two-species competition without inhibitor (classical case for short), where a basic assumption requires that equilibrium to be a repeller. Nevertheless, several results similar to those in the classical case, such as competitive exclusion and the existence of a "thin" separatrix, are obtained, but differently from the classical case, coexistence of the two species or extinction of one of them may depend on the initial conditions. As in almost all two species competition models, the strong monotonicity of the flow (with respect to a certain order on $\mathbb R^3$) is a key ingredient for establishing the main results of the paper.
keywords: convergence results competitive exclusion strong monotonicity Lotka-Volterra two-species competition inhibitor Poincaré-Bendixson Theorem.
CPAA
The Faber--Krahn inequality for random/nonautonomous parabolic equations
Janusz Mierczyński Wenxian Shen
The paper extends the Faber--Krahn inequality for elliptic and periodic parabolic problems to random and general nonautonomous parabolic problems. Under proper assumptions, it also provides necessary and sufficient conditions for the Faber-Krahn inequality being equality.
keywords: Schwarz symmetrization The Faber--Krahn inequality principal Lyapunov exponent recurrent function. principal spectrum Schwarz symmetrized domain
CPAA
Transition fronts in nonlocal Fisher-KPP equations in time heterogeneous media
Wenxian Shen Zhongwei Shen
The present paper is devoted to the study of transition fronts of nonlocal Fisher-KPP equations in time heterogeneous media. We first construct transition fronts with exact decaying rates as the space variable tends to infinity and with prescribed interface location functions, which are natural generalizations of front location functions in homogeneous media. Then, by the general results on space regularity of transition fronts of nonlocal evolution equations proven in the authors' earlier work ([25]), these transition fronts are continuously differentiable in space. We show that their space partial derivatives have exact decaying rates as the space variable tends to infinity. Finally, we study the asymptotic stability of transition fronts. It is shown that transition fronts attract those solutions whose initial data decays as fast as transition fronts near infinity and essentially above zero near negative infinity.
keywords: transition front Nonlocal Fisher-KPP equation stability. regularity
DCDS-S
Formulas for generalized principal Lyapunov exponent for parabolic PDEs
Janusz Mierczyński Wenxian Shen
An integral formula is given representing the generalized principal Lyapunov exponent for random linear parabolic PDEs. As an application, an upper estimate of the exponent is obtained.
keywords: Dirichlet form. measurable linear skew-product semiflow Random linear parabolic partial differential equation generalized principal Lyapunov exponent
DCDS
Transition fronts in nonlocal equations with time heterogeneous ignition nonlinearity
Wenxian Shen Zhongwei Shen

The present paper is devoted to the study of transition fronts in nonlocal reaction-diffusion equations with time heterogeneous nonlinearity of ignition type. It is proven that such an equation admits space monotone transition fronts with finite speed and space regularity in the sense of uniform Lipschitz continuity. Our approach is first constructing a sequence of approximating front-like solutions and then proving that the approximating solutions converge to a transition front. We take advantage of the idea of modified interface location, which allows us to characterize the finite speed of approximating solutions in the absence of space regularity, and leads directly to uniform exponential decaying estimates.

keywords: Transition front nonlocal equation ignition nonlinearity
DCDS
Global attractor and rotation number of a class of nonlinear noisy oscillators
Wenxian Shen
The current paper is concerned with the global dynamics of a class of nonlinear oscillators driven by real or white noises, of which a typical example is a shunted Josephson junction exposed to some random medium. Applying random dynamical systems theory, it is shown that a driven oscillator in the class under consideration with a tempered real noise has a one-dimensional global random attractor provided that the damping is not too small. Moreover, restricted to the global attractor, the oscillator induces a random dynamical system on $S^1$. It is then shown that there is a rotation number associated to the oscillator which characterizes the speed at which the solutions of the oscillator move around the global attractor. The results extend the existing ones for time periodic and quasi-periodic Josephson junctions and can be applied to Josephson junctions driven by white noises.
keywords: tempered function global random attractor invariant measure ergodic theorem random fixed point real or white noise random dynamical system rotation number. Josephson junction
DCDS-B
Spectraltheory for nonlocal dispersal operators with time periodic indefinite weight functions and applications
Wenxian Shen Xiaoxia Xie

In this paper, we study the spectral theory for nonlocal dispersal operators with time periodic indefinite weight functions subject to Dirichlet type, Neumann type and spatial periodic type boundary conditions. We first obtain necessary and sufficient conditions for the existence of a unique positive principal spectrum point for such operators. We then investigate upper bounds of principal spectrum points and sufficient conditions for the principal spectrum points to be principal eigenvalues. Finally we discuss the applications to nonlinear mathematical models from biology.

keywords: Nonlocal dispersal operator time periodic weight function principal spectrum point principal eigenvalue KPP equations
DCDS-B
Time averaging for nonautonomous/random linear parabolic equations
Janusz Mierczyński Wenxian Shen
Linear nonautonomous/random parabolic partial differential equations are considered under the Dirichlet, Neumann or Robin boundary conditions, where both the zero order coefficients in the equation and the coefficients in the boundary conditions are allowed to depend on time. The theory of the principal spectrum/principal Lyapunov exponents is shown to apply to those equations. In the nonautonomous case, the main result states that the principal eigenvalue of any time-averaged equation is not larger than the supremum of the principal spectrum and that there is a time-averaged equation whose principal eigenvalue is not larger than the infimum of the principal spectrum. In the random case, the main result states that the principal eigenvalue of the time-averaged equation is not larger than the principal Lyapunov exponent.
keywords: averaging. Nonautonomous linear partial differential equation of parabolic type principal Lyapunov exponent principal spectrum random linear partial differential equation of parabolic type
DCDS
Pullback attractors for nonautonomous and random parabolic equations on non-smooth domains
Jianhua Huang Wenxian Shen
The current paper is devoted to the study of pullback attractors for general nonautonomous and random parabolic equations on non-smooth domains $D$. Mild solutions are considered for such equations. We first extend various fundamental properties for solutions of smooth parabolic equations on smooth domains to solutions of general parabolic equations on non-smooth domains, including continuous dependence on parameters, monotonicity, and compactness, which are of great importance in their own. Under certain dissipative conditions on the nonlinear terms, we prove that mild solutions with initial conditions in $L_q(D)$ exist globally for $q$ » $1$. We then show that pullback attractors for nonautonomous and random parabolic equations on non-smooth domains exist in $L_q(D)$ for $1$ « $q$ < $\infty$.
keywords: Non-smooth domains nonautonomous dynamical systems random dynamical systems top Lyapunov exponents. pullback global attractors absorbing sets random parabolic equations nonautonomous parabolic equations

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