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Discrete and Continuous Dynamical Systems - S

October 2022 , Volume 15 , Issue 10

Issue on recent advances in partial differential equations and dynamical systems: Dedicated to Georg Hetzer, on the occasion of his 75th birthday

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Preface
Xiaoying Han, Amnon J. Meir and Paul G. Schmidt
2022, 15(10): i-x doi: 10.3934/dcdss.2022141 +[Abstract](372) +[HTML](181) +[PDF](3338.19KB)
Abstract:
$ \Sigma $-shaped bifurcation curves for classes of elliptic systems
Ananta Acharya, R. Shivaji and Nalin Fonseka
2022, 15(10): 2795-2806 doi: 10.3934/dcdss.2022067 +[Abstract](422) +[HTML](121) +[PDF](663.47KB)
Abstract:

We study positive solutions to classes of steady state reaction diffusion systems of the form:

where \begin{document}$ \lambda>0 $\end{document} is a positive parameter, \begin{document}$ \Omega $\end{document} is a bounded domain in \begin{document}$ \mathbb{R}^N $\end{document}; \begin{document}$ N > 1 $\end{document} with smooth boundary \begin{document}$ \partial \Omega $\end{document} or \begin{document}$ \Omega = (0, 1) $\end{document}, \begin{document}$ \frac{\partial z}{\partial \eta} $\end{document} is the outward normal derivative of \begin{document}$ z $\end{document}. Here \begin{document}$ f, g \in C^2[0, r) \cap C[0, \infty) $\end{document} for some \begin{document}$ r>0 $\end{document}. Further, we assume that \begin{document}$ f $\end{document} and \begin{document}$ g $\end{document} are increasing functions such that \begin{document}$ f(0) = 0 = g(0) $\end{document}, \begin{document}$ f'(0) = g'(0) = 1 $\end{document}, \begin{document}$ f''(0)>0, g''(0)>0 $\end{document}, and \begin{document}$ \lim\limits_{s \rightarrow \infty} \frac{f(Mg(s))}{s} = 0 $\end{document} for all \begin{document}$ M>0 $\end{document}. Under certain additional assumptions on \begin{document}$ f $\end{document} and \begin{document}$ g $\end{document} we prove that the bifurcation diagram for positive solutions of this system is at least \begin{document}$ \Sigma- $\end{document}shaped. We also discuss an example where \begin{document}$ f $\end{document} is sublinear at \begin{document}$ \infty $\end{document} and \begin{document}$ g $\end{document} is superlinear at \begin{document}$ \infty $\end{document} which satisfy our hypotheses.

A backward SDE method for uncertainty quantification in deep learning
Richard Archibald, Feng Bao, Yanzhao Cao and He Zhang
2022, 15(10): 2807-2835 doi: 10.3934/dcdss.2022062 +[Abstract](528) +[HTML](130) +[PDF](1759.39KB)
Abstract:

We develop a backward stochastic differential equation based probabilistic machine learning method, which formulates a class of stochastic neural networks as a stochastic optimal control problem. An efficient stochastic gradient descent algorithm is introduced with the gradient computed through a backward stochastic differential equation. Convergence analysis for stochastic gradient descent optimization and numerical experiments for applications of stochastic neural networks are carried out to validate our methodology in both theory and performance.

Stochastic energy balance climate models with Legendre weighted diffusion and an additive cylindrical Wiener process forcing
Gregorio Díaz and Jesús Ildefonso Díaz
2022, 15(10): 2837-2870 doi: 10.3934/dcdss.2021165 +[Abstract](605) +[HTML](256) +[PDF](497.13KB)
Abstract:

We consider a class of one-dimensional nonlinear stochastic parabolic problems associated to Sellers and Budyko diffusive energy balance climate models with a Legendre weighted diffusion and an additive cylindrical Wiener processes forcing. Our results use in an important way that, under suitable assumptions on the Wiener processes, a suitable change of variables leads the problem to a pathwise random PDE, hence an essentially "deterministic" formulation depending on a random parameter. Two applications are also given: the stability of solutions when the Wiener process converges to zero and the asymptotic behaviour of solutions for large time.

Bounded positive solutions for diffusive logistic equations with unbounded distributed limitations
Jesús Ildefonso Díaz and Jesús Hernández
2022, 15(10): 2871-2887 doi: 10.3934/dcdss.2022018 +[Abstract](413) +[HTML](149) +[PDF](384.84KB)
Abstract:

We establish the existence of bounded very weak solutions to a large class of stationary diffusive logistic equations with weights by constructing suitable sub and supersolutions. This class of problems corresponds to the case in which the absorption term dominates over the forcing term. The case of simultaneous singular nonlinearities and singular weights is also considered. This shows that if limitations in the growth of a population are distributed and unbounded, but satisfy some mild integrability assumption in terms of the distance to the boundary, solutions can still be bounded. The results extend several papers in the literature.

A nonautonomous chemostat model for the growth of gut microbiome with varying nutrient
Brittni Hall, Xiaoying Han, Peter E. Kloeden and Hans-Werner van Wyk
2022, 15(10): 2889-2908 doi: 10.3934/dcdss.2022075 +[Abstract](410) +[HTML](149) +[PDF](901.18KB)
Abstract:

A mathematical model describing the growth of gut microbiome inside and on the wall of the gut is developed based on the chemostat model with wall growth. Both the concentration and flow rate of the nutrient input are time-dependent, which results in a system of non-autonomous differential equations. First the stability of each meaningful equilibrium is studied for the autonomous counterpart. Then the existence of pullback attractors and its detailed structures for the nonautonomous system are investigated using theory and techniques of nonautonomous dynamical systems. In particular, sufficient conditions under which the microbiome vanishes or persists are constructed. Numerical simulations are provided to illustrate the theoretical results.

Pullback and forward dynamics of nonautonomous Laplacian lattice systems on weighted spaces
Xiaoying Han and Peter E. Kloeden
2022, 15(10): 2909-2927 doi: 10.3934/dcdss.2021143 +[Abstract](645) +[HTML](214) +[PDF](446.18KB)
Abstract:

A nonautonomous lattice system with discrete Laplacian operator is revisited in the weighted space of infinite sequences \begin{document}$ {{\ell_{\rho}^2}} $\end{document}. First the existence of a pullback attractor in \begin{document}$ {{\ell_{\rho}^2}} $\end{document} is established by utilizing the dense inclusion of \begin{document}$ \ell^2 \subset {{\ell_{\rho}^2}} $\end{document}. Moreover, the pullback attractor is shown to consist of a singleton trajectory when the lattice system is uniformly strictly contracting. Then forward dynamics is investigated in terms of the existence of a nonempty compact forward omega limit set. A general class of weights for the sequence space are adopted, instead of particular types of weights often used in the literature. The analysis presented in this work is more direct compare with previous studies.

On a global climate model with non-monotone multivalued coalbedo
Arturo Hidalgo and Lourdes Tello
2022, 15(10): 2929-2943 doi: 10.3934/dcdss.2022093 +[Abstract](301) +[HTML](123) +[PDF](373.24KB)
Abstract:

We are concerned with a global energy balance climate model formulated through a parabolic equation whose space domain is a manifold which simulates the Earth surface. The climate energy balance model includes the effect of coalbedo as one of the mean temperature feedback. We extend some mathematical results proved for maximal monotone coalbedo to the case where the coalbedo has not a monotone dependency on temperature. Numerical approximation is performed by the Finite Volume Method which allows to obtain and compare numerical solutions with different values of the coalbedo.

Optimized Ventcel-Schwarz waveform relaxation and mixed hybrid finite element method for transport problems
Thi-Thao-Phuong Hoang
2022, 15(10): 2945-2964 doi: 10.3934/dcdss.2022060 +[Abstract](526) +[HTML](293) +[PDF](519.66KB)
Abstract:

This paper is concerned with the optimized Schwarz waveform relaxation method and Ventcel transmission conditions for the linear advection-diffusion equation. A mixed formulation is considered in which the flux variable represents both diffusive and advective flux, and Lagrange multipliers are introduced on the interfaces between nonoverlapping subdomains to handle tangential derivatives in the Ventcel conditions. A space-time interface problem is formulated and is solved iteratively. Each iteration involves the solution of time-dependent problems with Ventcel boundary conditions in the subdomains. The subdomain problems are discretized in space by a mixed hybrid finite element method based on the lowest-order Raviart-Thomas space and in time by the backward Euler method. The proposed algorithm is fully implicit and enables different time steps in the subdomains. Numerical results with discontinuous coefficients and various Peclét numbers validate the accuracy of the method with nonconforming time grids and confirm the improved convergence properties of Ventcel conditions over Robin conditions.

The stability with the general decay rate of the solution for stochastic functional Navier-Stokes equations
Tongtong Liang
2022, 15(10): 2965-2980 doi: 10.3934/dcdss.2022127 +[Abstract](270) +[HTML](80) +[PDF](324.43KB)
Abstract:

This paper is concerned with the general stability of the solution to a stochastic functional 2D Navier-Stokes equation driven by a multiplicative white noise when the viscosity coefficient is time varying. First we give some sufficient conditions ensuring the existence and uniqueness of global solutions. Then the general stability of the solution in the sense of p-th (\begin{document}$ p\geq2 $\end{document}) moment is established. From this fact we further prove that the null solution is almost surely stable with the general decay rate. The convergence in probability of the solution is also analyzed.

Spreading speeds of a parabolic-parabolic chemotaxis model with logistic source on $ \mathbb{R}^{N} $
Wenxian Shen and Shuwen Xue
2022, 15(10): 2981-3002 doi: 10.3934/dcdss.2022074 +[Abstract](460) +[HTML](167) +[PDF](447.56KB)
Abstract:

The current paper is concerned with the spreading speeds of the following parabolic-parabolic chemotaxis model with logistic source on \begin{document}$ {{\mathbb R}}^{N} $\end{document},

where \begin{document}$ \chi, \ a,\ b,\ \lambda,\ \mu $\end{document} are positive constants. Assume \begin{document}$ b>\frac{N\mu\chi}{4} $\end{document}. Among others, it is proved that \begin{document}$ 2\sqrt{a} $\end{document} is the spreading speed of the global classical solutions of (1) with nonempty compactly supported initial functions, that is,

and

where \begin{document}$ (u(x,t;u_0,v_0), v(x,t;u_0,v_0)) $\end{document} is the unique global classical solution of (1) with \begin{document}$ u(x,0;u_0,v_0) = u_0 $\end{document}, \begin{document}$ v(x,0;u_0,v_0) = v_0 $\end{document}, and \begin{document}$ {\rm supp}(u_0) $\end{document}, \begin{document}$ {\rm supp}(v_0) $\end{document} are nonempty and compact. It is well known that \begin{document}$ 2\sqrt{a} $\end{document} is the spreading speed of the following Fisher-KPP equation,

Hence, if \begin{document}$ b>\frac{N\mu\chi}{4} $\end{document}, the chemotaxis neither speeds up nor slows down the spatial spreading in the Fisher-KPP equation.

Radially symmetric solutions for a Keller-Segel system with flux limitation and nonlinear diffusion
J. Ignacio Tello
2022, 15(10): 3003-3023 doi: 10.3934/dcdss.2022045 +[Abstract](472) +[HTML](128) +[PDF](403.23KB)
Abstract:

We consider a parabolic-elliptic system of partial differential equations with a chemotactic term in a \begin{document}$ N $\end{document}-dimensional unit ball "\begin{document}$ B $\end{document}" describing the behavior of a biological species "\begin{document}$ u $\end{document}" and a chemical stimuli "\begin{document}$ v $\end{document}". The system presents a sub-linear dependence of "\begin{document}$ \nabla v $\end{document}" in the chemotactic coefficient and a nonlinear diffusive term. The evolution of \begin{document}$ u $\end{document} is described by the equation

for a positive constant \begin{document}$ \chi $\end{document}. The concentration of the chemical substance \begin{document}$ v $\end{document} satisfies the linear elliptic equation

We consider the radially symmetric case and we prove the local existence of weak solutions for the mass accumulation function under assumption

for radial and regular initial data. Additionally, if the constrain

is satisfied, the solution globally exists in time.

Random attractors for stochastic delay wave equations on $ \mathbb{R}^n $ with linear memory and nonlinear damping
Jingyu Wang, Yejuan Wang, Lin Yang and Tomás Caraballo
2022, 15(10): 3025-3057 doi: 10.3934/dcdss.2021141 +[Abstract](722) +[HTML](238) +[PDF](546.77KB)
Abstract:

A non-autonomous stochastic delay wave equation with linear memory and nonlinear damping driven by additive white noise is considered on the unbounded domain \begin{document}$ \mathbb{R}^n $\end{document}. We establish the existence and uniqueness of a random attractor \begin{document}$ \mathcal{A} $\end{document} that is compact in \begin{document}$ C{([-h, 0];H^1(\mathbb{R}^n))}\times C{([-h, 0];L^2(\mathbb{R}^n))}\times L_\mu^2(\mathbb{R}^+;H^1(\mathbb{R}^n)) $\end{document} with \begin{document}$ 1\leqslant n \leqslant 3 $\end{document}.

Asymptotic behavior of nonlocal partial differential equations with long time memory
Jiaohui Xu, Tomás Caraballo and José Valero
2022, 15(10): 3059-3078 doi: 10.3934/dcdss.2021140 +[Abstract](783) +[HTML](284) +[PDF](431.04KB)
Abstract:

In this paper, it is first addressed the well-posedness of weak solutions to a nonlocal partial differential equation with long time memory, which is carried out by exploiting the nowadays well-known technique used by Dafermos in the early 70's. Thanks to this Dafermos transformation, the original problem with memory is transformed into a non-delay one for which the standard theory of autonomous dynamical system can be applied. Thus, some results about the existence of global attractors for the transformed problem are {proved}. Particularly, when the initial values have higher regularity, the solutions of both problems (the original and the transformed ones) are equivalent. Nevertheless, the equivalence of global attractors for both problems is still unsolved due to the lack of enough regularity of solutions in the transformed problem. It is therefore proved the existence of global attractors of the transformed problem. Eventually, it is highlighted how to proceed to obtain meaningful results about the original problem, without performing any transformation, but working directly with the original delay problem.

Traveling wave solutions of periodic nonlocal Fisher-KPP equations with non-compact asymmetric kernel
Aijun Zhang
2022, 15(10): 3079-3095 doi: 10.3934/dcdss.2022061 +[Abstract](423) +[HTML](192) +[PDF](405.85KB)
Abstract:

This paper deals with front propagation for nonlocal monostable equations in spatially periodic habitats. In the authors' earlier works, assuming the existence of principal eigenvalue, it is shown that there are periodic traveling wave solutions to a spatially periodic nonlocal monostable equation with symmetric and compact kernel connecting its unique positive stationary solution and the trivial solution in every direction with all propagating speeds greater than the spreading speed in that direction. In this paper, first assuming the existence of principal eigenvalue, we extend the results to the case that the kernel is asymmetric and supported on a non-compact region. In addition, without the assumption of the existence of principal eigenvalue, we explore the existence of semicontinuous traveling wave solutions.

2021 Impact Factor: 1.865
5 Year Impact Factor: 1.622
2021 CiteScore: 3.6

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