ISSN:

1937-1632

eISSN:

1937-1179

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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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**Abstract:**

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**Abstract:**

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

where

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**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.

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**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.

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**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.

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**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.

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**Abstract:**

A nonautonomous lattice system with discrete Laplacian operator is revisited in the weighted space of infinite sequences

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**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.

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**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.

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**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 (

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**Abstract:**

The current paper is concerned with the spreading speeds of the following parabolic-parabolic chemotaxis model with logistic source on

where

and

where

Hence, if

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**Abstract:**

We consider a parabolic-elliptic system of partial differential equations with a chemotactic term in a

for a positive constant

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.

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**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

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**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.

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**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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