Quasilinear Reaction-diffusion Systems Research Articles

On quasi-linear reaction diffusion systems arising from compartmental SEIR models

The global existence and boundedness of solutions to quasi-linear reaction-diffusion systems are investigated. The system arises from compartmental models describing the spread of infectious diseases proposed in Viguerie et al. (Appl Math Lett 111:106617, 2021); Viguerie et al. (Comput Mech 66(5):1131–1152, 2020), where the diffusion rate is assumed to depend on the total population, leading to quasilinear diffusion with possible degeneracy. The mathematical analysis of this model has been addressed recently in Auricchio et al. (Math Methods Appl Sci 46:12529–12548, 2023) where it was essentially assumed that all sub-populations diffuse at the same rate, which yields a positive lower bound of the total population, thus removing the degeneracy. In this work, we remove this assumption completely and show the global existence and boundedness of solutions by exploiting a recently developed Lp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^p$$\\end{document}-energy method. Our approach is applicable to a larger class of systems and is sufficiently robust to allow model variants and different boundary conditions.

A multiscale quasilinear system for colloids deposition in porous media: Weak solvability and numerical simulation of a near-clogging scenario

We study the weak solvability of a macroscopic, quasilinear reaction–diffusion system posed in a 2D porous medium which undergoes microstructural problems. The solid matrix of this porous medium is assumed to be made out of circles of not-necessarily uniform radius. The growth or shrinkage of these circles, which are governed by an ODE, has direct feedback to the macroscopic diffusivity via an additional elliptic cell problem.The reaction–diffusion system describes the macroscopic diffusion, aggregation, and deposition of populations of colloidal particles of various sizes inside a porous media made of prescribed arrangement of balls. The mathematical analysis of this two-scale problem relies on a suitable application of Schauder’s fixed point theorem which also provides a convergent algorithm for an iteration method to compute finite difference approximations of smooth solutions to our multiscale model. Numerical simulations illustrate the behavior of the local concentration of the colloidal populations close to clogging situations.

Quasilinear reaction diffusion systems with mass dissipation

<abstract><p>We study quasilinear reaction diffusion systems relative to the Shigesada-Kawasaki-Teramoto model. Nonlinearity standing for the external force is provided with mass dissipation. Estimate in several norms of the solution is provided under the restriction of diffusion coefficients, growth rate of reaction, and space dimension.</p></abstract>

Solvability of a coupled quasilinear reaction–diffusion system

The aim of this paper is to investigate the existence, uniqueness, and long-time behaviour of the classical solutions to a coupled system of four quasilinear parabolic equations. Two of them are solved in a domain and the other two are determined on the domain boundary. Such coupled systems arise in modelling of surface reactions between two reactants.

Wavefront solutions of degenerate quasilinear reaction–diffusion systems with mixed quasi-monotonicity

This paper is concerned with the existence of wavefront solutions to a system of degenerate quasilinear reaction–diffusion equations of mixed quasi-monotone properties in the form ∂ui∕∂t=∇⋅Diui∇ui+fiu−∞<x<∞,t>0for i=1,…,n. The important features of this system are that some of the diffusion coefficients Diui are density dependent and may vanish at certain value of ui, and that each function fi is quasi-monotone increasing for some components of u =u1,…,un and decreasing for other components of u. Such systems model reaction–diffusion processes with density driven diffusion mechanism. Under certain general conditions we prove the existence of a traveling wave solution that is between a pair of coupled upper and lower solutions. A predator–prey model with nonlinear diffusion is used as an illustration of application. The presence of wavefront solutions flowing toward the coexistence states is established by constructing appropriate upper and lower solutions.

Wavefront solutions of quasilinear reaction–diffusion systems with mixed quasi-monotonicity

ABSTRACTWe study the existence of wavefront solutions in a general class of mixed quasi-monotone reaction–diffusion systems with nonlinear diffusions in the form for . Each function is quasi-monotone increasing for some components of and decreasing for other components of . Such systems model reaction–diffusion processes with a density driven diffusion mechanism. Under certain general conditions we prove the existence of a traveling wave solution that is between a pair of coupled upper and lower solutions. As examples, we discuss a predator–prey model and a multi-species competition model, both with nonlinear diffusions. In both models, the presences of wavefront solutions flowing towards the coexistence states are established as illustrations for applications of our main result, with numerical simulations.

Dynamics of degenerate quasilinear reaction diffusion systems with nonnegative initial functions

This paper is concerned with a system of quasilinear reaction–diffusion equations with density dependent diffusion coefficients and mixed quasimonotone reaction functions. The equations are allowed to be degenerate and the boundary conditions are of the nonlinear type. The main goals are to prove the existence and uniqueness of the weak solution between a pair of coupled upper and lower solutions; show that the weak solution evolves into the classical solution, and analyze the asymptotic behavior of the solution using quasi-solutions of the steady-state system. The general results are applied to a degenerate Lotka–Volterra competition model. Conditions are given for the solution to exist globally, to evolve into the classical solution, and to be attracted into a sector formed by quasi-solutions of the elliptic system. Especially for the Neumann problem we give a simple condition for the solution to converge to a unique constant steady-state solution which is a global attractor.

A free boundary problem arising in the ecological models with N-species

This paper is concerned with the one-dimensional free boundary problem for quasilinear reaction-diffusion systems arising in the ecological models with N-species, where some of the species are made up of two separated groups and the mankind’s influence is taken into account. In the problem under consideration, there are n free boundaries, the coefficients of the equations are allowed to be discontinuous on the free boundaries and the reaction functions are mixed quasimonotone. The aim is to show the local existence of the solutions for the free boundary problem by the fixed point method, and the global existence and uniqueness of the solutions for the corresponding diffraction problem by the approximation and estimate methods.

Dynamics of Lotka–Volterra competition reaction–diffusion systems with degenerate diffusion

This paper is dealt with a class of Lotka–Volterra competition reaction–diffusion system and a competitor–competitor–mutualist system with density-dependent diffusion in a bounded domain under the three basic types of Dirichlet, Neumann and Robin boundary conditions. The governing equations for the competition system consist of an arbitrary number of degenerate quasilinear parabolic equations while the competitor–competitor–mutualist system involves three degenerate equations. The goal of the paper is to show: the existence of positive steady-state solutions or quasi-solutions, the existence and uniqueness of a classical global time-dependent solution, and the asymptotic behavior of the time-dependent solution in relation to the steady-state solutions or quasi-solutions. The above goals are achieved under a very simple condition on the reaction rates of the reaction functions and these results yield a global attractor and the coexistence of the competing species. In the case of Neumann boundary condition the system has a unique constant positive steady-state solution which is a global attractor of all the competing species. The above conclusions lead to some interesting distinct dynamic behavior between degenerate quasilinear reaction–diffusion systems and the corresponding semilinear reaction–diffusion systems in which some or all of the competing species may be in extinction.

Existence and Uniqueness of Weak Solution for a Class of Elliptic Reaction-Diffusion Systems

Abstract We present a simple proof of the existence and uniqueness of a weak solution for a class of quasilinear elliptic reaction-diffusion systems. The proof is based on an iterative process and on some a priori estimates.

Concentration Phenomena in a Nonlocal Quasi-linear Problem Modelling Phytoplankton I: Existence

We study the positive steady state of a quasi-linear reaction-diffusion system in one space dimension introduced by Klausmeier and Litchman for the modelling of the distributions of phytoplankton biomass and its nutrient. The system has nonlocal dependence on the biomass function, and it has a biomass-dependent drifting term describing the active movement of the biomass towards the location of the optimal growth condition. We obtain complete descriptions of the profile of the solutions when the coefficient of the drifting term is large, rigorously proving the numerically observed phenomenon of concentration of biomass for this model. Our theoretical results reveal four critical numbers for the model not observed before and offer several further insights into the problem being modelled. This is Part I of a two-part series, where we obtain nearly optimal existence and nonexistence results. The asymptotic profile of the solutions is studied in the separate Part II.

Concentration Phenomena in a Nonlocal Quasi-linear Problem Modelling Phytoplankton II: Limiting Profile

This is Part II of our study on the positive steady state of a quasi-linear reaction-diffusion system in one space dimension introduced by Klausmeier and Litchman for the modelling of the distributions of phytoplankton biomass and its nutrient. In Part I, we proved nearly optimal existence and nonexistence results. In Part II, we obtain complete descriptions of the profile of the solutions when the coefficient of the drifting term is large, rigorously proving the numerically observed phenomenon of concentration of biomass for this model. Moreover, we reveal four critical numbers for the model and provide further insights to the problem being modelled.

A remark on a nonlinear system of reaction-diffusion equations

This study concerns the existence of positive weak solutions for a quasilinear reaction-diffusion system with Dirichlet boundary conditions,

A quasilinear reaction-diffusion system coupled via nonlocal sources

In this paper, we establish the critical exponent and the blow-up rate for a quasilinear reaction-diffusion system with nonlocal nonlinear sources and inner absorptions, subject to homogeneous Dirichlet conditions and nonnegative initial data. It is found that the critical exponent is determined by the interaction among all the six nonlinear exponents from all the three kinds of the nonlinearities. While the blow-up rate is independent of the nonlinear diffusion exponents due to the effects of the coupled nonlocal sources. Two kinds of characteristic algebraic systems are introduced to get simple descriptions for the critical exponent and the blow-up rate, respectively. We prove moreover that the blow-up could be global due to the nonlocality of the nonlinear sources.

Diffraction problems for quasilinear reaction–diffusion systems

The aim of this paper is to investigate existence, uniqueness and qualitative properties of the weak and piecewise classical solutions of the diffraction problem for weakly coupled reaction–diffusion systems. Monotony and stability with respect to the time variable of the weak solutions is proved. As an application is considered a model reaction–diffusion system describing an enzyme sensor for concentration of phenol in the chemical industry.

Blow‐up estimates for a quasi‐linear reaction–diffusion system

AbstractIn this paper, some sufficient conditions under which the quasilinear elliptic system ‐div(∣∇u∣p‐2∇u) = uv, ‐div(∣∇u∣q‐2∇u) = uv in ℝN(N≥3) has no radially symmetric positive solution is derived. Then by using this non‐existence result, blow‐up estimates for a class of quasilinear reaction–diffusion systems ut = div (∣∇u∣p‐2∇u)+uv,vt = div(∣∇v∣q‐2∇v) +uv with the homogeneous Dirichlet boundary value conditions are obtained. Copyright © 2003 John Wiley &amp; Sons, Ltd.

Nonexistence of positive solutions to a quasilinear elliptic system and blow-up estimates for a non-Newtonian filtration system

The prior estimate and decay property of positive solutions are derived for a system of quasilinear elliptic differential equations first. Then the result of nonexistence for a differential equation system of radially nonincreasing positive solutions is implied. By using this nonexistence result, blow-up estimates for a class of quasilinear reaction-diffusion systems (non-Newtonian filtration systems) are established to extend the result of semilinear reaction-diffusion (Fujita type) systems.

Nonexistence of positive solutions to a quasilinear elliptic system and blow-up estimates for a quasilinear reaction–diffusion system

The prior estimate and decay property of positive solutions are derived for a system of quasilinear elliptic differential equations first. Then, the nonexistence result for radially nonincreasing positive solutions of the system is implied. By using this nonexistence result, blow-up estimates for a class of quasilinear reaction–diffusion systems (non-Newtonian filtration systems) are established to extend the result for semilinear reaction–diffusion systems (Fujita type).

Non-existence of positive solution to a quasilinear elliptic system and blow-up estimates for a reaction-diffusion system

The non-existence result for radially nonincreasing positive solutions of the system is implied. By using this non-existence result, blow-up estimates for a class of quasilinear reaction-diffusion systems (non-Newtonian filtration systems) are established to extend the result for semilinear reaction-diffusion systems (Fujita type).

Blow-up estimates for a non-newtonian filtration system

The prior estimate and decay property of positive solutions are derived for a system of quasi-linear elliptic differential equations first. Hence, the result of non-existence for differential equation system of radially nonincreasing positive solutions is implied. By using this non-existence result, blow-up estimates for a class quasi-linear reaction-diffusion systems (non-Newtonian filtration systems) are established, which extends the result of semi-linear reaction-diffusion (Fujita type) systems.