Quasilinear Reaction Diffusion Research Articles

Semilinear approximations of quasilinear parabolic equations with applications

An approach to solvability of certain quasilinear parabolic equations is presented by approximating the quasilinear equation under consideration with a parameter family of semilinear problems with stronger linear fractional diffusion term. Defined on arbitrarily long time intervals, solutions to the original problem are found as a suitable limit of global solutions to those semilinear approximations. The method is applied to nonlinear parabolic Kirchhoff equation, quasilinear reaction–diffusion equation, and critical 2D surface quasi‐geostrophic equation associated with Dirichlet boundary conditions.

Eternal solutions in exponential self-similar form for a quasilinear reaction-diffusion equation with critical singular potential

Eternal solutions in exponential self-similar form for a quasilinear reaction-diffusion equation with critical singular potential

Existence and Stability of Solutions with Internal Transition Layer for the Reaction–Diffusion–Advection Equation with a KPZ-Nonlinearity

We study a boundary value problem for a quasilinear reaction–diffusion–advection ordinary differential equation with a KPZ-nonlinearity containing the squared gradient of the unknown function. The noncritical and critical cases of existence of an internal transition layer are considered. An asymptotic approximation to the solution is constructed, and the asymptotics of the transition layer point is determined. Existence theorems are proved using the asymptotic method of differential inequalities, the Lyapunov asymptotic stability of solutions is proved by the narrowing barrier method, and instability theorems are proved with the use of unordered upper and lower solutions.

Qualitative properties of solutions to a reaction-diffusion equation with weighted strong reaction

We study the existence and qualitative properties of solutions to the Cauchy problem associated to the quasilinear reaction-diffusion equation $$ \partial_tu=\Delta u^m+(1+|x|)^{\sigma}u^p, $$ posed for \((x,t)\in\mathbb{R}^N\times(0,\infty)\), where \(m>1\), \(p\in(0,1)\) and \(\sigma>0\). Initial data are taken to be bounded, non-negative and compactly supported. In the range when \(m+p\geq2\), we prove existence of local solutions with a finite speed of propagation of their supports for compactly supported initial conditions. We also show in this case that, for a given compactly supported initial condition, there exist infinitely many solutions to the Cauchy problem, by prescribing the evolution of their interface. In the complementary range \(m+p<2\), we obtain new Aronson-Benilan estimates satisfied by solutions to the Cauchy problem, which are of independent interest as a priori bounds for the solutions. We apply these estimates to establish infinite speed of propagation of the supports of solutions if \(m+p<2\), that is, \(u(x,t)>0\) for any \(x\in\mathbb{R}^N\), \(t>0\), even in the case when the initial condition \(u_0\) is compactly supported. For more information see https://ejde.math.txstate.edu/Volumes/2023/72/abstr.html

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.

Nonstationary incompressible Navier-Stokes system governed by a quasilinear reaction-diffusion equation

In this paper, we consider a new class of a dynamic system which couplesa nonstationary incompressible Navier-Stokes equation involving nonmonotone and multivalued boundary 条件 of subdifferential type,and a generalized quasilinear reaction-diffusion equation with the Neumann boundary condition.The variational formulation of the system is provided in the form of an evolutionary hemivariational inequality coupled with the nonlinear parabolic equation. The main result concerning with the existence of global weak solutions to the system is proved.Firstly, the backward Euler technique and the feedback iterative method are used to develop a temporally semi-discrete approximate 问题. Next, invoking a surjectivity result for weakly-weakly upper semicontinuous multivalued operators, the existence and $a$ $priori$ estimates for the 解 to the semi-discrete approximate 问题 are established. Finally, using a limiting procedure for solutions to the approximate 问题, a 定理 on the existence of weak solutions to the system is obtained.

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>

Blow up profiles for a quasilinear reaction-diffusion equation with weighted reaction

We perform a thorough study of the blow up profiles associated to the following second order reaction-diffusion equation with non-homogeneous reaction:∂tu=∂xx(um)+|x|σup, in the range of exponents 1<p<m and σ>0. We classify blow up solutions in self-similar form, that are likely to represent typical blow up patterns for general solutions. We thus show that the non-homogeneous coefficient |x|σ has a strong influence on the qualitative aspects related to the finite time blow up. More precisely, for σ∼0, blow up profiles have similar behavior to the well-established profiles for the homogeneous case σ=0, and typically global blow up occurs, while for σ>0 sufficiently large, there exist blow up profiles for which blow up occurs only at space infinity, in strong contrast with the homogeneous case. This work is a part of a larger program of understanding the influence of unbounded weights on the blow up behavior for reaction-diffusion equations.

Regularity of stable solutions to quasilinear elliptic equations on Riemannian models

We investigate the regularity of semi-stable, radially symmetric, and decreasing solutions for a class of quasilinear reaction-diffusion equations in the inhomogeneous context of Riemannian manifolds. We prove uniform boundedness, Lebesgue and Sobolev estimates for this class of solutions for equations involving the p-Laplace Beltrami operator and locally Lipschitz non-linearity. We emphasize that our results do not depend on the boundary conditions and the specific form of the non-linearities and metric. Moreover, as an application, we establish regularity of the extremal solutions for equations involving the p-Laplace Beltrami operator with zero Dirichlet boundary conditions.

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.

Robust distributed control of quasilinear reaction–diffusion equations via infinite-dimensional sliding modes

This paper investigates the problem of robust tracking control for quasilinear reaction–diffusion partial differential equations subject to external unknown perturbations. The considered class of equations is quite general, and includes classical equations such as the heat equation or the Fisher–KPP equation as special cases. Global practical stabilization of the tracking error system is established under mild conditions on the disturbance term using a regularized infinite-dimensional sliding-mode controller. Extensive simulations support and validate the theoretical results.

Blow Up Profiles for a Quasilinear Reaction–Diffusion Equation with Weighted Reaction with Linear Growth

We study the blow up profiles associated to the following second order reaction-diffusion equation with non-homogeneous reaction: $$ \partial_tu=\partial_{xx}(u^m) + |x|^{\sigma}u, $$ with $\sigma>0$. Through this study, we show that the non-homogeneous coefficient $|x|^{\sigma}$ has a strong influence on the blow up behavior of the solutions. First of all, it follows that finite time blow up occurs for self-similar solutions $u$, a feature that does not appear in the well known autonomous case $\sigma=0$. Moreover, we show that there are three different types of blow up self-similar profiles, depending on whether the exponent $\sigma$ is closer to zero or not. We also find an explicit blow up profile. The results show in particular that \emph{global blow up} occurs when $\sigma>0$ is sufficiently small, while for $\sigma>0$ sufficiently large blow up \emph{occurs only at infinity}, and we give prototypes of these phenomena in form of self-similar solutions with precise behavior. This work is a part of a larger program of understanding the influence of non-homogeneous weights on the blow up sets and rates.

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.

Upper and lower bounds for the blow-up time in quasilinear reaction diffusion problems

In this paper, we consider a quasilinear reaction diffusion equation with Neumann boundary conditions in a bounded domain. Basing on Sobolev inequality and differential inequality technique, we obtain upper and lower bounds for the blow-up time of the solution. An example is also given to illustrate the abstract results obtained of this paper.

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.

Blow-up analysis in quasilinear reaction–diffusion problems with weighted nonlocal source

In this paper, we consider the blow-up of solutions to a class of quasilinear reaction–diffusion problems g(u)t=∇⋅ρ|∇u|2∇u+a(x)f(u) in Ω×(0,t∗),∂u∂ν+γu=0 on ∂Ω×(0,t∗),u(x,0)=u0(x) in Ω¯,where Ω is a bounded convex domain in Rn(n≥2), weighted nonlocal source satisfies a(x)f(u(x,t))≤a1+a2u(x,t)p∫Ωu(x,t)ldxm, and a1,a2,p,l, and m are positive constants. By utilizing a differential inequality technique and maximum principles, we establish conditions to guarantee that the solution remains global or blows up in a finite time. Moreover, an upper and a lower bound for blow-up time are derived. Furthermore, two examples are given to illustrate the applications of obtained results.

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.

Blow-up problems for quasilinear reaction diffusion equations with weighted nonlocal source

Blow-up problems for quasilinear reaction diffusion equations with weighted nonlocal source

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 cooperation systems governed by degenerate quasilinear reaction–diffusion equations

This paper deals with a class of Lotka–Volterra cooperation system where the densities of the cooperating species are governed by a finite number of degenerate reaction–diffusion equations. Three basic types of Dirichlet, Neumann, and Robin boundary conditions and two types of reaction functions, with and without saturation, are considered. The aim of the paper is to show the existence of positive minimal and maximal steady-state solutions, including the uniqueness of the positive solution, the existence and uniqueness of a global time-dependent solution, and the asymptotic behavior of the time-dependent solution in relation to the steady-state solutions. Some very simple conditions on the physical parameters for the above objectives are obtained. Also discussed is the finite-time blow up property of the time-dependent solution and the non-existence of positive steady-state solution for the system with Neumann boundary condition.