Quasilinear Reaction Diffusion Equations 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.

Existence of blow-up self-similar solutions for the supercritical quasilinear reaction-diffusion equation

Existence of blow-up self-similar solutions for the supercritical quasilinear reaction-diffusion equation

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

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

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.

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.

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.

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

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.

Blow-Up of Solutions for a Class of Reaction-Diffusion Equations with a Gradient Term under Nonlinear Boundary Condition

The blow-up of solutions for a class of quasilinear reaction-diffusion equations with a gradient term ut = div(a(u)b(x)▽u)+ f (x;u; |▽u|2; t) under nonlinear boundary condition ¶u=¶n+g(u) = 0 are studied. By constructing a new auxiliary function and using Hopf’s maximum principles, we obtain the existence theorems of blow-up solutions, upper bound of blow-up time, and upper estimates of blow-up rate. Our result indicates that the blow-up time T* may depend on a(u), while being independent of g(u) and f .

Global existence and blow-up solutions for quasilinear reaction–diffusion equations with a gradient term

In this work, we study the blow-up and global solutions for a quasilinear reaction–diffusion equation with a gradient term and nonlinear boundary condition: { ( g ( u ) ) t = Δ u + f ( x , u , | ∇ u | 2 , t ) in D × ( 0 , T ) , ∂ u ∂ n = r ( u ) on ∂ D × ( 0 , T ) , u ( x , 0 ) = u 0 ( x ) > 0 in D ¯ , where D ⊂ R N is a bounded domain with smooth boundary ∂ D . Through constructing suitable auxiliary functions and using maximum principles, the sufficient conditions for the existence of a blow-up solution, an upper bound for the “blow-up time”, an upper estimate of the “blow-up rate”, the sufficient conditions for the existence of the global solution, and an upper estimate of the global solution are specified under some appropriate assumptions on the nonlinear system functions f , g , r , and initial value u 0 .

Non-linear reaction–diffusion systems with variable diffusivities: Lie symmetries, ansätze and exact solutions

This work first considers the classical Lie symmetry analysis of a class of systems of two quasilinear reaction–diffusion equations having variable diffusivities. Subsequently, non-Lie reductions to systems of first order ordinary differential equations are obtained for a subclass of these systems. In particular, families of exact solutions of a diffusive Lotka–Volterra type system are constructed.

Multinonlinear interactions in quasi-linear reaction-diffusion equations with nonlinear boundary flux

This paper deals with multinonlinearity interactions for more general quasi-linear reaction-diffusion models, where three nonlinear mechanisms-nonlinear diffusion, nonlinear reaction (or absorption), and nonlinear boundary flux-are included. The absorption/ boundary flux balance is studied to get the global solvability conditions as well as the blow-up criterion for the model with boundary flux and absorption. Specifically, the global solvability conditions established in this paper for the model with boundary flux and reaction are both necessary and sufficient.

Blowup in diffusion equations: A survey

This paper deals with quasilinear reaction-diffusion equations for which a solution local in time exists. If the solution ceases to exist for some finite time, we say that it blows up. In contrast to linear equations blowup can occur even if the data are smooth and well-defined for all times. Depending on the equation either the solution or some of its derivatives become singular. We shall concentrate on those cases where the solution becomes unbounded in finite time. This can occur in quasilinear equations if the heat source is strong enough. There exist many theoretical studies on the question on the occurrence of blowup. In this paper we shall recount some of the most interesting criteria and most important methods for analyzing blowup. The asymptotic behavior of solutions near their singularities is only completely understood in the special case where the source is a power. A better knowledge would be useful also for their numerical treatment. Thus, not surprisingly, the numerical analysis of this type of problems is still at a rather early stage. The goal of this paper is to collect some of the known results and algorithms and to direct the attention to some open problems.

Invariant manifolds and bifurcation for quasilinear reaction-diffusion systems

IN THIS PAPER we consider parameter-dependent quasilinear reaction-diffusion equations on a bounded domain of R”. It is our purpose to study the dynamical behavior of solutions to these equations. In particular, we are interested in the behavior of the solution set if the parameter is changed. To be more precise, let us assume that Sz C R” is a bounded domain having a smooth boundary. Then we consider the parameter-dependent equation