Degenerate Reaction-diffusion Equations Research Articles

Numerical Investigation of Some Reductions for the Gatenby–Gawlinski Model

Two (consecutive) reductions of the complete Gatenby–Gawlinski model for cancer invasion are proposed in order to investigate the mathematical framework, mainly from a computational perspective. After a brief overview of the full model, we proceed by examining the case of a two-equations-based and one-equation-based reduction, both obtained by means of a quasi-steady-state assumption. We focus on invasion fronts, exploiting a numerical strategy based on a finite volume approximation, and perform corresponding computational simulations to study the sharpness/smoothness of the traveling waves. Then, we employ a space-averaged wave speed estimate—referred to as the LeVeque–Yee formula—to quantitatively approach the propagation phenomenon. Concerning the one-equation-based model, we propose a scalar degenerate reaction-diffusion equation, which proves to be effective in order to qualitatively recover the typical trends arising from the Gatenby–Gawlinski model. Finally, we carry out some numerical tests in a specific case where the analytical solution is available.

Interior Hölder continuity for singular-degenerate porous medium type equations with an application to a biofilm model

We show interior Hölder continuity for a class of quasi-linear degenerate reaction-diffusion equations. The diffusion coefficient in the equation has a porous medium type degeneracy and its primitive has a singularity. The reaction term is locally bounded except in zero. The class of equations we analyze is motivated by a model that describes the growth of biofilms. Our method is based on the original proof of interior Hölder continuity for the porous medium equation. We do not restrict ourselves to solutions that are limits in the weak topology of a sequence of approximate continuous solutions of regularized problems, which is a common assumption.

Long time behavior of a degenerate NPZ model with spatial heterogeneity

This work proposes a nutrient–phytoplankton–zooplankton (NPZ) model which is governed by a partially degenerate reaction–diffusion equation. Due to the noncompactness of solution map, the Kuratowski measure is introduced to prove the existence of the global attractor. By an innovative construction of Lyapunov functions, sufficient conditions are given to obtain the global attractivity of steady states of the system with spatial parameters.

Global solutions for bistable degenerate reaction–diffusion equation with time-delay and nonlocal effect

In this paper, we prove the existence, uniqueness and regularity of global solution to the bistable reaction–diffusion equation with degenerate diffusion and nonlocal time-delay. The adopted approach is the Holmgren’s approximation scheme combining the compactness analysis.

Traveling wave phenomena of a nonlocal reaction-diffusion equation with degenerate nonlinearity

This paper deals with traveling wave phenomena of a degenerate reaction-diffusion equation with the nonlocal effect. We study the existence of traveling wave solutions which may be non-monotonic based on the two-point boundary value problem and Schauder’s fixed point theorem. We are excited to find that the unknown positive steady state is exactly a unique positive equilibrium for the large wave speed and the monotonicity of traveling waves depends on the wave speed. On the other hand, we demonstrate the existence of monotone traveling wave solutions by using the monotone iteration and operator theory, and investigate how wave profile (e.g. being non-monotone or periodic) is affected by the enhancement of nonlocality, parameter values and variation of initial values for two different forms of convolution kernel functions.

Analysis of a diffusive host-pathogen model with standard incidence and distinct dispersal rates

Abstract This paper concerns with detailed analysis of a reaction-diffusion host-pathogen model with space-dependent parameters in a bounded domain. By considering the fact the mobility of host individuals playing a crucial role in disease transmission, we formulate the model by a system of degenerate reaction-diffusion equations, where host individuals disperse at distinct rates and the mobility of pathogen is ignored in the environment.We first establish the well-posedness of the model, including the global existence of solution and the existence of the global compact attractor. The basic reproduction number is identified, and also characterized by some equivalent principal spectral conditions, which establishes the threshold dynamical result for pathogen extinction and persistence. When the positive steady state is confirmed, we investigate the asymptotic profiles of positive steady state as host individuals disperse at small and large rates. Our result suggests that small and large diffusion rate of hosts have a great impacts in formulating the spatial distribution of the pathogen.

Adaptive ADI Numerical Analysis of 2D Quenching-Type Reaction: Diffusion Equation with Convection Term

An adaptive high-order difference solution about a 2D nonlinear degenerate singular reaction-diffusion equation with a convection term is initially proposed in the paper. After the first and the second central difference operator approximating the first-order and the second-order spatial derivative, respectively, the higher-order spatial derivatives are discretized by applying the Taylor series rule and the temporal derivative is discretized by using the Crank–Nicolson (CN) difference scheme. An alternating direction implicit (ADI) scheme with a nonuniform grid is built in this way. Meanwhile, accuracy analysis declares the second order in time and the fourth order in space under certain conditions. Sequentially, the high-order scheme is performed on an adaptive mesh to demonstrate quenching behaviors of the singular parabolic equation and analyse the influence of combustion chamber size on quenching. The paper displays rationally that the proposed scheme is practicable for solving the 2D quenching-type problem.

Nonnegative controllability for a class of nonlinear degenerate parabolic equations with application to climate science

We consider a nonlinear degenerate reaction-diffusion equation. First we prove that if the initial state is nonnegative, then the solution remains nonnegative for all time. Then we prove the approximate controllability between nonnegative states via multiplicative controls, this is done using the reaction coefficient as control.
 For more information see https://ejde.math.txstate.edu/Volumes/2020/59/abstr.html

Numerical simulation of mathematical heart model in COMSOL

Electrical activity is essential for the cardiac cell to perform its function. Mathematical modeling of cardiac electrical activity is performed from the cell, tissue and organ levels through to the body surface level. The electrical activity of the cardiac as a whole is thus characterized by a complex multiscale structure. The most complete model of such a complex setting is the anisotropic bidomain model that consists of a system of two degenerate parabolic reaction diffusion equations describing the intra and extracellular potentials in the cardiac muscle, coupled with a system of ordinary deferential equations describing the ionic currents flowing through the cellular membrane. This study describes an anatomically realistic 3D Bidomain model of whole-heart electrical activity. The heart was embedded in a human torso, incorporating spontaneous activation with heterogeneous action potential (AP) morphologies throughout the heart. The aim of this study is the development of a geometrically simple and computationally efficient 3D model of heart. In this paper a finite element formulation, model and simulation of Bidomain equation has been conducted. The FitzHugh-Nagumo (FHN) equations were incorporated into Bidomain model of cardiac electrical activity, which was comprised of a simplified geometry of the whole heart with the torso as an extracellular volume conductor. Laplace equation for the torso also considered. Simplified 3D cardiac model was implemented using COMSOL Multiphysics 5.0 finite element software. Electrical potential at different point on torso is measured. Propagation of electrical excitation on heart surface is also observed. This study represents the first stage toward the development of an accurate computer model of heart activation.  Â

Multiplicative controllability for nonlinear degenerate parabolic equations between sign-changing states

In this paper we study the global approximate multiplicative controllability for nonlinear degenerate parabolic Cauchy problems. In particular, we consider a one-dimensional semilinear degenerate reaction-diffusion equation in divergence form governed via the coefficient of the reaction term (bilinear or multiplicative control). The above one-dimensional equation is degenerate since the diffusion coefficient is positive on the interior of the spatial domain and vanishes at the boundary points. Furthermore, two different kinds of degenerate diffusion coefficient are distinguished and studied in this paper: the weakly degenerate case, that is, if the reciprocal of the diffusion coefficient is summable, and the strongly degenerate case, that is, if that reciprocal isn’t summable. In our main result we show that the above systems can be steered from an initial continuous state that admits a finite number of points of sign change to a target state with the same number of changes of sign in the same order. Our method uses a recent technique introduced for uniformly parabolic equations employing the shifting of the points of sign change by making use of a finite sequence of initial-value pure diffusion problems. Our interest in degenerate reaction-diffusion equations is motivated by the study of some energy balance models in climatology (see,e.g., the Budyko-Sellers model) and some models in population genetics (see,e.g., the Fleming-Viot model).

Stochastically forced cardiac bidomain model

The bidomain system of degenerate reaction–diffusion equations is a well-established spatial model of electrical activity in cardiac tissue, with “reaction” linked to the cellular action potential and “diffusion” representing current flow between cells. The purpose of this paper is to introduce a “stochastically forced” version of the bidomain model that accounts for various random effects. We establish the existence of martingale (probabilistic weak) solutions to the stochastic bidomain model. The result is proved by means of an auxiliary nondegenerate system and the Faedo–Galerkin method. To prove convergence of the approximate solutions, we use the stochastic compactness method and Skorokhod–Jakubowski a.s. representations. Finally, via a pathwise uniqueness result, we conclude that the martingale solutions are pathwise (i.e., probabilistic strong) solutions.

Evolution of interfaces for the nonlinear double degenerate parabolic equation of turbulent filtration with absorption

We prove the short-time asymptotic formula for the interfaces and local solutions near the interfaces for the nonlinear double degenerate reaction–diffusion equation of turbulent filtration with strong absorption ut=(|(um)x|p−1(um)x)x−buβ,mp>1,β>0.Full classification is pursued in terms of the nonlinearity parameters m,p,β and asymptotics of the initial function near its support. Numerical analysis using a weighted essentially nonoscillatory (WENO) scheme with interface capturing is implemented, and comparison of numerical and analytical results is presented.

Existence and Stability of Traveling Waves for Degenerate Reaction–Diffusion Equation with Time Delay

This paper deals with the existence and stability of traveling wave solutions for a degenerate reaction–diffusion equation with time delay. The degeneracy of spatial diffusion together with the effect of time delay causes us the essential difficulty for the existence of the traveling waves and their stabilities. In order to treat this case, we first show the existence of smooth- and sharp-type traveling wave solutions in the case of $$c\ge c^*$$ for the degenerate reaction–diffusion equation without delay, where $$c^*>0$$ is the critical wave speed of smooth traveling waves. Then, as a small perturbation, we obtain the existence of the smooth non-critical traveling waves for the degenerate diffusion equation with small time delay $$\tau >0$$ . Furthermore, we prove the global existence and uniqueness of $$C^{\alpha ,\beta }$$ -solution to the time-delayed degenerate reaction–diffusion equation via compactness analysis. Finally, by the weighted energy method, we prove that the smooth non-critical traveling wave is globally stable in the weighted $$L^1$$ -space. The exponential convergence rate is also derived.

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.

On a class of nonlocal degenerate reaction-diffusion equations with localized nonlinear diffusion term

We study a class of second order nonlocal degenerate semilinear reaction-diffusion equations with localized nonlinear diffusion term. Under a set of conditions on the localized nonlinear diffusivity and nonlinear nonlocal source term, we prove global existence and uniqueness result in the whole of some weighted Sobolev’s spaces. Furthermore, we prove nonexistence of smooth solution or blow-up of solution under some other set of conditions. Finally, we give illustrative examples for which our results apply.

Convective combustion in porous media: singular limit of high activation energy

In this paper, we consider a system of degenerate reaction diffusion equations which describes a convective (pressure driven) regime of combustion in porous media. The goal of this paper is to study the behaviour of this system in the limit of high activation energy. We show that the limit solution for this problem in arbitrary spatial dimension solves a parabolic equation with memory term similar to one arising in solid combustion. Moreover, under the additional assumption of the solution being time increasing, we prove that the limit problem coincides with Stefan problem for supercooled water with spatially inhomogeneous coefficients. We also obtain the precise limit problem for (not necessarily planar) travelling waves in any dimension.

An adaptive splitting approach for the quenching solution of reaction–diffusion equations over nonuniform grids

The numerical solution of a nonlinear degenerate reaction–diffusion equation of the quenching type is investigated. While spatial derivatives are discretized over symmetric nonuniform meshes, a Peaceman–Rachford splitting method is employed to advance solutions of the semidiscretized system. The temporal step is determined adaptively through a suitable arc-length monitor function. A criterion is derived to ensure that the numerical solution acquired preserves correctly the positivity and monotonicity of the analytical solution. Weak stability is proven in a von Neumann sense via the ∞-norm. Computational examples are presented to illustrate our results.

A Revisit of the Semi-Adaptive Method for Singular Degenerate Reaction-Diffusion Equations

AbstractThis article discusses key characteristics of a semi-adaptive finite difference method for solving singular degenerate reaction-diffusion equations. Numerical stability, monotonicity, and convergence are investigated. Numerical experiments illustrate the discussion. The study reconfirms and improves several of our earlier results.

Dynamical Structure of Some Nonlinear Degenerate Diffusion Equations

We consider degenerate reaction diffusion equations of the form ut = Δum + f(x, u), where f(x, u) ~ a(x)up with 1 ≤ p 0 at least in some part of the spatial domain, so that \({u \equiv 0}\) is an unstable stationary solution. We prove that the unstable manifold of the solution \({u \equiv 0 }\) has infinite Hausdorff dimension, even if the spatial domain is bounded. This is in marked contrast with the case of non-degenerate semilinear equations. The above result follows by first showing the existence of a solution that tends to 0 as \({t\to -\infty}\) while its support shrinks to an arbitrarily chosen point x* in the region where a(x) > 0, then superimposing such solutions, to form a family of solutions of arbitrarily large number of free parameters. The construction of such solutions will be done by modifying self-similar solutions for the case where a is a constant.

On stability of two degenerate reaction–diffusion systems

In this paper, we construct a partially degenerate reaction–diffusion equation subject to the Neumann boundary condition and show that the zero solution is asymptotically stable but not exponentially asymptotically stable. In this way, we solve an open problem proposed by Casten and Holland (1977) [4]. Moreover, we give the exponential asymptotic stability of the zero solution to a totally degenerate system with cross-diffusion effects, which cannot be determined by a simple spectral analysis based on the well developed semigroup theory.