System Of Nonlinear Parabolic Equations Research Articles

Turing patterns, Lengyel–Epstein systems and Faber splines

This paper deals with the Lengyel–Epstein CIMA (chlorite-iodide-malonic acid) system of non-linear parabolic equations in the context of function spaces, especially of Hölder-Zygmund type. We discuss in particular the size of Turing patterns (if occur) in

Stochastic models for forward systems of nonlinear parabolic equations

Stochastic differential equations give a constructive way to obtain a large number of stochastic processes. They can be used as well to simulate these processes numerically. As a by-product we can construct numerically solutions of the Cauchy problem for some parabolic equations and systems. In this paper we derive systems of stochastic equations for stochastic processes underlying systems of coupled parabolic equations which are particular cases of cross-diffusion systems of PDEs. Stochastic processes satisfying these stochastic systems allow to obtain probabilistic representations of classical and weak solutions of the Cauchy problem for original PDE systems and can be used to simulate the PDE solutions.

Cosymmetry approach and mathematical modeling of species coexistence in a heterogeneous habitat

<p style='text-indent:20px;'>We explore an approach based on the theory of cosymmetry to model interaction of predators and prey in a two-dimensional habitat. The model under consideration is formulated as a system of nonlinear parabolic equations with spatial heterogeneity of resources and species. Firstly, we analytically determine system parameters, for which the problem has a nontrivial cosymmetry. To this end, we formulate cosymmetry relations. Next, we employ numerical computations to reveal that under said cosymmetry relations, a one-parameter family of steady states is formed, which may be characterized by different proportions of predators and prey. The numerical analysis is based on the finite difference method (FDM) and staggered grids. It allows to follow the transformation of spatial patterns with time. Eventually, the destruction of the continuous family of equilibria due to mistuned parameters is analyzed. To this end, we derive the so-called cosymmetric selective equation. Investigation of the selective equation gives an insight into scenarios of local competition and coexistence of species, together with their connection to the cosymmetry relations. When the cosymmetry relation is only slightly violated, an effect we call 'memory on the lost family' may be observed. Indeed, in this case, a slow evolution takes place in the vicinity of the lost states of equilibrium.

Periodic Variations of an Autowave Structure in Two-dimensional System of Parabolic Equations

The work is aimed to study front solutions of a nonlinear system of parabolic equations in a two-dimensional region. The system can be considered as a mathematical model describing an abrupt change in physical characteristics of spatially heterogeneous media. We consider a system with small parameters raised to the different powers at a differential operator, that represents the difference of typical processes speeds for the system components. The study of the system is conducted by using the contrast structures theory methods, which allowed us to obtain conditions for the existence of front solutions contained in the neighborhood of a closed curve, to determine the front velocity depending on time and coordinate along the front curve, and to obtain the zero-order and the first-order terms of the asymptotic approximation to the solution. The scope of the system includes the description of autowave solutions in the field of ecology, biophysics, combustion physics and chemical kinetics. The approximate solution allows us to choose the model parameters so that the result corresponds to the processes observed, to explain and describe the characteristics of the solutions with sharp gradients, to create models with stable solutions and thereby to simplify the numerical analysis. Note that the numerical experiment for the two-dimensional spatial models requires a considerable amount of processing power and the use of parallel computing techniques and does not allow to effectively analyze and modify the model. In this paper, we obtain the asymptotic approximation that is to be justified, which can be done by the method of differential inequalities.

A variational approach to symmetry, monotonicity, and comparison for doubly-nonlinear equations

We advance a variational method to prove qualitative properties such as symmetries, monotonicity, upper and lower bounds, sign properties, and comparison principles for a large class of doubly-nonlinear evolutionary problems including gradient flows, some nonlocal problems, and systems of nonlinear parabolic equations. Our method is based on the so-called Weighted-Energy-Dissipation (WED) variational approach. This consists in defining a global parameter-dependent functional over entire trajectories and proving that its minimizers converge to solutions to the target problem as the parameter goes to zero. Qualitative properties and comparison principles can be easily proved for minimizers of the WED functional and, by passing to the limit, for the limiting problem. Several applications of the abstract results to systems of nonlinear PDEs and to fractional/nonlocal problems are presented. Eventually, we present some extensions of this approach in order to deal with rate-independent systems and hyperbolic problems.

Positive Solution of a Nonlinear Parabolic System Arising in Grain Drying

Coupled system of nonlinear parabolic equations for grain drying is proposed and the existence and uniqueness theorem of classical solutions is proved by using the upper and lower solutions technique. The long-time behaviour of the solution is also investigated.

Image filtering with the use of anisotropic diffusion

A system of nonlinear parabolic equations describing the evolution of a color image is considered. The existence and uniqueness of a global solution to the mixed problem for this system is proved.

An ignition-temperature model with two free interfaces in premixed flames

In this paper we consider an ignition-temperature zero-order reaction model of thermo-diffusive combustion. This model describes the dynamics of thick flames, which have recently received considerable attention in the physical and engineering literature. The model admits a unique (up to translations) planar travelling wave solution. This travelling wave solution is quite different from those usually studied in combustion theory. The main qualitative feature of this travelling wave is that it has two interfaces: the ignition interface where the ignition temperature is attained and the trailing interface where the concentration of deficient reactants reaches zero. We give a new mathematical framework for studying the cellular instability of such travelling front solutions. Our approach allows the analysis of a free boundary problem to be converted into the analysis of a boundary value problem having a fully nonlinear system of parabolic equations. The latter is very suitable for both mathematical and numerical analysis. We prove the existence of a critical Lewis number such that the travelling wave solution is stable for values of Lewis number below the critical one and is unstable for Lewis numbers that exceed this critical value. Finally, we discuss the results of numerical simulations of a fully nonlinear system that describes the perturbation dynamics of planar fronts. These simulations reveal, in particular, some very interesting ‘two-cell’ steady patterns of curved combustion fronts.

Monotone iterative ADI method for solving coupled systems of nonlinear parabolic equations

The paper deals with numerical solution of coupled systems of nonlinear parabolic equations based on a nonlinear ADI scheme. The convergence of the nonlinear ADI scheme to the continuous solution is proved. A monotone iterative ADI method is constructed. The existence and uniqueness of a solution of the nonlinear ADI scheme are established. An analysis of convergence of the monotone iterative ADI method to the solution of the nonlinear ADI scheme on the whole time interval is given. Numerical experiments are presented.

Parabolic Bellman-Systems with Mean Field Dependence

We consider the necessary conditions for Nash-points of Vlasov-McKean functionals $$\mathcal {J}^{i}[\mathbf{v}]=\int _{Q}mf^{i}(\cdot ,m,\mathbf{v})\,dx\,dt$$Ji[v]=?Qmfi(·,m,v)dxdt ($$i=1,...,N$$i=1,...,N). The corresponding payoffs $$f^{i}$$fi depend on the controls $$\mathbf{v}$$v and, in addition, on the field variable $$m=m(\mathbf{v})$$m=m(v). The necessary conditions lead to a coupled forward-backward system of nonlinear parabolic equations, motivated by stochastic differential games. The payoffs may have a critical nonlinearity of quadratic growth and any polynomial growth w.r.t. m is allowed as long as it can be dominated by the controls in a certain sense. We show existence and regularity of solutions to these mean-field-dependent Bellman systems by a purely analytical approach, no tools from stochastics are needed.

Dynamics of strongly coupled spatially distributed logistic equations with delay

The dynamics of a system of two logistic delay equations with spatially distributed coupling is studied. The coupling coefficient is assumed to be sufficiently large. Special nonlinear systems of parabolic equations are constructed such that the behavior of their solutions is determined in the first approximation by the dynamical properties of the original system.

Integration of Newton Linearization into the Time Discretization of Initial-Boundary-Value Problems

We construct a one-step recursive scheme for the integration of the Cauchy problem for large systems of ordinary differential equations appearing after the space semidiscretization of the initial-boundary-value problems for systems of nonlinear parabolic equations. The main specific feature of the construction of this scheme is connected with balancing of the orders of the error of piecewise linear approximation with respect to time and the error of Newton linearization. The indicated feature enables us to construct a numerical predictor-corrector-type scheme with weight parameter. With the help of the principle of contracting mappings, we establish sufficient conditions for the correctness of discretized problems. It is shown that, in the case of sufficiently high regularity of the desired solution of the Cauchy problem, the proposed one-step recursive scheme can attain the quadratic rate of convergence of approximations to this solution. We present the results of numerical experiments characterizing the proposed scheme by comparing with the Runge–Kutta schemes of different orders and its application in modeling the reaction of oxidation of carbon monoxide on the platinum surface.

Численное исследование сосуществования популяций в одной экологической нише

The predator-prey interactions on the spatial heterogeneous two-dimensional area are described. The model is written as a system of nonlinear parabolic equations for two closely related predator populations and two prey populations competing for the general resource. It is shown that under certain relationships between the parameters and the variable natural habitat resource functions, the model belongs to the class of the cosymmetric dynamical systems. In this case, there is a continuous family of stationary distributions of the coexistent populations. The simulation experiment is based on the method of straight lines, and on the scheme of staggered grids. The balance method is used for the approximation in spatial variables of the task on a rectangular area. The results showing the model capabilities for describing the formation of the population stationary distributions are presented. The formation of the biological structures is studied under the growth parameter heterogeneity; the conditions for the coexistence of closely related types are analyzed.

Non-existence of Global Solutions to a System of Fractional Diffusion Equations

We present non-existence results for a system of non-linear parabolic equations with non-local diffusions posed either in $\mathbb{R}^{N} $ or in a bounded domain. Our method of proof is based on the test function method with a judicious choice of the test function.

Existence of solutions for a nonlinear system of parabolic equations with gradient flow structure

We prove global existence of weak solutions of a variant of the parabolic-parabolic Keller–Segel model for chemotaxis on the whole space \({{\mathbb {R}}^d}\) for \(d\ge 3\) with a supercritical porous-medium diffusion exponent and an external drift. The structure of the equations allow the chemotactic drift to be seen both as attraction and repulsion. The method of proof relies on the inherent gradient flow structure of this system with respect to a coupled Wasserstein-\(L^2\) metric. Additional regularity estimates are derived from the dissipation of an entropy functional.

Reduction of weakly nonlinear parabolic partial differential equations

It is known that the Swift-Hohenberg equation \documentclass[12pt]{minimal}\begin{document}$\partial u/\partial t = -(\partial _x^2 + 1)^2u + \varepsilon (u-u^3)$\end{document}∂u/∂t=−(∂x2+1)2u+ɛ(u−u3) can be reduced to the Ginzburg-Landau equation (amplitude equation) \documentclass[12pt]{minimal}\begin{document}$\partial A/\partial t\break = 4\partial _x^2 A + \varepsilon (A-3A|A|^2)$\end{document}∂A/∂t=4∂x2A+ɛ(A−3A|A|2) by means of the singular perturbation method. This means that if ɛ &amp;gt; 0 is sufficiently small, a solution of the latter equation provides an approximate solution of the former one. In this paper, a reduction of a certain class of a system of nonlinear parabolic equations \documentclass[12pt]{minimal}\begin{document}$\partial u/\partial t = \mathcal {P}u + \varepsilon f(u)$\end{document}∂u/∂t=Pu+ɛf(u) is proposed. An amplitude equation of the system is defined and an error estimate of solutions is given. Further, it is proved under certain assumptions that if the amplitude equation has a stable steady state, then a given equation has a stable periodic solution. In particular, near the periodic solution, the error estimate of solutions holds uniformly in t &amp;gt; 0.

A control problem for a cross-diffusion system in a nonhomogeneous medium

We study a model, motivated by a bioremediation process, describing a cross-diffusion movement of a bacteria population b attracted by a chemoattractant signal c, in a nonhomogeneous stratified medium with n layers. We assume that this reaction–diffusion process is characterized by a low rate of degradation and a low diffusion coefficient of the chemoattractant, expressed in the model by a small parameter ϵ. The model consists of n systems of nonlinear parabolic equations with transmission conditions between layers. We prove a global-in-time solution for the asymptotic model setup with respect to the small parameter of the problem, for arbitrarily large initial data. Next, we deal with the control problem focusing mainly on the reduction of the chemoattractant concentration, by acting upon the initial distribution of the bacteria population b0. To this end, we prove the existence of a solution to the control problem and determine the optimality conditions.

Hodograph Transformations and Cauchy Problem to Systems of Nonlinear Parabolic Equations

In this paper, we provide a method to solve the Cauchy problem of systems of quasi‐linear parabolic equations, such systems can be transformed to the systems of linear parabolic equations with variable coefficients via the hodograph transformations. Our approach to solve the linear systems with variable coefficients is to use their fundamental solutions, which are constructed by using the Lie's symmetry method. In consequence, we can derive explicit solutions to the Cauchy problem of the quasi‐linear systems in terms of the solutions of the linear systems and the hodograph transformations relating to the quasi‐linear and the linear systems.

Probabilistic approach to viscosity solutions of the Cauchy problems for systems if fully nonlinear parabolic equations

In this paper, we discuss a probabilistic approach to construction of a viscosity solution of the Cauchy problem for a system of nonlinear parabolic equations. Our approach is based on reduction of the original problem to a system of quasilinear parabolic equation in the first step and to a system of fully coupled forward-backward stochastic differential equations in the second step. Solution of the stochastic problem allows us to construct a probabilistic representation of a viscosity solution of the original problem and state conditions which ensure the existence and uniqueness of this solution.

One-dimensional two-phase generalized Forchheimer flows of incompressible fluids

We derive a nonlinear system of parabolic equations to describe the one-dimensional two-phase generalized Forchheimer flows of incompressible, immiscible fluids in porous media, with the presence of capillary forces. Under relevant constraints on relative permeabilities and capillary pressure, non-constant steady state solutions are found and classified into sixteen types according to their monotonicity and asymptotic behavior. For a steady state whose saturation can never attain either value 0 or 1, we prove that it is stable with respect to a certain weight. This weight is a function comprised of the steady state, relative permeabilities and capillary pressure. The proof is based on specific properties of the steady state, weighted maximum principle and Bernstein’s estimate.