Reaction-diffusion Equations Research Articles

Multi-resolution partial differential equations preserved learning framework for spatiotemporal dynamics

Traditional data-driven deep learning models often struggle with high training costs, error accumulation, and poor generalizability in complex physical processes. Physics-informed deep learning (PiDL) addresses these challenges by incorporating physical principles into the model. Most PiDL approaches regularize training by embedding governing equations into the loss function, yet this depends heavily on extensive hyperparameter tuning to weigh each loss term. To this end, we propose to leverage physics prior knowledge by “baking” the discretized governing equations into the neural network architecture via the connection between the partial differential equations (PDE) operators and network structures, resulting in a PDE-preserved neural network (PPNN). This method, embedding discretized PDEs through convolutional residual networks in a multi-resolution setting, largely improves the generalizability and long-term prediction accuracy, outperforming conventional black-box models. The effectiveness and merit of the proposed methods have been demonstrated across various spatiotemporal dynamical systems governed by spatiotemporal PDEs, including reaction-diffusion, Burgers’, and Navier-Stokes equations.

Comments to ‘Stability of traveling waves for deterministic and stochastic delayed reaction-diffusion equation based on phase shift’

In Liu et al. (2023), the authors set out to prove orbital stability for traveling waves in a (stochastic) delay reaction–diffusion equation. For the deterministic proof, they follow a phase-tracking technique as developed by Zumbrun and Howard (1998) [1] and for the stochastic version, they use a stochastic phase-tracking technique as developed by Hamster and Hupkes (2019, 2020a,b) [2,3] (which in turn is based on the aforementioned technique by Howard and Zumbrun). Note that the authors do not acknowledge the origins of their techniques.In the deterministic case, stability results are known (Mei, 2009) [4], but the application of phase-tracking to this problem appears to be new. However, there is a very subtle interplay between the time-dependent phase and the delay in the delay equation which is not accounted for, nor is explained or proved if the chosen phase has the same properties in the general framework of delay semigroups as for regular semigroups.The proof of the stochastic case of course has the same issues, but let us assume that the issues can be resolved, then there are still several issues with the proof. Especially, the phase tracking introduces terms that depend on the gradient of the solution which must be bounded in the H1 norm. This introduces singularities in the bounds that do not show in these proofs.

A Mixed Finite Element Method for the Multi-Term Time-Fractional Reaction–Diffusion Equations

In this work, a fully discrete mixed finite element (MFE) scheme is designed to solve the multi-term time-fractional reaction–diffusion equations with variable coefficients by using the well-known L1 formula and the Raviart–Thomas MFE space. The existence and uniqueness of the discrete solution is proved by using the matrix theory, and the unconditional stability is also discussed in detail. By introducing the mixed elliptic projection, the error estimates for the unknown variable u in the discrete L∞(L2(Ω)) norm and for the auxiliary variable λ in the discrete L∞((L2(Ω))2) and L∞(H(div,Ω)) norms are obtained. Finally, three numerical examples are given to demonstrate the theoretical results.

Variational approaches to Dirichlet gradient-type systems on the Sierpiński gasket

In this paper, we study the multiple solutions of parametric quasi-linear systems of the gradient-type on the Sierpiński gasket arising in physical problems leading to nonlinear models involving reaction-diffusion equations, in problems on elastic fractal media or fluid flow through fractal regions. By using some critical point theorems, we give some new results to establish one, two and three weak solutions for our system. Finally, we give four examples to illustrate the main results.

Rate of convergence for reaction–diffusion equations with nonlinear Neumann boundary conditions and $${\mathcal {C}}^1$$ variation of the domain

Rate of convergence for reaction–diffusion equations with nonlinear Neumann boundary conditions and $${\mathcal {C}}^1$$ variation of the domain

Wave Solutions for a (2 + 1)-Dimensional Burgers–KdV Equation with Variable Coefficients via the Functional Expansion Method

A (2 + 1)-dimensional fourth order Burgers–KdV equation with variable coefficients (vcBKdV) is studied here and interesting wave-type solutions with variable amplitudes and velocities are reported. The model has been not previously studied in the chosen form and it presents a twofold interest: as a model describing a rich variety of phenomena and as a higher-order equation solving difficulties generated by the presence of the variable coefficients. The novelty of our approach is related to the use of the functional expansion, a solving method based on an auxiliary equation that generalizes other approaches, such as, for example, the G′G proved here. We use a similarity reduction with a nonlinear wave variable that leads to a determining system that it is not usually algebraic, but an over-determined system of partial differential equations. It depends on 14 constant or functional parameters and can generate much richer classes of solutions. Three such classes of solutions, corresponding to the case when a specific form of the generalized reaction–diffusion equation is used as auxiliary equation, are considered. The influence on the dynamical behavior of two important factors, the choices of the auxiliary equation and the form of solution, are studied by providing graphical representations of specific solutions for various values of the parameters.

Monotonicity of entire solutions to reaction-diffusion equations involving fractional p-Laplacian

Abstract We obtain the one-dimensional symmetry and monotonicity of the entire positive solutions to some reaction-diffusion equations involving fractional p-Laplacian by virtue of the sliding method. More precisely, we consider the following problem { ∂ ⁡ u ∂ ⁡ t ⁢ ( x , t ) + ( - Δ ) p s ⁢ u ⁢ ( x , t ) = f ⁢ ( t , u ⁢ ( x , t ) ) , ( x , t ) ∈ Ω × ℝ , u ⁢ ( x , t ) > 0 , ( x , t ) ∈ Ω × ℝ , u ⁢ ( x , t ) = 0 , ( x , t ) ∈ Ω c × ℝ , \left\{\begin{aligned} \displaystyle{}\frac{\partial u}{\partial t}(x,t)+(-% \Delta)_{p}^{s}u(x,t)&\displaystyle=f(t,u(x,t)),&\hskip 10.0pt(x,t)&% \displaystyle\in\Omega\times\mathbb{R},\\ \displaystyle u(x,t)&\displaystyle>0,&\hskip 10.0pt(x,t)&\displaystyle\in% \Omega\times\mathbb{R},\\ \displaystyle u(x,t)&\displaystyle=0,&\hskip 10.0pt(x,t)&\displaystyle\in% \Omega^{c}\times\mathbb{R},\end{aligned}\right. where s ∈ ( 0 , 1 ) {s\in(0,1)} , p ≥ 2 {p\geq 2} , ( - Δ ) p s {(-\Delta)_{p}^{s}} is the fractional p-Laplacian, f ⁢ ( t , u ) {f(t,u)} is some continuous function, the domain Ω ⊂ ℝ n {\Omega\subset\mathbb{R}^{n}} is unbounded and Ω c = ℝ n ∖ Ω {\Omega^{c}=\mathbb{R}^{n}\setminus\Omega} . Firstly, we establish a maximum principle involving the parabolic p-Laplacian operator. Then, under certain conditions of f, we prove the asymptotic behavior of solutions far away from the boundary uniformly in t ∈ ℝ {t\in\mathbb{R}} . Finally, the sliding method is implemented to derive the monotonicity and uniqueness of the bounded positive entire solutions. To our best knowledge, there has not been any results on the symmetry and monotonicity properties of solutions to the parabolic fractional p-Laplacian equations before.

A Harnack inequality for a class of 1D nonlinear reaction–diffusion equations and applications to wave solutions

In this paper, a differential-geometric method is applied to build some Li–Yau–Hamilton-type Harnack inequalities for the positive solutions to a one spatial dimensional nonlinear reaction–diffusion equation in a plane geometry. The class of reaction–diffusion equation that is considered here contains several important equations some of which are Newel–Whitehead–Segel, Allen–Cahn and Fisher–KPP equations. The Harnack inequalities so derived are used to discuss some other important properties of positive solutions and in the characterization of positive wave solutions.

Schauder estimates for stationary and evolution equations associated to stochastic reaction-diffusion equations driven by colored noise

We consider stochastic reaction-diffusion equations with colored noise on the space of real-valued and continuous functions on a compact subset of ℝ d for d = 1 , 2 , 3 . We prove Schauder-type estimates, which will depend on the color of the noise, for the stationary and evolution problems associated with the corresponding transition semigroup.

Hausdorff and fractal dimensions of attractors for functional differential equations in Banach spaces

The main objective of this paper is to obtain estimations of Hausdorff dimension as well as fractal dimension of global attractors and pullback attractors for both autonomous and nonautonomous functional differential equations (FDEs) in Banach spaces. New criterions for the finite Hausdorff dimension and fractal dimension of attractors in Banach spaces are firslty proposed by combining the squeezing property and the covering of finite subspace of Banach spaces, which generalize the method established in Hilbert spaces. In order to surmount the barrier caused by the lack of orthogonal projectors with finite rank, which is the key tool for proving the squeezing property of partial differential equations in Hilbert spaces, we adopt the state decomposition of phase space based on the exponential dichotomy of the studied FDEs to obtain similar squeezing property. The theoretical results are applied to a retarded nonlinear reaction-diffusion equation and a non-autonomous retarded functional differential equation in the natural phase space, for which explicit bounds of dimensions that do not depend on the entropy number but only depend on the spectrum of the linear parts and Lipschitz constants of the nonlinear parts are obtained.

Mixed finite element methods for nonlinear reaction–diffusion equations with interfaces

We develop mixed finite element methods for nonlinear reaction–diffusion equations with interfaces which have Robin-type interface conditions. We introduce the velocity of chemicals as new variables and reformulate the governing equations. The stability of semidiscrete solutions, existence and the a priori error estimates of fully discrete solutions are proved by fixed point theorem and continuous/discrete Grönwall inequalities. Numerical results illustrating our theoretical analysis are included.

Turing patterns in domains with periodic inhomogeneities; a homogenization approach

In this work, we study reaction–diffusion equations with space-dependent, differentiable, and ɛ-periodic diffusion coefficients, using the asymptotic homogenization technique. The importance of this problem is that experimental evidence suggests that in some biological systems, spatial inhomogeneities are important to regulate spatial patterns, and several pattern-forming systems are regulated by a stratified domain that determines the transport properties. Here, we consider the case when the stratification is periodic with a period ɛ; consequently, the diffusion coefficient is a space-dependent ɛ-periodic function. A methodology based on the asymptotic homogenization technique was developed to solve this problem, and the conditions for Turing instability were found in terms of the effective coefficients of the equivalent homogenized domain. Additionally, extensive numerical calculations were performed to validate the results predicted by the homogenization procedure; one- and two-dimensional examples are presented. We conclude that the homogenization method is useful in the field of pattern formation to handle problems involving spatial inhomogeneities or stratified domains. In addition, the proposed generalization of Turing analysis for a homogenized problem enriches the field of pattern formation.

A numerical method for singularly perturbed convection–diffusion–reaction equations on polygonal meshes

A numerical method for singularly perturbed convection–diffusion–reaction equations on polygonal meshes

An efficient numerical method based on Chelyshkov operation matrix for solving a type of time-space fractional reaction diffusion equation

An efficient numerical method based on Chelyshkov operation matrix for solving a type of time-space fractional reaction diffusion equation

Unconditionally Positive, Explicit, Fourth Order Method for the Diffusion- and Nagumo-Type Diffusion–Reaction Equations

Unconditionally Positive, Explicit, Fourth Order Method for the Diffusion- and Nagumo-Type Diffusion–Reaction Equations

Random exponential attractor for a stochastic reaction-diffusion equation in $L^{2p}(D)$

AbstractIn this paper, we establish some sufficient conditions for the existence of a random exponential attractor for a random dynamical system in a Banach space. As an application, we consider a stochastic reaction-diffusion equation with multiplicative noise. We show that the random dynamical system $\phi(t,\omega)$ ϕ ( t , ω ) generated by this stochastic reaction-diffusion equation is uniformly Fréchet differentiable on a positively invariant random set in $L^{2p}(D)$ L 2 p ( D ) and satisfies the conditions of the abstract result, then we obtain the existence of a random exponential attractor in $L^{2p}(D)$ L 2 p ( D ) , where p is the growth of the nonlinearity satisfying $1< p\leq 3$ 1 < p ≤ 3 .

Dynamical behavior of a temporally discrete non-local reaction-diffusion equation on bounded domain

This paper focuses on the study of global dynamics of a class of temporally discrete non-local reaction-diffusion equations on bounded domains. Similar to classical reaction diffusion equations and integro-difference equations, temporally discrete reaction-diffusion equations can also be used to describe the dispersal phenomena in population dynamics. In this paper, we first derived a temporally discrete reaction diffusion equation model with time delay and nonlocal effects to model the evolution of a single species population with age-structured located in a bounded domain. By establishing a new maximum principle and applying the monotone iteration method, the global stabilities of the trivial solution and the positive steady state solution are obtained respectively under some appropriate assumptions.

A kinetic non-steady state analysis of immobilized enzyme systems with external mass transfer resistance

<abstract><p>The goal of this paper is to utilize the homotopy perturbation method (HPM) and Laplace transform to provide an approximate analytical expression to the non-linear time-dependent reaction diffusion equation arising in a mathematical model of an immobilized enzyme system with external mass transfer resistance. This mathematical model is a non-steady, non-linear reaction diffusion equation based on Michaelis–Menten kinetics. Approximate analytical expressions are also provided for various geometries of the enzyme catalytic pellets, namely, planar, cylindrical, and spherical. Obtained semi-analytical expressions are proven to fit for all the parameters appearing in the system and for all the geometries of enzyme catalytic pellets. When comparing the numerical and approximate analytical solutions, satisfactory results are obtained. Also, approximate analytical expressions of the effectiveness factor (EF) of the immobilized system are presented, and the effect of parameters on the EF is also analyzed.</p></abstract>

Curved fronts in time-periodic reaction-diffusion equations with monostable nonlinearity

Recently, it was found that there exists various curved fronts with nonplanar level sets, which has been attracting increasing attention. In this paper, we study periodic traveling curved fronts for reaction-diffusion equations with time-periodic monostable nonlinearity in $ \mathbb{R}^2 $. By using the comparison principle and constructing super-sub solutions, we first establish the existence of periodic traveling curved fronts. Furthermore, we show that when the given perturbation decays at infinity under some weighted sense, the periodic traveling curved front is the globally asymptotic stability by introducing mild subsolutions.

ON THE CORRECTNESS OF ONE CLASS OF FOURTH-ORDERSINGULAR DIFFERENTIAL EQUATIONS

The differential equation of fourth-order with variable smooth coefficients, given on the real axis, is investigated. The coefficients of the equation can be unbounded, and the sign of the potential (the lowest coefficient) is not defined. The case is considered that the intermediate term containing the second derivative of the desired function, as an operator, does not obey the operator of the equation. It is known that fourth-order differential equations with variable coefficients are used in various problems of mathematical physics. The simplest biharmonic equation with minor terms is important for its applications in the theory of elasticity, the mechanics of elastic plates and the slow flow of viscous liquids. However, it is a very special case of a general equation with variable coefficients. Some problems of stochastic analysis, oscillation theory, biology and mathematical finance lead to the equation we are studying. In addition, fourth–order singular differential equations are often used as regularizers in the study of lower-order equations, for example, reaction-diffusion equations. In the paper, sufficient solvability conditions of the equation and the maximal regularity inequality of a strong generalized solution are obtained. The restrictions are formulated in terms of the coefficients themselves and rather weak. They do not contain conditions for any derivatives of coefficients and represent relations between the orders of growth of coefficients of different orders at infinity. The weight estimation of the norms of a solution established by us allows us to additionally apply the methods of the theory of functional spaces to the study of further properties of the solution, for example, the approximation of its elements by finite-dimensional spaces.