Blow‐up solutions for localized reaction‐diffusion equations with variable exponents

This paper deals with radial solutions to localized reaction‐diffusion equations with variable exponents, subject to homogeneous Dirichlet boundary conditions. The global existence versus blow‐up criteria are studied in terms of the variable exponents. We proposed that the maximums of variable exponents are the key clue to determine blow‐up classifications and describe blow‐up rates for positive solutions. Copyright © 2011 John Wiley & Sons, Ltd.     

Blow‐up profiles of solutions for the exponential reaction‐diffusion equation

We consider the blow‐up of solutions for a semilinear reaction‐diffusion equation with exponential reaction term. It is known that certain solutions that can be continued beyond the blow‐up time possess a non‐constant self‐similar blow‐up profile. Our aim is to find the final time blow‐up profile for such solutions. The proof is based on general ideas using semigroup estimates. The same approach works also for the power nonlinearity. Copyright © 2011 John Wiley & Sons, Ltd.     

Blow‐up rate estimates for a parabolic system with multiple nonlinearities

In this paper, we study the blow‐up behaviors for the solutions of parabolic systems u_t=Δu+δ₁e, v_t=Δv+µ₁u in ?×(0, T) with nonlinear boundary conditions Here δ_i?0, µ_j?0, p_i?0, q_j?0 and at least one of δ_iµ_jp_iq_j>0(i, j=1, 2). We prove that the solutions will blow up in finite time for suitable ‘large’ initial values. The exact blow‐up rates are also obtained. Copyright © 2009 John Wiley & Sons, Ltd.     

Global a posteriori error estimates for convection–reaction–diffusion problems

In this note we propose a nonstandard technique for constructing global a posteriori error estimates for the stationary convection–reaction–diffusion equation. In order to estimate the approximation error in appropriate weighted energy norms, which measures the overall quality of the approximations, the underlying bilinear form is decomposed into several terms which can be directly computed or easily estimated from above using elementary tools of functional analysis. Several auxiliary parameters are introduced to construct such a splitting and tune the resulting upper error bound. It is demonstrated how these parameters can be chosen in some natural and convenient way for computations so that the weighted energy norm of the error is almost recovered, which shows that the estimates proposed are, in fact, quasi-sharp. The presented methodology is completely independent of numerical techniques used to compute approximate solutions. In particular, it is applicable to approximations which fail to satisfy the Galerkin orthogonality, e.g., due to an inconsistent stabilization, flux limiting, low-order quadrature rules, round-off and iteration errors etc. Moreover, the only constant that appears in the proposed error estimates is of global nature and comes from the Friedrichs–Poincaré inequality.     

Quenching for a reaction–diffusion system with logarithmic singularity

In this paper, we study the quenching phenomenon for a reaction–diffusion system with singular logarithmic source terms and positive Dirichlet boundary conditions. Some sufficient conditions for quenching of the solutions in finite time are obtained, and the blow-up of time-derivatives at the quenching point is verified. Furthermore, under appropriate hypotheses, the non-simultaneous quenching of the system is proved, and the estimates of quenching rate is given.     

Large‐time behavior of a two‐scale semilinear reaction–diffusion system for concrete sulfatation

We study the large‐time behavior of (weak) solutions to a two‐scale reaction–diffusion system coupled with a nonlinear ordinary differential equations modeling the partly dissipative corrosion of concrete (or cement)‐based materials with sulfates. We prove that as t → ∞ , the solution to the original two‐scale system converges to the corresponding two‐scale stationary system. To obtain the main result, we make use essentially of the theory of evolution equations governed by subdifferential operators of time‐dependent convex functions developed combined with a series of two‐scale energy‐like time‐independent estimates. Copyright © 2014 John Wiley & Sons, Ltd.     

Stability and asymptotic behaviour in a reaction–diffusion system

This paper is concerned with some qualitative analysis for a coupled system of five reaction–diffusion equations which arises from a physiology model. The uniform boundedness of the time-dependent solution is obtained under various boundary conditions. Sufficient conditions are also given to ensure the asymptotic stability of the non-negative steady-state solutions under Dirichlet or Robin boundary condition for each component. Under homogeneous Neumann boundary condition for some components the time-dependent solution is proven to converge to a constant steady state determined by the initial functions.     

Global boundedness of solutions to a reaction–diffusion system

We obtain in this paper the global boundedness of solutions to a Fujita‐type reaction–diffusion system. This global boundedness results from diffusion effect, homogeneous Dirichlet boundary value conditions and appropriate reactions. Copyright © 1999 John Wiley & Sons, Ltd.     

Algebraic multigrid preconditioners for the bidomain reaction–diffusion system

The so-called bidomain system is possibly the most complete model for the cardiac bioelectric activity. It consists of a reaction–diffusion system, modeling the intra, extracellular and transmembrane potentials, coupled through a nonlinear reaction term with a stiff system of ordinary differential equations describing the ionic currents through the cellular membrane. In this paper we address the problem of efficiently solving the large linear system arising in the finite element discretization of the bidomain model, when a semiimplicit method in time is employed. We analyze the use of structured algebraic multigrid preconditioners on two major formulations of the model, and report on our numerical experience under different discretization parameters and various discontinuity properties of the conductivity tensors. Our numerical results show that the less exercised formulation provides the best overall performance on a typical simulation of the myocardium excitation process.     

Blow‐up solutions to Landau–Lifshitz–Maxwell systems

The Landau–Lifshitz–Gilbert equation describes the evolution of spin fields in continuum ferromagnetics. The present paper consists of two parts. The first one is to prove the local existence of smooth solution to the Landau–Lifshitz–Maxwell systems in dimensions three. The second is to prove the finite time blow up of solutions for these systems. It states that for suitably chosen initial data, the short time smooth solutions to the Landau–Lifshitz–Maxwell equations do blow up at finite time. Copyright © 2011 John Wiley & Sons, Ltd.     

Blow‐up phenomena for a pseudo‐parabolic equation

By the means of a differential inequality technique, we obtain a lower bound for blow‐up time if p and the initial value satisfy some conditions. Also, we establish a blow‐up criterion and an upper bound for blow‐up time under some conditions as well as a nonblow‐up and exponential decay under some other conditions. Copyright © 2014 John Wiley & Sons, Ltd.     

Time‐weighted blow‐up rates and pointwise profile for single‐point blow‐up solutions in reaction–diffusion equations

This paper deals with asymptotic behavior for blow‐up solutions to time‐weighted reaction–diffusion equations u_t=Δu+e^αtv^p and v_t=Δv+e^βtu^q, subject to homogeneous Dirichlet boundary. The time‐weighted blow‐up rates are defined and obtained by ways of the scaling or auxiliary‐function methods for all α, . Aiding by key inequalities between components of solutions, we give lower pointwise blow‐up profiles for single‐point blow‐up solutions. We also study the solutions of the system with variable exponents instead of constant ones, where blow‐up rates and new blow‐up versus global existence criteria are obtained. Time‐weighted functions influence critical Fujita exponent, critical Fujita coefficient and formulae of blow‐up rates, but they do not limit the order of time‐weighted blow‐up rates and pointwise profile near blow‐up time. Copyright © 2017 John Wiley & Sons, Ltd.     

A parameter‐robust numerical method for a system of reaction–diffusion equations in two dimensions

A system of M(≥ 2) coupled singularly perturbed linear reaction–diffusion equations is considered on the unit square. Under certain hypotheses on the coupling, a maximum principle is established for the differential operator. The relationship between compatibility conditions at the corners of the square and the smoothness of the solution on the closed domain is fully described. A decomposition of the solution of the system is constructed. A finite‐difference method for the solution of the system on a Shishkin mesh is presented, and it is proved that the computed solution is second‐order accurate (up to a logarithmic factor). Numerical results are given to support this result and to investigate the effect of weaker compatibility assumptions on the data. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

A parameter-uniform Schwarz method for a coupled system of reaction–diffusion equations

We consider an arbitrarily sized coupled system of one-dimensional reaction–diffusion problems that are singularly perturbed in nature. We describe an algorithm that uses a discrete Schwarz method on three overlapping subdomains, extending the method in [H. MacMullen, J.J.H. Miller, E. O’Riordan, G.I. Shishkin, A second-order parameter-uniform overlapping Schwarz method for reaction-diffusion problems with boundary layers, J. Comput. Appl. Math. 130 (2001) 231–244] to a coupled system. On each subdomain we use a standard finite difference operator on a uniform mesh. We prove that when appropriate subdomains are used the method produces ε-uniform results. Furthermore we improve upon the analysis of the above-mentioned reference to show that, for small ε, just one iteration is required to achieve the expected accuracy.     

Nonstandard finite difference schemes for reaction‐‐diffusion equations having linear advection

We extend previous work on nonstandard finite difference schemes for one‐space dimension, nonlinear reaction–diffusion PDEs to the case where linear advection is included. The use of a positivity condition allows the determination of a functional relation between the time and space step‐sizes, and provides schemes that are explicit. The Fisher equation is used to illustrate the method. © 2000 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 16: 361–364, 2000     

Wave simulations of Gray‐Scott reaction‐diffusion system

In this work, we study the numerical simulation of the one‐dimensional reaction‐diffusion system known as the Gray‐Scott model. This model is responsible for the spatial pattern formation, which we often meet in nature as the result of some chemical reactions. We have used the trigonometric quartic B‐spline (T4B) functions for space discretization with the Crank‐Nicolson method for time integration to integrate the nonlinear reaction‐diffusion equation into a system of algebraic equations. The solutions of the Gray‐Scott model are presented with different wave simulations. Test problems are chosen from the literature to illustrate the stationary waves, pulse‐splitting waves, and self‐replicating waves.     

Blow‐up analysis for a class of nonlinear reaction diffusion equations with Robin boundary conditions

This paper is devoted to the study of the blow‐up phenomena of following nonlinear reaction diffusion equations with Robin boundary conditions: Here, is a bounded convex domain with smooth boundary. With the aid of a differential inequality technique and maximum principles, we establish a blow‐up or non–blow‐up criterion under some appropriate assumptions on the functions f,g,ρ,k, and u₀. Moreover, we dedicate an upper bound and a lower bound for the blow‐up time when blowup occurs.     

Positive solutions and dynamics of a finite difference reaction–diffusion system

We investigate the dynamics and methods of computation for some nonlinear finite difference systems that are the discretized equations of a time-dependent and a steady-state reaction–diffusion problem. The formulation of the discrete equations for the time-dependent problem is based on the implicit method for parabolic equations, and the computational algorithm is based on the method of monotone iterations using upper and lower solutions as the initial iterations. The monotone iterative method yields improved upper and lower bounds of the solution in each iteration, and the sequence of iterations converges monotonically to a solution for both the time-dependent and the steady-state problems. An important consequence of this method is that it leads to a bifurcation point that determines the dynamic behavior of the time-dependent problem in relation to the corresponding steady-state problem. This bifurcation point also determines whether the steady-state problem has one or two non-negative solutions, and is explicitly given in terms of the physical parameters of the system and the type of boundary conditions. Numerical results are presented for both the time-dependent and the steady-state problems under various boundary conditions, including a test problem with known analytical solution. These numerical results exhibit the predicted dynamic behavior of the time-dependent solution given by the theoretical analysis. Also discussed are the numerical stability of the computational algorithm and the convergence of the finite difference solution to the corresponding continuous solution of the reaction–diffusion problem. © 1993 John Wiley & Sons, Inc.     

A second‐order linearized difference scheme for a strongly coupled reaction‐diffusion system

This article deals with the numerical solution to some models described by the system of strongly coupled reaction–diffusion equations with the Neumann boundary value conditions. A linearized three‐level scheme is derived by the method of reduction of order. The uniquely solvability and second‐order convergence in L₂‐norm are proved by the energy method. A numerical example is presented to demonstrate the accuracy and efficiency of the proposed method. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007