Semilinear Reaction-diffusion Equations Research Articles

A high-order compact difference method and its Richardson extrapolation for semi-linear reaction–diffusion equations with piecewise continuous argument in diffusion term

In this paper, for the initial–boundary value problems of semi-linear reaction–diffusion equations with piecewise continuous argument in spatial derivative, we suggest Crank–Nicolson method, high-order compact difference (HOCD) method and HOCD-based Richardson extrapolation (RHOCD) method. Under the appropriate conditions, it is proved that HOCD (resp. RHOCD) method has the computational accuracy O(τ2+h4) (resp. O(τ4+h4)). This shows that RHOCD method improves the calculation accuracy of HOCD method in temporal direction. Moreover, we also analyze the stability of HOCD method and thus derive a global stability criterion of this method. Finally, with a series of numerical experiments, we further confirm the computational effectiveness and theoretical accuracy of the concerned methods.

Input-to-state practical stability criteria of non-autonomous infinite-dimensional systems

We develop tools for the investigation of input-to-state practical stability (ISpS) and integral input-to-state practical stability (iISpS) of non-autonomous infinite-dimensional systems in Banach spaces. Sufficient conditions of ISpS and iISpS are given based on indefinite Lyapunov functions. The practical stability analysis is accomplished with the help of scalar practical stable functions. Then, a construction of ISpS Lyapunov function for a class of non-autonomous evolutions equations is provided in Hilbert spaces. We propose the ISpS Lyapunov methodology to make it suitable for the analysis of ISpS w.r.t. inputs from -spaces. Furthermore, we illustrate the theory with an example of a semi-linear reaction-diffusion equation.

Blow-up time analysis of parabolic equations with variable nonlinearities

The blow-up of a class of nondegenerate parabolic equations in a bounded domain will be considered in the sense that the nonnegative diffusion coefficient allowed us to discuss our blow-up results differently. We prove blow-up results and give a new lower and upper bounds of the blow-up time to a class of semilinear reaction-diffusion equations and a nonlinear equation governed by the -Laplacian, for .

Asymptotic behavior for parabolic equations with interior degeneracy

The long time behavior of a class of degenerate parabolic equations in a bounded domain will be considered in the sense that the nonnegative diffusion coefficient a(x) is allowed to vanish in a set of positive measure in the interior of the domain. We prove decay rates for a class of semilinear reaction diffusion equations and a nonlinear equation governed by the p-Laplacian, for p>2.

An inverse source problem of a semilinear time-fractional reaction–diffusion equation

In this paper, we investigate an inverse problem of determining the time-dependent source coefficient in a semilinear reaction–diffusion equation involving the Caputo fractional time derivative of order in the case of nonlocal boundary and integral overdetermination conditions. We prove the existence and uniqueness of the classical solution by Fourier analysis and the iteration method. Moreover, we show the continuous dependence of this solution upon the additional data of the inverse problem.

Analysis and implementation of a computational technique for a coupled system of two singularly perturbed parabolic semilinear reaction–diffusion equations having discontinuous source terms

This article constructs and analyzes a numerical method for a time-dependent weakly coupled system of two singularly perturbed semilinear reaction–diffusion equations. The source terms in both equations have discontinuities in the spatial variables along the interface x=d,d∈Ω≔(0,1). The highest order spatial derivatives in the first and second equations are multiplied by positive perturbation parameters ɛ1 and ɛ2, respectively, which could be arbitrarily small. In the solution of the considered problem, boundary and interior layers appear in narrow regions of the domain. According to the boundary and interior layers, the solution is decomposed into regular and singular components, and precise bounds on the solution and its derivatives are given. The domain is discretized using an appropriate Shishkin mesh. For the mesh points which are not on the interface, the problem is discretized using a central difference method in space and backward Euler in time; for the mesh points which are on the interface, a special finite difference scheme is constructed. Parameters-uniform ((ɛ1,ɛ2)-uniform) error estimates in “maximum norm” have been obtained. It is proved that the method is parameters-uniformly convergent of first-order in time and almost second-order in space concerning perturbation parameters. Numerical experiments are conducted to demonstrate the efficiency of the method.

Approximate controllability of the semilinear reaction-diffusion equation governed by a multiplicative control

<p style='text-indent:20px;'>In this paper we are concerned with the approximate controllability of a multidimensional semilinear reaction-diffusion equation governed by a multiplicative control, which is locally distributed in the reaction term. For a given initial state we provide sufficient conditions on the desirable state to be approximately reached within an arbitrarily small time interval. Our approaches are based on linear semigroup theory and some results on uniform approximation with smooth functions.</p>

Well-posedness of singular-degenerate porous medium type equations and application to biofilm models

We show the well-posedness for a large class of degenerate parabolic equations with an additional singularity and mixed Dirichlet-Neumann boundary conditions on bounded Lipschitz domains. The proof is based on an L1-contraction result. In addition, we analyze systems where degenerate equations are coupled to semilinear reaction diffusion equations. This setting includes mathematical models for biofilm growth which are the motivation for our analysis.

Exact controllability of a semilinear reaction–diffusion equation governed by a bilinear control

This paper investigates the exact controllability of a semilinear reaction–diffusion equation governed by a multiplicative control. Here, the reaction term leads to an unbounded control operator. We will first study the well-posedness and then proceed to the question of exact steering for the system at hand. Our approaches are based on a fixed point argument and the linearization method. The obtained results are further analysed through a numerical example.

Pointwise error estimates for a system of two singularly perturbed time‐dependent semilinear reaction–diffusion equations

We present a finite difference method for a system of two singularly perturbed initial‐boundary value semilinear reaction–diffusion equations. The highest order derivatives are multiplied by small perturbation parameters of different magnitudes. The problem is discretized using a central difference scheme in space and backward difference scheme in time on a Shishkin mesh. The convergence analysis has been given, and it has been established that the method enjoys almost second‐order parameter‐uniform convergence in space and first‐order in time. Numerical experiments are conducted to demonstrate the efficiency of the method.

A new two-grid mixed finite element analysis of semi-linear reaction–diffusion equation

A new mixed finite element method is established for solving the semi-linear reaction–diffusion equation based on two-grid discretization. Here, a symmetric positive definite mixed finite element method is constructed for semi-linear problem, and the corresponding error estimates in L2 and Lp-norms are considered. Then, the two-grid techniques with and without one Newton iteration correction are applied for the proposed mixed element procedure to improve the efficiency of computation. The corresponding convergence of the proposed algorithms is analyzed, and the asymptotically optimal approximation is achieved with the relation H=O(h12). Finally, numerical examples are provided to confirm theoretical results.

奇异摄动反应扩散方程的后验误差估计及自适应算法

An adaptive grid method for singularly perturbed semilinear reactiondiffusion equations was studied. The equation was discretized by means of the upwind finite difference scheme on an arbitrary nonuniform mesh. A posteriori error estimation of the presented numerical scheme was built. In turn, the posteriori error bound was derived, and the adaptive grid generation algorithm was designed. Numerical experiments prove the effectiveness of the proposed adaptive grid method.

Hyers-Ulam stability of a nonautonomous semilinear equation with fractional diffusion

AbstractIn this paper, we study the Hyers-Ulam stability of a nonautonomous semilinear reaction-diffusion equation. More precisely, we consider a nonautonomous parabolic equation with a diffusion given by the fractional Laplacian. We see that such a stability is a consequence of a Gronwall-type inequality.

A fully discrete [formula omitted]-method for solving semi-linear reaction–diffusion equations with time-variable delay

In this paper, a fully discrete θ-method with 0≤θ≤1 is suggested to solve the initial–boundary value problem of semi-linear reaction–diffusion equations with time-variable delay. Under some appropriate conditions, a novel global stability criterion of the method is derived and it is shown that this method has the computational accuracy O(τ2+h2)(resp.O(τ+h2)) when θ=12(resp.θ≠12), where h and τ denote spatial and temporal stepsizes, respectively. Moreover, with some numerical experiments, the theoretical accuracy and global stability of the method are further illustrated.

Maximum Norm a Posteriori Error Estimation for a System of Singularly Perturbed Semilinear Reaction-Diffusion Equations

In this paper, a system of singularly perturbed semilinear reaction-diffusion equations is discretized on an arbitrary nonuniform mesh using an upwind finite difference scheme. Based on the quadratic polynomial interpolation function, a second-order maximum norm a posteriori error estimation is derived. Then, this bound is used to design an adaptive grid generation algorithm. Numerical results are given to verify the effectiveness of the presented adaptive grid method.<br />

Random attractors for semilinear reaction-diffusion equation with distribution derivatives and multiplicative noise on \(\mathbb{R}^{n}\)

Random attractors for semilinear reaction-diffusion equation with distribution derivatives and multiplicative noise on \(\mathbb{R}^{n}\)

Continuity of a Semi-Linear Fractional Reaction-Diffusion Equation System with Non-Local

Continuity of a Semi-Linear Fractional Reaction-Diffusion Equation System with Non-Local

Controllability of some bilinear and semilinear parabolic problems

Abstract We present in this paper a survey of recent results on the controllability of the parabolic system governed by bilinear control. We first discuss the problem of global controllability which corresponds to the question of whether the solution of the system can be driven to a given state at a some finite time by means of a control. We give some results on the global controllability of bilinear and semilinear reaction-diffusion equations. After this we introduce the case of partial controllability with the control acting on a subregion of the domain. Illustrative examples are also provided.

A posteriori error estimates for the monodomain model in cardiac electrophysiology

We consider the monodomain model, a system of a parabolic semilinear reaction–diffusion equation coupled with a nonlinear ordinary differential equation, arising from the (simplified) mathematical description of the electrical activity of the heart. We derive a posteriori error estimators accounting for different sources of error (space/time discretization and linearization). We prove reliability and efficiency (this latter under a suitable assumption) of the error indicators. Finally, numerical experiments assess the validity of the theoretical results.

Stable blowup for the supercritical Yang–Mills heat flow

In this paper, we consider the heat flow for Yang–Mills connections on $\mathbb{R}^5 \times SO(5)$. In the $SO(5)$-equivariant setting, the Yang–Mills heat equation reduces to a single semilinear reaction-diffusion equation for which an explicit self-similar blowup solution was found by Weinkove [“Singularity formation in the Yang-Mills flow”, Calc. Var. Partial Differential Equations, 19(2):211–220, 2004]. We prove the nonlinear asymptotic stability of this solution under small perturbations. In particular, we show that there exists an open set of initial conditions in a suitable topology such that the corresponding solutions blow up in finite time and converge to a non-trivial self-similar blowup profile on an unbounded domain. Convergence is obtained in suitable Sobolev norms and in $L^{\infty}$.