scholarly journals On Global Attractors for a Class of Reaction-Diffusion Equations on Unbounded Domains with Some Strongly Nonlinear Weighted Term

2015 ◽  
Vol 2015 ◽  
pp. 1-10
Author(s):  
Jin Zhang ◽  
Chengkui Zhong
Keyword(s):  
Global Attractor ◽  
Equilibrium Points ◽  
Unbounded Domains ◽  
Reaction Diffusion ◽  
Index Theory ◽  
Global Attractors ◽  
Reaction Diffusion Equations ◽  
Reaction Diffusion Equation ◽  
Infinite Dimensional ◽  
Strongly Nonlinear

We consider the existence and properties of the global attractor for a class of reaction-diffusion equation∂u/∂t-Δu-u+κ(x)|u|p-2u+f(u)=0,  in  Rn×R+;  u(x,0)=u0(x),  in  Rn. Under some suitable assumptions, we first prove that the problem has a global attractorAinL2(Rn). Then, by using theZ2-index theory, we verify thatAis an infinite dimensional set and it contains infinite distinct pairs of equilibrium points.

scholarly journals On Lr-regularity of global attractors generated by strong solutions of reaction-diffusion equations

2016 ◽  
Vol 1 (2) ◽  
pp. 375-390 ◽  
Author(s):  
José Valero
Keyword(s):  
Nonlinear Term ◽  
Mild Solutions ◽  
Reaction Diffusion ◽  
Strong Solutions ◽  
Global Attractors ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Reaction Diffusion Equation ◽  
The Cauchy Problem ◽  
Weak Attractor

AbstractIn this paper we prove that the global attractor generated by strong solutions of a reaction-diffusion equation without uniqueness of the Cauchy problem is bounded in suitable Lr-spaces. In order to obtain this result we prove first that the concepts of weak and mild solutions are equivalent under an appropriate assumption.Also, when the nonlinear term of the equation satisfies a supercritical growth condition the existence of a weak attractor is established.

scholarly journals Regularity of random attractors for stochastic reaction-diffusion equations on unbounded domains

2015 ◽  
Vol 16 (01) ◽  
pp. 1650006 ◽  
Author(s):  
Bao Quoc Tang
Keyword(s):  
Diffusion Equation ◽  
Unbounded Domains ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Time Dependent ◽  
Diffusion Equations ◽  
Reaction Diffusion Equation ◽  
Random Attractors ◽  
External Forces ◽  
Stochastic Reaction

The existence of a unique random attractors in [Formula: see text] for a stochastic reaction-diffusion equation with time-dependent external forces is proved. Due to the presence of both random and non-autonomous deterministic terms, we use a new theory of random attractors which is introduced in [B. Wang, J. Differential Equations 253 (2012) 1544–1583] instead of the usual one. The asymptotic compactness of solutions in [Formula: see text] is established by combining “tail estimate” technique and some new estimates on solutions. This work improves some recent results about the regularity of random attractors for stochastic reaction-diffusion equations.

scholarly journals Attractor of reaction-diffusion equations in Banach spaces

2001 ◽  
Vol 2 (1) ◽  
pp. 77
Author(s):  
José Valero
Keyword(s):  
Fractal Dimension ◽  
Phase Space ◽  
Banach Spaces ◽  
Global Attractor ◽  
Differential Inclusions ◽  
Reaction Diffusion ◽  
Global Attractors ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Dissipative Operators

In this paper we prove first some abstract theorems on existence of global attractors for differential inclusions generated by w-dissipative operators. Then these results are applied to reaction-diffusion equations in which the Babach space L<sub>p </sub>is used as phase space. Finally, new results concerning the fractal dimension of the global attractor in the space L<sub>2</sub> are obtained.<br /><sub> </sub>

scholarly journals Asymptotic behavior for the semilinear reaction–diffusion equations with memory

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Jiangwei Zhang ◽  
Yongqin Xie ◽  
Qingqing Luo ◽  
Zhipiao Tang
Keyword(s):  
Global Attractor ◽  
Polynomial Growth ◽  
General Setting ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Reaction Diffusion Equation ◽  
Fading Memory ◽  
Evolutionary Equations ◽  
Asymptotic Compactness ◽  
With Memory

AbstractIn this paper, we consider the dynamics of a reaction–diffusion equation with fading memory and nonlinearity satisfying arbitrary polynomial growth condition. Firstly, we prove a criterion in a general setting as an alternative method (or technique) to the existence of the bi-spaces attractors for the nonlinear evolutionary equations (see Theorem 2.14). Secondly, we prove the asymptotic compactness of the semigroup on $L^{2}(\varOmega )\times L_{\mu }^{2}(\mathbb{R}; H_{0}^{1}( \varOmega ))$L2(Ω)×Lμ2(R;H01(Ω)) by using the contractive function, and the global attractor is confirmed. Finally, the bi-spaces global attractor is obtained by verifying the asymptotic compactness of the semigroup on $L^{p}( \varOmega )\times L_{\mu }^{2}(\mathbb{R}; H_{0}^{1}(\varOmega ))$Lp(Ω)×Lμ2(R;H01(Ω)) with initial data in $L^{2}(\varOmega )\times L_{\mu }^{2}(\mathbb{R}; H_{0} ^{1}(\varOmega ))$L2(Ω)×Lμ2(R;H01(Ω)).

Nonlinear solution of the reaction–diffusion equation using a two-step third–fourth-derivative block method

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Oluwaseun Adeyeye ◽  
Ali Aldalbahi ◽  
Jawad Raza ◽  
Zurni Omar ◽  
Mostafizur Rahaman ◽  
...  
Keyword(s):  
Diffusion Equation ◽  
Reaction Diffusion ◽  
Numerical Approach ◽  
Reaction Diffusion Equations ◽  
Reaction Diffusion Equation ◽  
Block Method ◽  
Wide Range ◽  
Higher Derivatives ◽  
Fourth Derivative ◽  
Improved Accuracy

AbstractThe processes of diffusion and reaction play essential roles in numerous system dynamics. Consequently, the solutions of reaction–diffusion equations have gained much attention because of not only their occurrence in many fields of science but also the existence of important properties and information in the solutions. However, despite the wide range of numerical methods explored for approximating solutions, the adoption of block methods is yet to be investigated. Hence, this article introduces a new two-step third–fourth-derivative block method as a numerical approach to solve the reaction–diffusion equation. In order to ensure improved accuracy, the method introduces the concept of nonlinearity in the solution of the linear model through the presence of higher derivatives. The method obtained accurate solutions for the model at varying values of the dimensionless diffusion parameter and saturation parameter. Furthermore, the solutions are also in good agreement with previous solutions by existing authors.

Existence of bistable waves for a nonlocal and nonmonotone reaction-diffusion equation

2019 ◽  
Vol 150 (2) ◽  
pp. 721-739
Author(s):  
Sergei Trofimchuk ◽  
Vitaly Volpert
Keyword(s):  
Diffusion Equation ◽  
A Priori Estimates ◽  
Travelling Waves ◽  
A Priori ◽  
Unbounded Domains ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equation ◽  
Nonlocal Nonlinearity ◽  
Estimates Of Solutions ◽  
Priori Estimates

AbstractReaction-diffusion equation with a bistable nonlocal nonlinearity is considered in the case where the reaction term is not quasi-monotone. For this equation, the existence of travelling waves is proved by the Leray-Schauder method based on the topological degree for elliptic operators in unbounded domains and a priori estimates of solutions in properly chosen weighted spaces.

scholarly journals A Modified Techniques of Fractional-Order Cauchy-Reaction Diffusion Equation via Shehu Transform

2021 ◽  
Vol 2021 ◽  
pp. 1-15
Author(s):  
Mounirah Areshi ◽  
A. M. Zidan ◽  
Rasool Shah ◽  
Kamsing Nonlaopon
Keyword(s):  
Fractional Order ◽  
Reaction Diffusion ◽  
Approximate Solutions ◽  
Transformation Method ◽  
Reaction Diffusion Equations ◽  
Reaction Diffusion Equation ◽  
Closed Form Solutions ◽  
Homotopy Perturbation ◽  
In Series ◽  
The Given

In this article, the iterative transformation method and homotopy perturbation transformation method are applied to calculate the solution of time-fractional Cauchy-reaction diffusion equations. In this technique, Shehu transformation is combined of the iteration and the homotopy perturbation techniques. Four examples are examined to show validation and the efficacy of the present methods. The approximate solutions achieved by the suggested methods indicate that the approach is easy to apply to the given problems. Moreover, the solution in series form has the desire rate of convergence and provides closed-form solutions. It is noted that the procedure can be modified in other directions of fractional order problems. These solutions show that the current technique is very straightforward and helpful to perform in applied sciences.

scholarly journals Regularity of Global Attractor for the Reaction-Diffusion Equation

2012 ◽  
Vol 2012 ◽  
pp. 1-16
Author(s):  
Hong Luo
Keyword(s):  
Sobolev Space ◽  
Diffusion Equation ◽  
Existence Theorem ◽  
Global Attractor ◽  
Reaction Diffusion ◽  
Iteration Procedure ◽  
Reaction Diffusion Equation ◽  
Bounded Subset ◽  
Regularity Estimates ◽  
Regularity Of Global Attractor

By using an iteration procedure, regularity estimates for the linear semigroups, and a classical existence theorem of global attractor, we prove that the reaction-diffusion equation possesses a global attractor in Sobolev spaceHkfor allk>0, which attracts any bounded subset ofHk(Ω) in theHk-norm.

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