The Experts below are selected from a list of 208827 Experts worldwide ranked by ideXlab platform
Ming Mei - One of the best experts on this subject based on the ideXlab platform.
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existence and stability of traveling waves for degenerate reaction diffusion Equation with time delay
Journal of Nonlinear Science, 2018Co-Authors: Rui Huang, Ming Mei, Chunhua Jin, Jingxue YinAbstract:This paper deals with the existence and stability of traveling wave solutions for a degenerate reaction–diffusion Equation with time delay. The degeneracy of spatial diffusion together with the effect of time delay causes us the essential difficulty for the existence of the traveling waves and their stabilities. In order to treat this case, we first show the existence of smooth- and sharp-type traveling wave solutions in the case of \(c\ge c^*\) for the degenerate reaction–diffusion Equation without delay, where \(c^*>0\) is the critical wave speed of smooth traveling waves. Then, as a small perturbation, we obtain the existence of the smooth non-critical traveling waves for the degenerate diffusion Equation with small time delay \(\tau >0\). Furthermore, we prove the global existence and uniqueness of \(C^{\alpha ,\beta }\)-solution to the time-delayed degenerate reaction–diffusion Equation via compactness analysis. Finally, by the weighted energy method, we prove that the smooth non-critical traveling wave is globally stable in the weighted \(L^1\)-space. The exponential convergence rate is also derived.
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traveling wavefronts for time delayed reaction diffusion Equation ii nonlocal nonlinearity
Journal of Differential Equations, 2009Co-Authors: Ming Mei, Chikun Lin, Chitien LinAbstract:Abstract In this paper, we study a class of time-delayed reaction–diffusion Equation with local nonlinearity for the birth rate. For all wavefronts with the speed c > c ∗ , where c ∗ > 0 is the critical wave speed, we prove that these wavefronts are asymptotically stable, when the initial perturbation around the traveling waves decays exponentially as x → − ∞ , but the initial perturbation can be arbitrarily large in other locations. This essentially improves the stability results obtained by Mei, So, Li and Shen [M. Mei, J.W.-H. So, M.Y. Li, S.S.P. Shen, Asymptotic stability of traveling waves for the Nicholson's blowflies Equation with diffusion, Proc. Roy. Soc. Edinburgh Sect. A 134 (2004) 579–594] for the speed c > 2 D m ( e p − d m ) with small initial perturbation and by Lin and Mei [C.-K. Lin, M. Mei, On travelling wavefronts of the Nicholson's blowflies Equations with diffusion, submitted for publication] for c > c ∗ with sufficiently small delay time r ≈ 0 . The approach adopted in this paper is the technical weighted energy method used in [M. Mei, J.W.-H. So, M.Y. Li, S.S.P. Shen, Asymptotic stability of traveling waves for the Nicholson's blowflies Equation with diffusion, Proc. Roy. Soc. Edinburgh Sect. A 134 (2004) 579–594], but inspired by Gourley [S.A. Gourley, Linear stability of travelling fronts in an age-structured reaction–diffusion population model, Quart. J. Mech. Appl. Math. 58 (2005) 257–268] and based on the property of the critical wavefronts, the weight function is carefully selected and it plays a key role in proving the stability for any c > c ∗ and for an arbitrary time-delay r > 0 .
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stability of strong travelling waves for a non local time delayed reaction diffusion Equation
Proceedings of The Royal Society A: Mathematical Physical and Engineering Sciences, 2008Co-Authors: Ming MeiAbstract:The paper is concerned with a non-local time-delayed reaction–diffusion Equation. We prove the (time) asymptotic stability of a travelling wavefront without a smallness assumption on its wavelength, i.e. the so-called strong wavefront, by means of the (technical) weighted energy method, when the initial perturbation around the wave is small. The exponential convergent rate is also given. Selection of a suitable weight plays a key role in the proof.
Juan Soler - One of the best experts on this subject based on the ideXlab platform.
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pattern formation in a flux limited reaction diffusion Equation of porous media type
Inventiones Mathematicae, 2016Co-Authors: Juan Calvo, Juan Campos, Vicent Caselles, Oscar Sanchez, Juan SolerAbstract:A non-linear PDE featuring flux limitation effects together with those of the porous media Equation (non-linear Fokker–Planck) is presented in this paper. We analyze the balance of such diverse effects through the study of the existence and qualitative behavior of some admissible patterns, namely traveling wave solutions, to this singular reaction–diffusion Equation. We show the existence and qualitative behavior of different types of traveling waves: classical profiles for wave speeds high enough, and discontinuous waves that are reminiscent of hyperbolic shock waves when the wave speed lowers below a certain threshold. Some of these solutions are of particular relevance as they provide models by which the whole solution (and not just the bulk of it, as it is the case with classical traveling waves) spreads through the medium with finite speed.
Wenqiang Zhao - One of the best experts on this subject based on the ideXlab platform.
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strong begin document l 2 l gamma cap h_0 1 end document continuity in initial data of nonlinear reaction diffusion Equation in any space dimension
Electronic Research Archive, 2020Co-Authors: Hongyong Cui, Peter E Kloeden, Wenqiang ZhaoAbstract:In this paper we study the continuity in initial data of a classical Reaction-Diffusion Equation with arbitrary \begin{document}$ p>2 $\end{document} order nonlinearity and in any space dimension \begin{document}$ N \geqslant 1 $\end{document} . It is proved that the weak solutions can be \begin{document}$ (L^2, L^\gamma\cap H_0^1) $\end{document} -continuous in initial data for arbitrarily large \begin{document}$ \gamma \geqslant 2 $\end{document} (independent of the physical parameters of the system), i.e., can converge in the norm of any \begin{document}$ L^\gamma\cap H_0^1 $\end{document} as the corresponding initial values converge in \begin{document}$ L^2 $\end{document} . In fact, the system is shown to be \begin{document}$ (L^2, L^\gamma\cap H_0^1) $\end{document} -smoothing in a H \begin{document}$ \ddot{\rm o} $\end{document} lder way. Applying this to the global attractor we find that, with external forcing only in \begin{document}$ L^2 $\end{document} , the attractor \begin{document}$ \mathscr{A} $\end{document} attracts bounded subsets of \begin{document}$ L^2 $\end{document} in the norm of any \begin{document}$ L^\gamma\cap H_0^1 $\end{document} , and that every translation set \begin{document}$ \mathscr{A}-z_0 $\end{document} of \begin{document}$ \mathscr{A} $\end{document} for any \begin{document}$ z_0\in \mathscr{A} $\end{document} is a finite dimensional compact subset of \begin{document}$ L^\gamma\cap H_0^1 $\end{document} . The main technique we employ is a combination of a Moser iteration and a decomposition of the nonlinearity, by which the interpolation inequalities are avoided and the new continuity result is obtained without any restrictions on the order \begin{document}$ p>2 $\end{document} of the nonlinearity and the space dimension \begin{document}$ N \geqslant 1 $\end{document} .
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strong l 2 l gamma cap h_0 1 continuity in initial data of nonlinear reaction diffusion Equation in any space dimension
arXiv: Dynamical Systems, 2019Co-Authors: Hongyong Cui, Peter E Kloeden, Wenqiang ZhaoAbstract:In this paper, we study the continuity in initial data of a classical Reaction-Diffusion Equation with arbitrary $p>2$ order nonlinearity and in any space dimension $N\geq 1$. It is proved that the weak solutions can be $(L^2, L^\gamma\cap H_0^1)$-continuous in initial data for any $\gamma\geq 2$ (independent of the physical parameters of the system), i.e., can converge in the norm of any $L^\gamma\cap H_0^1$ as the corresponding initial values converge in $L^2$. Applying this to the global attractor we find that, with external forcing only in $ L^2$, the attractor $\mathscr{A}$ attracts bounded subsets of $L^2$ in the norm of any $L^\gamma\cap H_0^1$, and that every translation set $\mathscr{A}-z_0$ of $\mathscr{A}$ for any $z_0 \in \mathscr{A}$ is a finite dimensional compact subset of $L^\gamma\cap H_0^1$. The main technique we employ is a combination of the mathematical induction and a decomposition of the nonlinearity by which the continuity result is strengthened to $(L^2, L^\gamma \cap H_0^1)$-continuity and, since interpolation inequalities are avoided, the restriction on space dimension is removed.
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l2 lp random attractors for stochastic reaction diffusion Equation on unbounded domains
Nonlinear Analysis-theory Methods & Applications, 2012Co-Authors: Wenqiang ZhaoAbstract:Abstract In this paper, the existence of ( L 2 , L p ) -random attractor is established for a stochastic reaction–diffusion Equation on the whole space R N . This random attractor is a compact and invariant tempered set which attracts every tempered random subset of L 2 in the topology of L p . The nonlinearity f is supposed to satisfy some growth of arbitrary order p − 1 , where p ≥ 2 . The ( L 2 , L p ) -asymptotic compactness of the random dynamical system is proved by an asymptotic a priori estimate of the unbounded part of solutions.
Wenping Chen - One of the best experts on this subject based on the ideXlab platform.
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finite difference spectral approximations for the distributed order time fractional reaction diffusion Equation on an unbounded domain
Journal of Computational Physics, 2016Co-Authors: Hu Chen, Wenping ChenAbstract:The numerical approximation of the distributed order time fractional Reaction-Diffusion Equation on a semi-infinite spatial domain is discussed in this paper. A fully discrete scheme based on finite difference method in time and spectral approximation using Laguerre functions in space is proposed. The scheme is unconditionally stable and convergent with order O ( ? 2 + Δ α 2 + N ( 1 - m ) / 2 ) , where ?, Δα, N, and m are the time-step size, step size in distributed-order variable, polynomial degree, and regularity in the space variable of the exact solution, respectively. A pseudospectral scheme is also proposed and analyzed. Some numerical examples are presented to demonstrate the efficiency of the proposed scheme.
Jinfeng Wang - One of the best experts on this subject based on the ideXlab platform.
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an h 1 galerkin mixed finite element method for time fractional reaction diffusion Equation
Journal of Applied Mathematics and Computing, 2015Co-Authors: Yang Liu, Jinfeng WangAbstract:In this article, an \(H^1\)-Galerkin mixed finite element (MFE) method for solving time fractional reaction–diffusion Equation is presented. The optimal time convergence order \(O(\varDelta t^{2-\alpha })\) and the optimal spatial rate of convergence in \(H^1\) and \(L^2\)-norms for variable \(u\) and its gradient \(\sigma \) are derived. Moreover, some numerical results are shown to support our theoretical analysis.
