S. H. Rasouli
This paper is concerned with the existence of positive solutions for a fractional population model with the homogeneous Dirichlet condition on the exterior of a bounded domain. The approach is based on the sub-super solutions method. Our results generalize some recent results in the literature.
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We first formulate an inverse problem for a linear fractional Lamé system. We determine the Lamé parameters from exterior partial measurements of the Dirichlet-to-Neumann map. We further study an inverse obstacle problem as well as an inverse problem for a nonlinear fractional Lamé system. Our arguments are based on the unique continuation property for the fractional operator as well as the associated Runge approximation property.
We establish the well-posedness of an initial-boundary value problem for a general class of time-fractional, advection-diffusion-reaction equations, allowing space- and time-dependent coefficients as well as initial data that may have low regularity. Our analysis relies on novel energy methods in combination with a fractional Gronwall inequality and properties of fractional integrals.
We study the time asymptotic propagation of solutions to the reaction-diffusion cooperative systems with fractional diffusion. We prove that the propagation speed is exponential in time, and we find the precise exponent of propagation. This exponent depends on the smallest index of the fractional laplacians and on the principal eigenvalue of the matrix DF(0) where F is the reaction term. We also note that this speed does not depend on the space direction.
Tools for stability analysis of fractional reaction diffusion systems
本文针对时间分数阶导数阶数在(0,1)的抽象分数阶反应扩散方程证明线性化原理，并将其应用于具体方程，得到分数阶情形下的图灵不稳定性对应结果。
We consider time-fractional parabolic equations with a Caputo time derivative of order $α\in(0,1)$. For such equations, we give an elementary proof of the weak maximum principle under no assumptions on the sign of the reaction coefficient. This proof is also extended for weak solutions, as well as for various types of boundary conditions and variable-coefficient variable-order multiterm time-fractional parabolic equations.
On a fractional $p$-Laplacian problem in the whole space and with singular reaction
研究全空间上带有奇异反应项的分数阶 $p$-Laplacian 问题，利用截断论证、变分方法和先验估计，证明了正弱解的存在性及其在无穷远处逐点衰减。
We introduce the study of isolated singularities for a semilinear equation involving the fractional Laplacian. In conformal geometry, it is equivalent to the study of singular metrics with constant fractional curvature. Our main ideas are: first, to set the problem into a natural geometric framework, and second, to perform some kind of phase portrait study for this non-local ODE.
We study a nonlinear elliptic equation driven by the degenerate fractional p-Laplacian, with Dirichlet type condition and a jumping reaction, i.e., (p-1)-linear both at infinity and at zero but with different slopes crossing the principal eigenvalue. Under two different sets of hypotheses, entailing different types of asymmetry, we prove the existence of at least two nontrivial solutions. Our method is based on degree theory for monotone operators and nonlinear fractional spectral theory.
We study a Dirichlet problem driven by the degenerate fractional p-Laplacian and involving a nonlinear reaction, which depends on a positive parameter. The reaction is assumed to be (p-1)-sublinear near the origin and (p-1)-superlinear at infinity (including the concave-convex case). Following a variational approach based on a combination of critical point theory and suitable truncation techniques, we prove a bifurcation-type result for the existence of positive solutions.
In this article, we will analyze the existence of Peregrine type solutions for the fractional diffusion reaction equation by applying Splitting-type methods. These functions that have two main characteristics, they are direct sum of functions of periodic type and functions that tend to zero at infinity. Global existence results are obtained for each particular characteristic, for then finally combining both results.
Optimal Estimates on the Propagation of Reactions with Fractional Diffusion
本文研究了带分数阶扩散的反应方程，在不存在行波的情形下首次获得了波前解传播的最优界，且结果适用于局部化初始数据的传播。
In this paper, we consider a time-fractional reaction-diffusion system with the same nonlinearities of the Newton-Leipnik chaotic system. Through analytical tools and numerical results, we derive sufficient conditions for the asymptotic stability of the proposed model and show the existence of chaos. We also propose a nonlinear synchronization controller for a pair of systems and establish the local and global asymptotic convergence of the trajectories by means of fractional stability theory and the Lyapunov method.
Convergence of solutions for a reaction-diffusion problem with fractional Laplacian
本文研究无界区域上带分数阶Laplacian的非局部反应扩散方程，证明分数阶参数趋于1时其解收敛于经典标准Laplacian方程的解，还探究了全局吸引子存在性，为该类问题的解收敛性研究提供了首个相关结果。
We investigate the behavior of the time derivatives of the solution to a linear time-fractional, advection-diffusion-reaction equation, allowing space- and time-dependent coefficients as well as initial data that may have low regularity. Our focus is on proving estimates that are needed for the error analysis of numerical methods. The nonlocal nature of the fractional derivative creates substantial difficulties compared with the case of a classical parabolic PDE. In our analysis, we rely on novel energy methods in combination with a fractional Gronwall inequality and certain properties of frac…
We provide an asymptotic analysis of a fractional Fisher-KPP type equation in periodic non-connected 1-dimensional media with Dirichlet conditions outside the domain. After demonstrating the existence and uniqueness of a non-trivial bounded stationary state $n\_+$ , we prove that the stable state $n\_+$ invades the unstable state 0 with a speed which is exponential in time.
This paper is concerned with diffusion-reaction equations where the classical diffusion term, such as the Laplacian operator, is replaced with a singular integral term, such as the fractional Laplacian operator. As far as the reaction term is concerned, we consider bistable non-linearities. After properly rescaling (in time and space) these integro-differential evolution equations, we show that the limits of their solutions as the scaling parameter goes to zero exhibit interfaces moving by anisotropic mean curvature. The singularity and the unbounded support of the potential at stake are both …
We consider a Dirichlet type problem for a nonlinear, nonlocal equation driven by the degenerate fractional p-Laplacian, whose reaction combines a sublinear term depending on a positive parameter and an asymmetric perturbation (superlinear at positive infinity, at most linear at negative infinity). By means of critical point theory and Morse theory, we prove that, for small enough values of the parameter, such problem admits at least four nontrivial solutions: two positive, one negative, and one nodal. As a tool, we prove a Brezis-Oswald type comparison result.
We provide an asymptotic analysis of a non-local Fisher-KPP type equation in periodic media and with a non-local stable operator of order $α$ $\in$ (0, 2). We perform a long time-long range scaling in order to prove that the stable state invades the unstable state with a speed which is exponential in time.
Fractional Dirichlet problems with singular and non-locally convective reaction
本文研究由分数阶$(p,q)$-Laplacian驱动、反应项具有弱奇异性和非局部对流性的Dirichlet问题，利用上下解方法结合变分技巧、截断论证与不动点理论证明了正弱解的存在性。
In this paper, we study the problem of finding the solution of a multi-dimensional time fractional reactiondiffusion equation with nonlinear source from the final value data. We prove that the present problem is not well-posed. Then regularized problems are constructed using the truncated expansion method (in the case of two-dimensional) and the quasi-boundary value method (in the case of multi-dimensional). Finally, convergence rates of the regularized solutions are given and investigated numerically.
We analyze the fundamental solution of a time-fractional problem, establishing existence and uniqueness in an appropriate functional space. We also focus on the one-dimensional spatial setting in the case in which the time-fractional exponent is equal to, or larger than, $\frac12$. In this situation, we prove that the speed of invasion of the fundamental solution is at least `almost of square root type', namely it is larger than~$ct^β$ for any given~$c>0$ and~$β\in\left(0,\frac12\right)$.
Asymptotic behavior of the fundamental solution of space-time fractional equations with a reaction term
本文研究带反应项的时空分数阶偏微分方程，利用从属原理刻画其基本解的入侵速度，拓展了此前仅适用于一维空间、部分分数阶参数的相关结果，适用范围覆盖任意维数、任意分数阶参数及任意多项式入侵速度。
Global Solutions of Coupled Nonlocal Parabolic Systems Arising from Reversible Chemical Reactions
本文通过Lyapunov函数和最大原理研究了源自可逆化学反应的耦合非局部抛物系统全局解的存在性。
In this paper, we propose the invariant subspace approach to find exact solutions of time-fractional partial differential equations (PDEs) with time delay. An algorithmic approach of finding invariant subspaces for the generalized non-linear time-fractional reaction-diffusion equations with time delay is presented. We show that the fractional reaction-diffusion equations with time delay admit several invariant subspaces which further yields several distinct analytical solutions. We also demonstrate how to derive exact solutions for time-fractional PDEs with multiple time delays. Finally, we ex…
We prove a uniqueness theorem for the obstacle problem for linear equations involving the fractional Laplacian with zero Dirichlet exterior condition. The problem under consideration arises as the limit of some logistic-type equations. Our result extends (and slightly strengthens) the known corresponding results for the classical Laplace operator with zero boundary condition. Our proof, as compared with the known proof for the classical Laplace operator, is entirely new, and is based on the probabilistic potential theory. Its advantage is that it may be applied to a wide class of integro-diffe…
This work investigates how we can extend the invariant subspace method to two-dimensional time-fractional non-linear PDEs. More precisely, the systematic study has been provided for constructing the various dimensions of the invariant subspaces for the two-dimensional time-fractional generalized convection-reaction-diffusion-wave equation along with the initial conditions for the first time. Additionally, the special types of the above-mentioned equation are discussed through this method separately such as reaction-diffusion-wave equation, convection-diffusion-wave equation and diffusion-wave …
In this paper, Belousov-Zhabotinsky (B-Z) reaction model with Caputo fractional time derivative is investigated by the fractional reduced differential transform method (FRDTM) methods, an iterative technique. The outcome using FRDTM method reveals an efficiency with high accuracy and minimal computations for numerical solutions. Moreover, the solution profiles which demonstrate the behavior of the obtained result are presented.
Well-Posedness for Fractional Reaction-Diffusion Systems with Mass Dissipation in $\mathbb R^N$
本文研究全空间中带质量耗散的分数阶反应扩散系统的适定性，证明了二次非线性项下解的全局唯一有界性，以及超二次非线性项满足中间和条件时解的全局存在性，拓展了相关理论到非局部扩散无界域情形。
The quantitative convergence to equilibrium for reaction-diffusion systems arising from complex balanced chemical reaction networks with mass action kinetics is studied by using the so-called entropy method. In the first part of the paper, by deriving explicitly the entropy dissipation, we show that for complex balanced systems without boundary equilibria, each trajectory converges exponentially fast to the unique complex balance equilibrium. Moreover, a constructive proof is proposed to explicitly estimate the rate of convergence in the special case of a cyclic reaction. In the second part of…
This paper deals with the investigation of the computational solutions of an unified fractional reaction-diffusion equation, which is obtained from the standard diffusion equation by replacing the time derivative of first order by the generalized fractional time-derivative defined by Hilfer (2000), the space derivative of second order by the Riesz-Feller fractional derivative and adding the function phi(x,t) which is a nonlinear function overning reaction. The solution is derived by the application of the Laplace and Fourier transforms in a compact and closed form in terms of the H-function. T…
The global boundedness and asymptotic behavior are investigate for the solution of time-space fractional non-local reaction-diffusion equation (TSFNRDE) $$ \frac{\partial^{α}u}{\partial t^{α}}=-(-Δ)^{s} u+μu^{2}(1-kJ*u)-γu, \qquad(x,t)\in\mathbb{R}^{N}\times(0,+\infty),$$ where $s\in(0,1),α\in(0,1), N \leq 2$. The operator $\partial_{t}^{α}$ is the Caputo fractional derivative, which $-(-Δ)^{s}$ is the fractional Laplacian operator. For appropriate assumptions on $J$, it is proved that for homogeneous Dirichlet boundary condition, this problem admits a global bounded weak solution for $N=1$, w…
We prove a contraction in $L^1$ property for the solutions of a nonlinear reaction--diffusion system whose special cases include intercellular transport as well as reversible chemical reactions. Assuming the existence of stationary solutions we show that the solutions stabilize as $t$ tends to infinity. Moreover, in the special case of linear reaction terms, we prove the existence and the uniqueness (up to a multiplicative constant) of the stationary solution.
Topological singularities arising from fractional-gradient energies
该研究证明平面正则区域上经标度的Ginzburg-Landau型泛函相对于雅可比平坦收敛Γ收敛于涡旋型能量，通过与含Riesz位势的标准泛函比较得紧性与Γ-下极限，用连接截断类涡旋函数的直接论证得Γ-上极限。
This article provides techniques of raising the regularity of fractional order equations and resolves fundamental questions on the one-dimensional homogeneous boundary-value problem of skewed (double-sided) fractional diffusion advection reaction equation (FDARE) with variable coefficients on the bounded interval. The existence of the true (classical) solution together with norm estimation is established and the precise regularity bound is found; also, the structure of the solution is unraveled, capturing the essence of regularity, singularity, and other features of the solution. The key analy…
Analysis and numerics of traveling waves for asymmetric fractional reaction-diffusion equations
本文研究非对称分数反应扩散方程中行波的存在性、唯一性及稳定性，提出数值方法并探讨其扩展应用。
Long-time behaviour of a nonlocal stochastic fractional reaction--diffusion equation arising in tumour dynamics
本文研究了源于肿瘤动力学的非局部随机分数反应-扩散方程的长期行为，通过分析噪声对肿瘤扩散和灭绝的影响，揭示了异常扩散与分数噪声的相互作用机制。
We perform an asymptotic analysis of models of population dynamics with a fractional Laplacian and local or nonlocal reaction terms. The first part of the paper is devoted to the long time/long range rescaling of the fractional Fisher-KPP equation. This rescaling is based on the exponential speed of propagation of the population. In particular we show that the only role of the fractional Laplacian in determining this speed is at the initial layer where it determines the thickness of the tails of the solutions. Next, we show that such rescaling is also possible for models with non-local reactio…
For the non-local space-time reaction-diffusion equation involving fractional $p$-Laplacian \begin{equation*} \begin{cases} \frac{\partial^{α}u}{\partial t^{α}}+(-Δ)_{p}^{s} u=μu^{2}(1-kJ*u)-γu,&(x,t)\in\mathbb{R}^{N}\times(0,T)\\ u(x,0)=u_{0}(x),& x\in\mathbb{R}^{N} \end{cases} \end{equation*} $μ>0 ,k>0,γ\geq 1,α\in(0,1),s\in(0,1),1<p$, we consider for $N\leq2$ the problem of finding a global boundedness of the weak solution by virtue of Gagliardo-Nirenberg inequality and fractional Duhamel's formula. Moreover, we prove such weak solution converge to $0$ exponentially or locally uniformly as …
We prove that the steady state of a class of multidimensional reaction-diffusion systems is asymptotically stable at the intersection of unweighted space and exponentially weighted Sobolev spaces, and pay particular attention to a special case, namely, systems of equations that arise in combustion theory. The steady-state solutions considered here are the end states of the traveling fronts associated with the systems, and thus the present results complement recent papers \cite{GLS1, GLS2, GLS3, GLSR, GLY} that study the stability of traveling fronts.
We will look at reaction-diffusion type equations of the following type, $$\partial^β_tV(t,x)=-(-Δ)^{α/2} V(t,x)+I^{1-β}_t[V(t,x)^{1+η}].$$ We first study the equation on the whole space by making sense of it via an integral equation. Roughly speaking, we will show that when $0<η\leqη_c$, there is no global solution other than the trivial one while for $η>η_c$, non-trivial global solutions do exist. We also study the equation on a bounded domain with Dirichlet boundary condition and show that the presence of the time derivative induces a significant change in the behaviour of the solution.
Uniqueness of traveling fronts in premixed flames with stepwise ignition-temperature kinetics and fractional reaction order
本文研究具有点火温度动力学与分数反应阶数的预混火焰反应-扩散系统，证明行波锋面解在平移意义下唯一，并通过渐近与数值分析推测传播速度随各参数递减。
We study the nonlinear fractional reaction diffusion equation $\partial_{t}u + (-Δ)^{s} u= f(t,x,u)$, $s\in(0,1)$ in a bounded domain $Ω$ together with Dirichlet boundary conditions on $\R^N \setminus Ω$. We prove asymptotic symmetry of nonnegative globally bounded solutions in the case where the underlying data obeys some symmetry and monotonicity assumptions. More precisely, we assume that $Ω$ is symmetric with respect to reflection at a hyperplane, say ${x_1=0}$, and convex in the $x_1$-direction, and that the nonlinearity $f$ is even in $x_1$ and nonincreasing in $|x_1|$. Under rather weak…
We study a Dirichlet-type boundary value problem for a pseudodifferential equation driven by the fractional Laplacian, proving the existence of three nonzero solutions. When the reaction term is sublinear at infinity, we apply the second deformation theorem and spectral theory. When the reaction term is superlinear at infinity, we apply the mountain pass theorem and Morse theory.
Existence and nonexistence of solutions for fractional elliptic equations arising from closed MEMS model
本文研究了源于封闭MEMS模型的分数椭圆方程解的存在性与不存在性，分析了边界衰减对解和拉紧电压的影响。
A class of parabolic reaction-diffusion systems governed by spectral fractional Laplacians : Analysis and numerical simulations
本文研究了一类由谱分数阶拉普拉斯算子支配的抛物反应扩散系统，证明了强解的全局存在性，并通过数值模拟探讨了相关理论问题。
On the solutions of coupled nonlinear time-fractional diffusion-reaction system with time delays
本文采用不变子空间法，在 Riemann–Liouville 与 Caputo 两种时间分数阶导数下研究含多时间延迟的耦合非线性扩散-反应系统，构造有限维不变向量空间并求其 IBVPs 的广义可分解析解。
Martingale Solutions of Fractional Stochastic Reaction-Diffusion Equations Driven by Superlinear Noise
本文证明了一类具有任意阶多项式增长伪单调漂移和超线性扩散项的随机方程鞅解的存在性，并应用于分数阶随机反应扩散方程，利用伪单调技术和Skorokhod-Jakubowski表示定理通过Galerkin近似取极限。
The initial inverse problem of finding solutions and their initial values ($t = 0$) appearing in a general class of fractional reaction-diffusion equations from the knowledge of solutions at the final time ($t = T$). Our work focuses on the existence and regularity of mild solutions in two cases: \begin{itemize} \item[--] The first case: The nonlinearity is globally Lipschitz and uniformly bounded which plays important roles in PDE theories, and especially in numerical analysis. \item[--] The second case: The nonlinearity is locally critical which widely arises from the Navier-Stokes, Schrödin…
Motivated by a nonlocal free boundary problem, we study uniform properties of solutions to a singular perturbation problem for a boundary-reaction-diffusion equation, where the reaction term is of combustion type. This boundary problem is related to the fractional Laplacian. After an optimal uniform Hölder regularity is shown, we pass to the limit to study the free boundary problem it leads to.
Similarity solutions for a class of Fractional Reaction-Diffusion equation
本研究针对半直线上带Riemann-Liouville分数阶导数的分数阶反应扩散方程，通过推导标度对称条件、引入相似变量将方程约化为分数阶常微分方程，匹配已知可解模型确定精确可解系统，给出相关示例并验证其对应超扩散系统。
The fractional reaction diffusion equation u_t + Au = g(u) is discussed, where A is a fractional differential operator on the real line with order αbetween 0 and 2, the C^1 function g vanishes at 0 and 1, and either g is non-negative on (0,1) or g < 0 near 0. In the case of non-negative g, it is shown that solutions with initial support on the positive half axis spread into the left half axis with unbounded speed if g satisfies some weak growth condition near 0 in the case α> 1, or if g is merely positive on a sufficiently large interval near 1 in the case α< 1. On the other hand, it shown tha…
Global existence for vector valued fractional reaction-diffusion equations
本文研究了无限维分数非自主反应扩散方程的初值问题，通过时间分割方法证明了解的全局存在性，并分析了解的渐进行为。
In this paper we consider an extension of the results in shape differentiation of semilinear equations with smooth nonlinearity presented in J.I. Díaz and D. Gómez-Castro: An Application of Shape Differentiation to the Effectiveness of a Steady State Reaction-Diffusion Problem Arising in Chemical Engineering. Electron. J. Differ. Equations in 2015 to the case in which the nonlinearities might be less smooth. Namely we will show that Gateaux shape derivatives exists when the nonlinearity is only Lipschitz continuous, and we will give a definition of the derivative when the nonlinearity has a bl…
In this paper we are interested in a degenerate parabolic system of reaction-diffusion equations arising in biology when studying cell adhesion at the protein level. In this modeling the unknown is the couple of the distribution laws of the freely diffusing proteins and of the fixed ones. Under sufficient conditions on the aggregation and unbinding probabilities, we prove the existence of solution of the considered system, as well as their positivity, boundedness and uniqueness. Moreover, we discuss the stability of the equilibrium solution. Finally, we show that the simpler and particular cho…
We consider the following fractional semilinear Neumann problem on a smooth bounded domain $Ω\subset\mathbb{R}^n$, $n\geq2$, $$\begin{cases} (-\varepsilonΔ)^{1/2}u+u=u^{p},&\hbox{in}~Ω,\\ \partial_νu=0,&\hbox{on}~\partialΩ,\\ u>0,&\hbox{in}~Ω, \end{cases}$$ where $\varepsilon>0$ and $1<p<(n+1)/(n-1)$. This is the fractional version of the semilinear Neumann problem studied by Lin--Ni--Takagi in the late 80's. The problem arises by considering steady states of the Keller--Segel model with nonlocal chemical concentration diffusion. Using the semigroup language for the extension method and variat…
By employing the N-barrier method developed in the paper, we establish a new N-barrier maximum principle for diffusive Lotka-Volterra systems of two competing species. As an application of this maximum principle, we show under certain conditions, the existence and nonexistence of traveling waves solutions for systems of three competing species. In addition, new $(1,0,0)$-$(u^{\ast},v^{\ast},0)$ waves are given in terms of the tanh function provided that the parameters satisfy certain conditions.
We deal with a Dirichlet problem driven by the degenerate fractional p-Laplacian and involving a nonlinear reaction which satisfies, among other hypotheses, a (p-1)-linear growth at infinity with non-resonance above the first eigenvalue. The energy functional governing the problem is thus noncoercive. Thus we focus on the behavior of the reaction near the origin, assuming that it has a (p-1)-sublinear growth at zero, vanishes at three points, and satisfies a reverse Ambrosetti-Rabinowitz condition. Under such assumptions, by means of critical point theory and Morse theory, and using suitably t…
Fractional elliptic reaction-diffusion systems with coupled gradient terms and different diffusion
本文研究带不同阶分数阶扩散与耦合梯度项的非局部椭圆系统非负解的存在性，通过加权Lebesgue空间下的新精细估计得到存在性结果，兼具理论意义。
The global boundedness and asymptotic behavior are investigated for the solutions of a nonlocal time fractional reaction-diffusion equation (NTFRDE) $$ \frac{\partial^{α}u}{\partial t^{α}}=Δu+μu^{2}(1-kJ*u)-γu, \qquad(x,t)\in\mathbb{R}^{N}\times(0,+\infty)$$ with $0<α<1,β, μ,k>0,N\leq 2$ and $u(x,0)=u_{0}(x)$. Under appropriate assumptions on $J$ and the property of time fractional derivative, it is proved that for any nonnegative and bounded initial conditions, the problem has a global bounded classical solution if $k^{*}=0$ for $N=1$ or $k^{*}=(μC^{2}_{GN}+1)η^{-1}$ for $N=2$, where $C_{GN}$…
Well-posedness and large deviations of fractional McKean-Vlasov stochastic reaction-diffusion equations on unbounded domains
本文研究无界区域上带任意次多项式漂移的分数阶McKean-Vlasov随机反应扩散方程，证明了其适定性与受控方程解的强收敛性，最终通过弱收敛方法在弱假设下建立了大偏差原理。
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