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Decay estimates for nonlocal problems via energy estimates
 Journal de Mathematiques Pures et Applique’es
"... Abstract. In this paper we study the applicability of energy methods to obtain bounds for the asymptotic decay of solutions to nonlocal diffusion problems. With these energy methods we can deal with nonlocal problems that not necessarily involve a convolution, that is, of the form ut(x, t) = Rd G(x ..."
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Cited by 18 (6 self)
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Abstract. In this paper we study the applicability of energy methods to obtain bounds for the asymptotic decay of solutions to nonlocal diffusion problems. With these energy methods we can deal with nonlocal problems that not necessarily involve a convolution, that is, of the form ut(x, t) = Rd G(x − y)(u(y, t) − u(x, t)) dy. For example, we will consider equations like, ut(x, t) =
Travelling fronts in asymmetric nonlocal reaction diffusion equation: The bistable and ignition case. Preprint du CMM
, 2006
"... ABSTRACT. This paper is devoted to the study of the travelling front solutions which appear in a nonlocal reactiondiffusion equations of the form ..."
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Cited by 13 (4 self)
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ABSTRACT. This paper is devoted to the study of the travelling front solutions which appear in a nonlocal reactiondiffusion equations of the form
Traveling waves for the KellerSegel system with Fisher birth terms
, 2007
"... We consider the traveling wave problem for the one dimensional KellerSegel system with a birth term of either a Fisher/KPP type or with a truncation for small population densities. We prove that there exists a solution under some stability conditions on the coefficients which enforce an upper bound ..."
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Cited by 11 (1 self)
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We consider the traveling wave problem for the one dimensional KellerSegel system with a birth term of either a Fisher/KPP type or with a truncation for small population densities. We prove that there exists a solution under some stability conditions on the coefficients which enforce an upper bound on the solution and ˙ H 1 (R) estimates. Solutions in the KPP case are built as a limit of traveling waves for the truncated birth rates (similar to ignition temperature in combustion theory). We also discuss some general bounds and long time convergence for the solution of the Cauchy problem and in particular linear and nonlinear stability of the nonzero steady state.
On bounded positive stationary solutions for a nonlocal FisherKPP equation, preprint
, 2014
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MAXIMUM AND ANTIMAXIMUM PRINCIPLES FOR SOME NONLOCAL DIFFUSION OPERATORS
"... Abstract. In this work we consider the maxiumum and antimaximum principles for the nonlocal Dirichlet problem J ∗ u − u+ λu+ h = RN J(x − y)u(y) dy − u(x) + λu(x) + h(x) = 0 in a bounded domain Ω, with u(x) = 0 in RN \ Ω. The kernel J in the convolution is assumed to be a continuous, compactly sup ..."
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Cited by 3 (2 self)
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Abstract. In this work we consider the maxiumum and antimaximum principles for the nonlocal Dirichlet problem J ∗ u − u+ λu+ h = RN J(x − y)u(y) dy − u(x) + λu(x) + h(x) = 0 in a bounded domain Ω, with u(x) = 0 in RN \ Ω. The kernel J in the convolution is assumed to be a continuous, compactly supported nonnegative function with unit integral. We prove that for λ < λ1(Ω), the solution verifies u> 0 in Ω if h ∈ L2(Ω), h ≥ 0, while for λ> λ1(Ω), and λ close to λ1(Ω), the solution verifies u < 0 in Ω, provided∫ Ω h(x)φ(x) dx> 0, h ∈ L∞(Ω). This last assumption is also shown to be optimal. The “Neumann ” version of the problem is also analyzed. 1.
TRANSITION FRONTS FOR INHOMOGENEOUS FISHERKPP REACTIONS AND NONLOCAL DIFFUSION
"... Abstract. We prove existence of and construct transition fronts for a class of reactiondiffusion equations with spatially inhomogeneous FisherKPP type reactions and nonlocal diffusion. Our approach is based on finding these solutions as perturbations of appropriate solutions to the linearization o ..."
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Cited by 3 (0 self)
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Abstract. We prove existence of and construct transition fronts for a class of reactiondiffusion equations with spatially inhomogeneous FisherKPP type reactions and nonlocal diffusion. Our approach is based on finding these solutions as perturbations of appropriate solutions to the linearization of the PDE at zero. Our work extends a method introduced by one of us to study such questions in the case of classical diffusion.
A LOGISTIC EQUATION WITH REFUGE AND NONLOCAL DIFFUSION
"... Abstract. In this work we consider the nonlocal stationary nonlinear problem (J ∗ u)(x) − u(x) = −λu(x) + a(x)up(x) in a domain Ω, with the Dirichlet boundary condition u = 0 in RN \ Ω and p> 1. The kernel J involved in the convolution (J ∗ u)(x) = ∫RN J(x − y)u(y) dy is a smooth, compactly su ..."
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Cited by 2 (0 self)
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Abstract. In this work we consider the nonlocal stationary nonlinear problem (J ∗ u)(x) − u(x) = −λu(x) + a(x)up(x) in a domain Ω, with the Dirichlet boundary condition u = 0 in RN \ Ω and p> 1. The kernel J involved in the convolution (J ∗ u)(x) = ∫RN J(x − y)u(y) dy is a smooth, compactly supported nonnegative function with unit integral, while the weight a(x) is assumed to be nonnegative and is allowed to vanish in a smooth subdomain Ω0 of Ω. Both when a(x) is positive and when it vanishes in a subdomain, we completely discuss the issues of existence and uniqueness of positive solutions, as well as their behavior with respect to the parameter λ. 1.
Planar traveling waves for nonlocal dispersion equation with monostable nonlinearity, Discret
 Contin. Dyn. Syst. A
"... (Communicated by Masaharu Taniguchi) Abstract. In this paper, we study a class of nonlocal dispersion equation with monostable nonlinearity in ndimensional spaceut − J ∗ u+ u+ d(u(t, x)) = Rn fβ(y)b(u(t − τ, x − y))dy, u(s, x) = u0(s, x), s ∈ [−τ, 0], x ∈ Rn, where the nonlinear functions d(u) and ..."
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(Communicated by Masaharu Taniguchi) Abstract. In this paper, we study a class of nonlocal dispersion equation with monostable nonlinearity in ndimensional spaceut − J ∗ u+ u+ d(u(t, x)) = Rn fβ(y)b(u(t − τ, x − y))dy, u(s, x) = u0(s, x), s ∈ [−τ, 0], x ∈ Rn, where the nonlinear functions d(u) and b(u) possess the monostable characters like FisherKPP type, fβ(x) is the heat kernel, and the kernel J(x) satisfies Ĵ(ξ) = 1 − Kξα + o(ξα) for 0 < α ≤ 2 and K> 0. After establishing the existence for both the planar traveling waves φ(x · e+ ct) for c ≥ c ∗ (c ∗ is the critical wave speed) and the solution u(t, x) for the Cauchy problem, as well as the comparison principles, we prove that, all noncritical planar wavefronts φ(x · e + ct) are globally stable with the exponential convergence rate t−n/αe−µτ t for µτ> 0, and the critical wavefronts φ(x · e + c∗t) are globally stable in the algebraic form t−n/α, and these rates are optimal. As application,we also automatically obtain the stability of traveling wavefronts to the classical FisherKPP dispersion equations. The adopted approach is Fourier transform and the weighted energy method with a suitably selected weight function.
Decay estimates for nonlinear nonlocal diffusion problems in the whole space
"... Abstract. In this paper we obtain bounds for the decay rate in the Lr(Rd)norm for the solutions to a nonlocal and nolinear evolution equation, namely, ut(x, t) = ..."
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Abstract. In this paper we obtain bounds for the decay rate in the Lr(Rd)norm for the solutions to a nonlocal and nolinear evolution equation, namely, ut(x, t) =
Asymptotic behaviour of solutions to evolution problems with nonlocal diffusion
"... Evolution equations and numerical analysis ..."
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