Global smooth solutions for a class of parabolic integrodifferential equations

Engler, Hans

doi:10.1090/s0002-9947-96-01472-9

Cited by 19 publications

(2 citation statements)

References 38 publications

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“…[23,Theorem 7.22] along with the anisotropic Sobolev embedding for q = n + 3 (see e.g. [12,Lemma A3]), implies that there exists a constant C A,δ,n,R > 0 that depends only on A, δ, n and R such that, for any (t, x) ∈ (t 0 , +∞) × R n , |∇u(t, x)| ≤ C A,δ,n,R u L q ([t−δ,t]×B R (x)) + u(1 − u) L q ([t−δ,t]×B R (x))…”

Section: Proof Of Lemma 41mentioning

confidence: 99%

Propagation in a Fisher-KPP equation with non-local advection

Hamel

Henderson

2020

Journal of Functional Analysis

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We investigate the influence of a general non-local advection term of the form K * u to propagation in the one-dimensional Fisher-KPP equation. This model is a generalization of the Keller-Segel-Fisher system. When K ∈ L 1 (R), we obtain explicit upper and lower bounds on the propagation speed which are asymptotically sharp and more precise than previous works. When K ∈ L p (R) with p > 1 and is non-increasing in (−∞, 0) and in (0, +∞), we show that the position of the "front" is of order O(t 1/p ) if p < ∞ and O(e λt ) for some λ > 0 if p = ∞ and K(+∞) > 0. We use a wide range of techniques in our proofs.

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Section: Proof Of Lemma 41mentioning

confidence: 99%

Propagation in a Fisher-KPP equation with non-local advection

Hamel

Henderson

2020

Journal of Functional Analysis

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“…For existence results related to integrodifferential equations of non convolution type or to integrodifferential equations whose source includes a delay term see [10,11,24]. For recent developments in non-Fickian diffusion and its applications to viscoelastic materials, we refer to [14,16] and references therein.…”

Section: Introductionmentioning

confidence: 99%

Convergence of nonlinear integrodifferential reaction-diffusion equations via Mosco×Γ-convergence

Anza Hafsa,

Mandallena,

Michaille

2022

Annales Mathématiques Blaise Pascal

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We study the convergence of sequences of nonlinear integrodifferential reaction-diffusion equations when the Fickian terms belong to a class of convex functionals defined on a Hilbert space, equipped with the Mosco-convergence, and the non Fickian terms belong to a class of convex functionals, whose restrictions to a compactly embedded subspace are equipped with the Γ-convergence. As a consequence we prove a homogenization theorem for this class under a stochastic homogenization framework.

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On a fractional partial differential equation with dominating linear part

Gripenberg

Londen²,

Prüß

1997

Math. Meth. Appl. Sci.

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It is proved that there is a (weak) solution of the equation ut=a*uxx+b*g(ux)x+f, on ℝ+ (where * denotes convolution over (−∞, t)) such that ux is locally bounded. Emphasis is put on having the assumptions on the initial conditions as weak as possible. The kernels a and b are completely monotone and if a(t)=t−α, b(t)=t−β, and g(ξ)∼sign(ξ)∣ξ∣γ for large ξ, then the main assumption is that α>(2γ+2)/(3γ+1)β+(2γ−2)/(3γ+1). © 1997 by B. G. Teubner Stuttgart–John Wiley & Sons Ltd.

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Global smooth solutions for a class of parabolic integrodifferential equations

Cited by 19 publications

References 38 publications

Propagation in a Fisher-KPP equation with non-local advection

Propagation in a Fisher-KPP equation with non-local advection

Convergence of nonlinear integrodifferential reaction-diffusion equations via Mosco×Γ-convergence

On a fractional partial differential equation with dominating linear part

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