The Debye system: existence and large time behavior of solutions

Biler, Piotr; Hebisch, Waldemar; Nadzieja, Tadeusz

doi:10.1016/0362-546x(94)90101-5

Cited by 243 publications

(192 citation statements)

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“…We refer the interested reader to [1,12,15,16,17,18,19,21] where other equations with fractal type diffusion have been considered. We also mention that analogous problems of (1.1) in bounded domains were studied in [2,3,4,5,6,7].…”

Section: B(u) = −C 1 μ(−δ)mentioning

confidence: 98%

“…In this case (1.1) can also be regarded as a simplification of the classical Keller-Segel model [14]. On the other hand, the repulsive case μ = 1 models the Brownian diffusion of charged particles with Coulomb repulsion (see [2]). The regime 0 < α < 2 was studied in [8] and it corresponds to the so-called anomalous diffusion which in probabilistic terms has a connection with stochastic equations driven by Lévy α-stable processes.…”

Section: B(u) = −C 1 μ(−δ)mentioning

confidence: 99%

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Exploding solutions for a nonlocal quadratic evolution problem

Liu¹,

Rodrigo²,

Zhang³

2010

Rev. Mat. Iberoam.

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We consider a nonlinear parabolic equation with fractional diffusion which arises from modelling chemotaxis in bacteria. We prove the wellposedness, continuation criteria and smoothness of local solutions. In the repulsive case we prove global wellposedness in Sobolev spaces. Finally in the attractive case, we prove that for a class of smooth initial data the L ∞ x -norm of the corresponding solution blows up in finite time. This solves a problem left open by Biler and Woyczyński [8].

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Section: B(u) = −C 1 μ(−δ)mentioning

confidence: 98%

Section: B(u) = −C 1 μ(−δ)mentioning

confidence: 99%

Exploding solutions for a nonlocal quadratic evolution problem

Liu¹,

Rodrigo²,

Zhang³

2010

Rev. Mat. Iberoam.

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“…[4]), and etc. We refer the reader to see [1][2][3]6,9,[11][12][13]17,18,21,22] and the references therein for previous works on this system of equations concerning existence of (large) weak solutions, (small and local) mild solutions, convergence rate estimates to stationary solutions of time-dependent solutions and other related topics.…”

Section: Introductionmentioning

confidence: 99%

Regularizing and decay rate estimates for solutions to the Cauchy problem of the Debye–Hückel system

Zhao

Cui

2011

Nonlinear Differ. Equ. Appl.

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Abstract. In this paper we establish some regularizing and decay rate estimates for mild solutions of the Debye-Hückel system. We prove that if the initial data belong to the critical Lebesgue spaceof the βth order spatial derivative of mild solutions are majorized by K1(K2|β|) |β| t − |β| 2 −1+ n 2q for some constants K1 and K2. These estimates particularly imply that mild solutions are analytic in the space variable, and provide decay estimates in the time variable for higher-order derivatives of mild solutions. We also prove that similar estimates also hold for mild solutions whose initial data belong to the critical homogeneous Besov spaceḂ −2+ n p p,∞ (R n ) ( n 2 < p < n). Mathematics Subject Classification (2010). 35B65, 35K45, 35K55.

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“…(2) Here ϕ : Ω → R is an unknown function from a bounded domain Ω of R n into R, n ≥ 2, f : R → R + is a given C 1 function and M > 0, p > 0 are given parameters. The physical motivations for the study of nonlocal elliptic problems come from statistical mechanics ( [2], [5], [6]), theory of electrolytes ( [4]), and theory of thermistors ( [7], [13]). …”

mentioning

confidence: 99%

“…The existence results can be proved using either the technique of sub-and supersolutions ( [3]), or variational methods ( [6], [8]), or topological methods ( [4], [10], [12], [15]), whereas the nonexistence results are a consequence of the Pohozaev identity ( [3]), or construction of some special subsolutions ( [3]). …”

mentioning

confidence: 99%