Global solutions for a nonlinear integral equation with a generalized heat kernel

Kazuhiro Ishige; Tatsuki Kawakami; Kanako Kobayashi

doi:10.3934/dcdss.2014.7.767

Global solutions for a nonlinear integral equation with a generalized heat kernel

Kazuhiro Ishige, Tatsuki Kawakami, Kanako Kobayashi

SCI - Mathematics

Research output: Contribution to journal › Article › peer-review

15 Citations (Scopus)

Abstract

We study the existence and the large time behavior of global-intime solutions of a nonlinear integral equation with a generalized heat kernel u(x, t) = ∫_RN G(x - y, t)φ(y)dy +∫^t₀ ZRN G(x - y, t - s)F(y, s, u(y, s), . . . ,δ^lu(y, s))dyds, where φ 2 ∈W^l,∞(R^N) and l ∈ {0, 1, . . . }. The arguments of this paper are applicable to the Cauchy problem for various nonlinear parabolic equations such as fractional semilinear parabolic equations, higher order semilinear parabolic equations and viscous Hamilton-Jacobi equations.

Original language	English
Pages (from-to)	767-783
Number of pages	17
Journal	Discrete and Continuous Dynamical Systems - Series S
Volume	7
Issue number	4
DOIs	https://doi.org/10.3934/dcdss.2014.7.767
Publication status	Published - 2014 Aug

Keywords

Generalized heat kernel
Global solutions
Nonlinear integral equation
Semilinear parabolic equations
Weak Lr space

ASJC Scopus subject areas

Analysis
Discrete Mathematics and Combinatorics
Applied Mathematics

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10.3934/dcdss.2014.7.767

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title = "Global solutions for a nonlinear integral equation with a generalized heat kernel",

abstract = "We study the existence and the large time behavior of global-intime solutions of a nonlinear integral equation with a generalized heat kernel u(x, t) = ∫RN G(x - y, t)φ(y)dy +∫t0 ZRN G(x - y, t - s)F(y, s, u(y, s), . . . ,δlu(y, s))dyds, where φ 2 ∈Wl,∞(RN) and l ∈ {0, 1, . . . }. The arguments of this paper are applicable to the Cauchy problem for various nonlinear parabolic equations such as fractional semilinear parabolic equations, higher order semilinear parabolic equations and viscous Hamilton-Jacobi equations.",

keywords = "Generalized heat kernel, Global solutions, Nonlinear integral equation, Semilinear parabolic equations, Weak Lr space",

author = "Kazuhiro Ishige and Tatsuki Kawakami and Kanako Kobayashi",

year = "2014",

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doi = "10.3934/dcdss.2014.7.767",

language = "English",

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pages = "767--783",

journal = "Discrete and Continuous Dynamical Systems - Series S",

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publisher = "American Institute of Mathematical Sciences",

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T1 - Global solutions for a nonlinear integral equation with a generalized heat kernel

AU - Ishige, Kazuhiro

AU - Kawakami, Tatsuki

AU - Kobayashi, Kanako

PY - 2014/8

Y1 - 2014/8

N2 - We study the existence and the large time behavior of global-intime solutions of a nonlinear integral equation with a generalized heat kernel u(x, t) = ∫RN G(x - y, t)φ(y)dy +∫t0 ZRN G(x - y, t - s)F(y, s, u(y, s), . . . ,δlu(y, s))dyds, where φ 2 ∈Wl,∞(RN) and l ∈ {0, 1, . . . }. The arguments of this paper are applicable to the Cauchy problem for various nonlinear parabolic equations such as fractional semilinear parabolic equations, higher order semilinear parabolic equations and viscous Hamilton-Jacobi equations.

AB - We study the existence and the large time behavior of global-intime solutions of a nonlinear integral equation with a generalized heat kernel u(x, t) = ∫RN G(x - y, t)φ(y)dy +∫t0 ZRN G(x - y, t - s)F(y, s, u(y, s), . . . ,δlu(y, s))dyds, where φ 2 ∈Wl,∞(RN) and l ∈ {0, 1, . . . }. The arguments of this paper are applicable to the Cauchy problem for various nonlinear parabolic equations such as fractional semilinear parabolic equations, higher order semilinear parabolic equations and viscous Hamilton-Jacobi equations.

KW - Generalized heat kernel

KW - Global solutions

KW - Nonlinear integral equation

KW - Semilinear parabolic equations

KW - Weak Lr space

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JF - Discrete and Continuous Dynamical Systems - Series S

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Global solutions for a nonlinear integral equation with a generalized heat kernel

Abstract

Keywords

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