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

12 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

Access to Document

10.3934/dcdss.2014.7.767

Fingerprint Dive into the research topics of 'Global solutions for a nonlinear integral equation with a generalized heat kernel'. Together they form a unique fingerprint.

Cite this

@article{b3da37de2b7d4c2aafb2c2d0310a27db,

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",

month = aug,

doi = "10.3934/dcdss.2014.7.767",

language = "English",

volume = "7",

pages = "767--783",

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

issn = "1937-1632",

publisher = "American Institute of Mathematical Sciences",

number = "4",

}

TY - JOUR

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

UR - http://www.scopus.com/inward/record.url?scp=84897745490&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84897745490&partnerID=8YFLogxK

U2 - 10.3934/dcdss.2014.7.767

DO - 10.3934/dcdss.2014.7.767

M3 - Article

AN - SCOPUS:84897745490

VL - 7

SP - 767

EP - 783

JO - Discrete and Continuous Dynamical Systems - Series S

JF - Discrete and Continuous Dynamical Systems - Series S

SN - 1937-1632

IS - 4

ER -

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

Abstract

Keywords

ASJC Scopus subject areas

Access to Document

Other files and links

Cite this