Inhomogeneous parabolic equations on unbounded metric measure spaces

Kenneth John Falconer; Jiaxin Hu; Yuhua Sun

doi:10.1017/S0308210511000539

Inhomogeneous parabolic equations on unbounded metric measure spaces

Kenneth John Falconer, Jiaxin Hu, Yuhua Sun

Pure Mathematics

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Abstract

We study the inhomogeneous semilinear parabolic equation

u_t= Δu + u^p + f(x),

with source term f independent of time and subject to f(x) ≥ 0 and with u(0, x) = φ(x) ≥ 0, for the very general setting of a metric measure space. By establishing Harnack-type inequalities in time t and some powerful estimates, we give sufficient conditions for non-existence, local existence and global existence of weak solutions, depending on the value of p relative to a critical exponent.

Original language	English
Pages (from-to)	1003-1025
Journal	Proceedings of the Royal Society of Edinburgh, Section A: Mathematics
Volume	142
Issue number	5
Early online date	20 Sept 2012
DOIs	https://doi.org/10.1017/S0308210511000539
Publication status	Published - Oct 2012

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(c) 2012 The Royal Society of Edinburgh
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title = "Inhomogeneous parabolic equations on unbounded metric measure spaces",

abstract = "We study the inhomogeneous semilinear parabolic equation ut = Δu + up + f(x), with source term f independent of time and subject to f(x) ≥ 0 and with u(0, x) = φ(x) ≥ 0, for the very general setting of a metric measure space. By establishing Harnack-type inequalities in time t and some powerful estimates, we give sufficient conditions for non-existence, local existence and global existence of weak solutions, depending on the value of p relative to a critical exponent.",

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N2 - We study the inhomogeneous semilinear parabolic equation ut = Δu + up + f(x), with source term f independent of time and subject to f(x) ≥ 0 and with u(0, x) = φ(x) ≥ 0, for the very general setting of a metric measure space. By establishing Harnack-type inequalities in time t and some powerful estimates, we give sufficient conditions for non-existence, local existence and global existence of weak solutions, depending on the value of p relative to a critical exponent.

AB - We study the inhomogeneous semilinear parabolic equation ut = Δu + up + f(x), with source term f independent of time and subject to f(x) ≥ 0 and with u(0, x) = φ(x) ≥ 0, for the very general setting of a metric measure space. By establishing Harnack-type inequalities in time t and some powerful estimates, we give sufficient conditions for non-existence, local existence and global existence of weak solutions, depending on the value of p relative to a critical exponent.

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UR - https://www.scopus.com/pages/publications/84870042560

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DO - 10.1017/S0308210511000539

M3 - Article

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EP - 1025

JO - Proceedings of the Royal Society of Edinburgh, Section A: Mathematics

JF - Proceedings of the Royal Society of Edinburgh, Section A: Mathematics

IS - 5

ER -

Inhomogeneous parabolic equations on unbounded metric measure spaces

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