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Inhomogeneous parabolic equations on unbounded metric measure spaces

Published online by Cambridge University Press:  20 September 2012

Kenneth J. Falconer ,
Jiaxin Hu  and
Yuhua Sun
Kenneth J. Falconer
Affiliation:
Mathematical Institute, University of St Andrews, North Haugh, St Andrews, Fife KY16 9SS, UK (kjf@st-andrews.ac.uk)
Jiaxin Hu
Affiliation:
Department of Mathematical Sciences, Tsinghua University, Beijing 100084, People's Republic of China (hujiaxin@mail.tsinghua.edu.cn; sunyh08@mails.tsinghua.edu.cn)
Yuhua Sun
Affiliation:
Department of Mathematical Sciences, Tsinghua University, Beijing 100084, People's Republic of China (hujiaxin@mail.tsinghua.edu.cn; sunyh08@mails.tsinghua.edu.cn)
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Abstract

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We study the inhomogeneous semilinear parabolic equation

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.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 142 , Issue 5 , October 2012 , pp. 1003 - 1025
Copyright
Copyright © Royal Society of Edinburgh 2012