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Critical exponents for a semilinear parabolic equation with variable reaction

Published online by Cambridge University Press:  20 September 2012

R. Ferreira
Affiliation:
Departamento de Matemática Aplicada, Universidad Complutense de Madrid, 28040 Madrid, Spain ([email protected])
A. de Pablo
Affiliation:
Departamento de Matemáticas, Universidad Carlos III de Madrid, 28911 Leganés, Spain ([email protected])
M. Pérez-LLanos
Affiliation:
Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049 Madrid, Spain ([email protected])
J. D. Rossi
Affiliation:
Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Ciudad Universitaria, Pabellón 1, 1428 Buenos Aires, Argentina ([email protected])

Abstract

We study the blow-up phenomenon for non-negative solutions to the following parabolic problem:

where 0 < p− = min p ≤ p(x) ≤ max p = p+ is a smooth bounded function. After discussing existence and uniqueness, we characterize the critical exponents for this problem. We prove that there are solutions with blow-up in finite time if and only if p+ > 1.

When Ω = ℝN we show that if p− > 1 + 2/N, then there are global non-trivial solutions, while if 1 < p− ≤ p+ ≤ 1 + 2/N, then all solutions to the problem blow up in finite time. Moreover, in the case when p− < 1 + 2/N < p+, there are functions p(x) such that all solutions blow up in finite time and functions p(x) such that the problem possesses global non-trivial solutions.

When Ω is a bounded domain we prove that there are functions p(x) and domains Ω such that all solutions to the problem blow up in finite time. On the other hand, if Ω is small enough, then the problem possesses global non-trivial solutions regardless of the size of p(x).

Type
Research Article
Copyright
Copyright © Royal Society of Edinburgh 2012

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