Electron. J. Differential Equations,
Vol. 2016 (2016), No. 239, pp. 1-13.
Finite time extinction for nonlinear fractional evolution equations
and related properties
Jesus Ildefonso Diaz, Teresa Pierantozzi, Luis Vazquez
Abstract:
The finite time extinction phenomenon (the solution reaches an
equilibrium after a finite time) is peculiar to certain nonlinear
problems whose solutions exhibit an asymptotic behavior entirely
different from the typical behavior of solutions associated to
linear problems. The main goal of this work is twofold. Firstly,
we extend some of the results known in the literature to the case
in which the ordinary time derivative is considered jointly with a
fractional time differentiation. Secondly, we consider the limit
case when only the fractional derivative remains. The latter is
the most extraordinary case, since we prove that the finite time
extinction phenomenon still appears, even with a non-smooth
profile near the extinction time.
Some concrete examples of quasi-linear partial differential
operators are proposed. Our results can also be applied in the
framework of suitable nonlinear Volterra integro-differential
equations.
Submitted January 8, 2015. Published August 31, 2016.
Math Subject Classifications: 47J35, 26A33, 47J20.
Key Words: Nonlinear evolution equations; fractional derivative;
finite time extinction.
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Jesús Ildefonso Díaz
Instituto de Matemática Interdisciplinar and Departamento
de Matemática Aplicada
Facultad de Ciencias Matemáticas
Universidad Complutense de Madrid (UCM), Spain
email: jidiaz@ucm.es
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Teresa Pierantozzi
Independent Model Validation Unit
1 Churchill Place, Barclays, E14 5HP, UK
email: terpiera@gmail.com
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Luis Vázquez
Instituto de Matemática Interdisciplinar and Departamento de
Matemática Aplicada
Facultad de Informática
Universidad Complutense de Madrid
28040 Madrid, Spain
email: lvazquez@fdi.ucm.es
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