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

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
Teresa Pierantozzi
Independent Model Validation Unit
1 Churchill Place, Barclays, E14 5HP, UK
email: terpiera@gmail.com
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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