Finite time extinction for nonlinear fractional evolution equations and related properties

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.

Finite time extinction for nonlinear fractional evolution equations and related properties.

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.

Recent landscape evolution at the "Barranco del Río Dulce Natural Park" (Spain). Landscape units and mapping

Landscape units based on the visual features of the relief have been distinguished in the “Barranco del Río Dulce Natural Park” (Spain). These units are geomorphic entities composed of several elementary landforms and characterized by a visual internal homogeneity, and contrast with other landscape units in their location, height, profile and gradients, reflecting their different evolution and genesis. Landscape units bear some subjectivity in their definition and in their boundary location due to the overlapping of geomorphic processes along time. Visual, compositional and conventional boundaries have been used for mapping. Neogene landscape evolution mainly occurred through thrust faulting at the Iberian Ranges-Tagus Basin boundary, driving tectonic uplift and erosion of the Ranges and correlative sedimentation in the Basin. Erosion of the Ranges occurred with the development of planation surfaces, leaving minor isolated reliefs in the upland plains landscape. The lowering of the base level, caused by the endorheic–exorheic transition of the Tagus Basin in the Pliocene, originates fluvial entrenchment and water table lowering with development of the first fluvial valleys and the capture of karstic depressions. Two subsequent phases of renewed fluvial incision (Pleistocene) lead to abandonment of some Pliocene valleys, fluvial captures, and development and reincision of tributaries