Filters

2014 - 201932020 - 20211

More filters

2019 - 201932020 - 20221
Reset filters

Settings

Subject: fractional operators × WGL Institute: WIAS ×

Search Results

Now showing 1 - 4 of 4
  • Thumbnail Image
    Item
    Well-posedness for a class of phase-field systems modeling prostate cancer growth with fractional operators and general nonlinearities
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2021) Colli, Pierluigi; Gilardi, Gianni; Sprekels, Jürgen
    This paper deals with a general system of equations and conditions arising from a mathematical model of prostate cancer growth with chemotherapy and antiangiogenic therapy that has been recently introduced and analyzed (see [P. Colli et al., Mathematical analysis and simulation study of a phase-field model of prostate cancer growth with chemotherapy and antiangiogenic therapy effects, Math. Models Methods Appl. Sci. bf 30 (2020), 1253--1295]). The related system includes two evolutionary operator equations involving fractional powers of selfadjoint, nonnegative, unbounded linear operators having compact resolvents. Both equations contain nonlinearities and in particular the equation describing the dynamics of the tumor phase variable has the structure of a Allen--Cahn equation with double-well potential and additional nonlinearity depending also on the other variable, which represents the nutrient concentration. The equation for the nutrient concentration is nonlinear as well, with a term coupling both variables. For this system we design an existence, uniqueness and continuous dependence theory by setting up a careful analysis which allows the consideration of nonsmooth potentials and the treatment of continuous nonlinearities with general growth properties.
  • Thumbnail Image
    Item
    Homogenization and Orowan's law for anisotropic fractional operators of any order
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2014) Patrizi, Stefania; Valdinoci, Enrico
    We consider an anisotropic Lévy operator Is of any order s 2 (0, 1) and we consider the homogenization properties of an evolution equation. The scaling properties and the effective Hamiltonian that we obtain is different according to the cases s < 1/2 and s > 1/2. In the isotropic onedimensional case, we also prove a statement related to the so-called Orowans law, that is an appropriate scaling of the effective Hamiltonian presents a linear behavior.
  • Thumbnail Image
    Item
    Chaotic orbits for systems of nonlocal equations
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Dipierro, Serena; Patrizi, Stefania; Valdinoci, Enrico
    We consider a system of nonlocal equations driven by a perturbed periodic potential. We construct multibump solutions that connect one integer point to another one in a prescribed way. In particular, heteroclinc, homoclinic and chaotic trajectories are constructed. This is the first attempt to consider a nonlocal version of this type of dynamical systems in a variational setting and the first result regarding symbolic dynamics in a fractional framework.
  • Thumbnail Image
    Item
    Continuity and density results for a one-phase nonlocal free boundary problem
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Dipierro, Serena; Valdinoci, Enrico
    We consider a one-phase nonlocal free boundary problem obtained by the superposition of a fractional Dirichlet energy plus a nonlocal perimeter functional. We prove that the minimizers are Hölder continuous and the free boundary has positive density from both sides. For this, we also introduce a new notion of fractional harmonic replacement in the extended variables and we study its basic properties.