2,032 research outputs found
Existence of weak solutions for general nonlocal and nonlinear second-order parabolic equations
In this article, we provide existence results for a general class of nonlocal
and nonlinear second-order parabolic equations. The main motivation comes from
front propagation theory in the cases when the normal velocity depends on the
moving front in a nonlocal way. Among applications, we present level-set
equations appearing in dislocations' theory and in the study of Fitzhugh-Nagumo
systems
Monotone systems involving variable-order nonlocal operators
In this paper, we study the existence and uniqueness of bounded viscosity solutions for parabolic Hamilton-Jacobi monotone systems in which the diffusion term is driven by variable-order nonlocal operators whose kernels depend on the space-time variable. We prove the existence of solutions via Perron's method, and considering Hamiltonians with linear and superlinear nonlinearities related to their gradient growth we state a comparison principle for bounded sub and supersolutions. Moreover, we present steady-state large time behavior with an exponential rate of convergence
Uniqueness Results for Nonlocal Hamilton-Jacobi Equations
We are interested in nonlocal Eikonal Equations describing the evolution of
interfaces moving with a nonlocal, non monotone velocity. For these equations,
only the existence of global-in-time weak solutions is available in some
particular cases. In this paper, we propose a new approach for proving
uniqueness of the solution when the front is expanding. This approach
simplifies and extends existing results for dislocation dynamics. It also
provides the first uniqueness result for a Fitzhugh-Nagumo system. The key
ingredients are some new perimeter estimates for the evolving fronts as well as
some uniform interior cone property for these fronts
Second-Order Elliptic Integro-Differential Equations: Viscosity Solutions' Theory Revisited
The aim of this work is to revisit viscosity solutions' theory for
second-order elliptic integro-differential equations and to provide a general
framework which takes into account solutions with arbitrary growth at infinity.
Our main contribution is a new Jensen-Ishii's Lemma for integro-differential
equations, which is stated for solutions with no restriction on their growth at
infinity. The proof of this result, which is of course a key ingredient to
prove comparison principles, relies on a new definition of viscosity solution
for integro-differential equation (equivalent to the two classical ones) which
combines the approach with test-functions and sub-superjets
On the properties of solutions of a cross-diffusion system with nonlinear boundary flux
In this paper, based on a self-similar analysis and the method of standard equations, the properties of a nonlinear cross-diffusion system coupled via nonlocal boundary conditions are studied. We are investigated the qualitative properties of solutions of a nonlinear system of parabolic equations of cross-diffusion in a medium coupled with nonlinear boundary conditions. It is proved that for certain values of the numerical parameters of the nonlinear cross-diffusion system of parabolic equations coupled via nonlinear boundary conditions, they may not have global solutions in time. Based on a self-similar analysis and the principle of comparing solutions, a critical exponent of the Fujita type and a critical value of global solvability are established. Using the comparison theorem, upper bounds for global solutions and lower bounds for blow-up solutions are obtained
Formation of clumps and patches in self-aggregation of finite size particles
New model equations are derived for dynamics of self-aggregation of
finite-size particles. Differences from standard Debye-Huckel and Keller-Segel
models are: a) the mobility of particles depends on the locally-averaged
particle density and b) linear diffusion acts on that locally-averaged particle
density. The cases both with and without diffusion are considered here.
Surprisingly, these simple modifications of standard models allow progress in
the analytical description of evolution as well as the complete analysis of
stationary states. When remains positive, the evolution of collapsed
states in our model reduces exactly to finite-dimensional dynamics of
interacting particle clumps. Simulations show these collapsed (clumped) states
emerging from smooth initial conditions, even in one spatial dimension. If
vanishes for some averaged density, the evolution leads to spontaneous
formation of \emph{jammed patches} (weak solution with density having compact
support). Simulations confirm that a combination of these patches forms the
final state for the system.Comment: 38 pages, 8 figures; submitted to Physica
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