Vanishing of solutions of diffusion equation with convection and absorption.
Variational analysis for the Black and Scholes equation with stochastic volatility
We propose a variational analysis for a Black and Scholes equation with stochastic volatility. This equation gives the price of a European option as a function of the time, of the price of the underlying asset and of the volatility when the volatility is a function of a mean reverting Orstein-Uhlenbeck process, possibly correlated with the underlying asset. The variational analysis involves weighted Sobolev spaces. It enables to prove qualitative properties of the solution, namely a maximum principle...
Variational Analysis for the Black and Scholes Equation with Stochastic Volatility
We propose a variational analysis for a Black and Scholes equation with stochastic volatility. This equation gives the price of a European option as a function of the time, of the price of the underlying asset and of the volatility when the volatility is a function of a mean reverting Orstein-Uhlenbeck process, possibly correlated with the underlying asset. The variational analysis involves weighted Sobolev spaces. It enables to prove qualitative properties of the solution, namely a maximum principle...
Variational problems with free boundaries for the fractional Laplacian
We discuss properties (optimal regularity, nondegeneracy, smoothness of the free boundary etc.) of a variational interface problem involving the fractional Laplacian; due to the nonlocality of the Dirichlet problem, the task is nontrivial. This difficulty is bypassed by an extension formula, discovered by the first author and Silvestre, which reduces the study to that of a codimension 2 (degenerate) free boundary.
Volume Filling Effect in Modelling Chemotaxis
The oriented movement of biological cells or organisms in response to a chemical gradient is called chemotaxis. The most interesting situation related to self-organization phenomenon takes place when the cells detect and response to a chemical which is secreted by themselves. Since pioneering works of Patlak (1953) and Keller and Segel (1970) many particularized models have been proposed to describe the aggregation phase of this process. Most of...
Weak entropic solution to a scalar hyperbolic-parabolic law.
In this paper we are interested in the Dirichlet problem of a hyperbolic-parabolic degenerate equation. Thanks to a global entropic formulation in the sense of F. Otto, we propose a result of existence and uniqueness of the entropic measure valued solution and of the entropic weak solution in the space DM2.
Weak Solutions for a Fourth Order Degenerate Parabolic Equation
We consider an initial-boundary value problem for a fourth order degenerate parabolic equation. Under some assumptions on the initial value, we establish the existence of weak solutions by the discrete-time method. The asymptotic behavior and the finite speed of propagation of perturbations of solutions are also discussed.
Weak Solutions for Nonlinear Parabolic Equations with Variable Exponents
In this work, we study the existence and uniqueness of weak solutions of fourth-order degenerate parabolic equation with variable exponent using the difference and variation methods.
Weak solutions for quasilinear degenerate parabolic systems.
Weak-strong uniqueness for a class of degenerate parabolic cross-diffusion systems
Bounded weak solutions to a particular class of degenerate parabolic cross-diffusion systems are shown to coincide with the unique strong solution determined by the same initial condition on the maximal existence interval of the latter. The proof relies on an estimate established for a relative entropy associated to the system.
Well-posedness for non-isotropic degenerate parabolic-hyperbolic equations
Асимптотика элементов аттрактора, соответствующих сингулярно возмущенным параболическом уравнениям
Весовая оценка градиентов для неотрицательных обобщенных решений квазилинейных параболических уравнений, допускающих двойное вырождение
Внутренние оценки решений квазипараболических систем
Гельдеровские оценки для квазилинейных вырождающихся параболических систем второго порядка
Гельдеровские оценки для квазилинейных параболических уравнений с двойным вырождением
Гельдеровские оценки для уравнений типа медленной и нормальной диффузии
Задача Коши для некоторых вырождающихся квазилинейных параболических уравнений с поглощением
Исследование разностных схем для квазилинейных вырождающихся параболических уравнений
Мгновенное возникновение особенностей у решения вырождающегося параболического уравнения.