Discrete and Continuous Dynamical Systems
June 2021 , Volume 41 , Issue 6
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A distributed optimal control problem for a diffuse interface model, which physical context is that of tumour growth dynamics, is addressed. The system we deal with comprises a Cahn–Hilliard equation for the tumour fraction coupled with a reaction-diffusion for a nutrient species surrounding the tumourous cells. The cost functional to be minimised possesses some objective terms and it also penalises long treatments time, which may affect harm to the patients, and big aggregations of tumourous cells. Hence, the optimisation problem leads to the optimal strategy which reduces the time exposure of the patient to the medication and at the same time allows the doctors to achieve suitable clinical goals.
The present work is devoted to study comparison and converse comparison theorems for diagonally quadratic BSDEs. We give sufficient and necessary conditions under which the comparison holds. Sufficient and necessary conditions for non-positive and non-negative solutions are presented.
We study geometric and algebraic geometric properties of the continuous and discrete Neumann systems on cotangent bundles of Stiefel varieties
The parabolic-parabolic Keller-Segel model of chemotaxis is shown to come up as the hydrodynamic system describing the evolution of the modulus square
We derive second order estimates for
There is a long standing conjecture that there are at least
In this paper, we study the following coupled nonlocal system
satisfying the additional conditions
In this paper we study flows having an isolated non-saddle set. We see that the global structure of a flow having an isolated non-saddle set
We consider the initial value problem associated to a coupled system of modified Korteweg-de Vries type equations
and prove the local well-posedness results for a given data in low regularity Sobolev spaces
In this paper, we investigate the non-autonomous stochastic evolution equations of parabolic type with nonlinear noise and nonlocal initial conditions in Hilbert spaces, where the operators in linear part depend on time
We introduce the notion of forward untangled Lagrangian representation of a measure-divergence vector-measure
In this paper we consider the homogenization problem for a nonlocal equation that involve different smooth kernels. We assume that the spacial domain is divided into a sequence of two subdomains
We prove that given a measure preserving system
First, we prove existence, nonnegativity, and pathwise uniqueness of martingale solutions to stochastic porous-medium equations driven by conservative multiplicative power-law noise in the Ito-sense. We rely on an energy approach based on finite-element discretization in space, homogeneity arguments and stochastic compactness. Secondly, we use Monte-Carlo simulations to investigate the impact noise has on waiting times and on free-boundary propagation. We find strong evidence that noise on average significantly accelerates propagation and reduces the size of waiting times – changing in particular scaling laws for the size of waiting times.
In this note we consider a symmetric Random Walk defined by a
We consider an optimization problem with volume constraint for an energy functional associated to an inhomogeneous operator with nonstandard growth. By studying an auxiliary penalized problem, we prove existence and regularity of solution to the original problem: every optimal configuration is a solution to a one phase free boundary problem—for an operator with nonstandard growth and non-zero right hand side—and the free boundary is a smooth surface.
We prove the existence of a bounded positive solution for the following stationary Schrödinger equation
We study well-posedness of the complex-valued modified KdV equation (mKdV) on the real line. In particular, we prove local well-posedness of mKdV in modulation spaces
The aim of this paper is to study (regional) fractional Poincaré type inequalities on unbounded domains satisfying the finite ball condition. Both existence and non existence type results on regional fractional inequality are established depending on various conditions on domains and on the range of
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