Inspired by a problem in steel metallurgy, we prove the existence, regularity, uniqueness, and continuous data dependence of solutions to a coupled parabolic system in a smooth bounded 3D domain, with nonlinear and nonhomogeneous boundary conditions. The nonlinear coupling takes place in the diffusion coefficient. The proofs are based on anisotropic estimates in tangential and normal directions...

The aim of this paper is to prove the existence, uniqueness, and continuous dependence upon the data of a generalized solution for certain singular parabolic equations with initial and nonlocal boundary conditions. The proof is based on an a priori estimate established in nonclassical function spaces, and on the density of the range of the operator corresponding to the abstract formulation of t...

Inspired by a problem in steel metallurgy, we prove the existence, regularity, uniqueness, and continuous data dependence of solutions to a coupled parabolic system in a smooth bounded 3D domain, with nonlinear and nonhomogeneous boundary conditions. The nonlinear coupling takes place in the di usion coefcient. The proofs are based on anisotropic estimates in tangential and normal directions, a...

We introduce the notion of a generalized iterated function system GIFS , which is a finite family of functions fk : X →X, where X, d is a metric space and m ∈ N. In case that X, d is a compact metric space and the functions fk are contractions, using some fixed point theorems for contractions from X to X, we prove the existence of the attractor of such a GIFS and its continuous dependence in th...

In this article, we consider the robustness of a nonuniform (μ, ν) trichotomy in Banach spaces, in the sense that the existence of such a trichotomy for a given linear equation persists under sufficiently small linear perturbations. The continuous dependence with the perturbation of the constants in the notion of trichotomy is studied, and the related robustness of strong (μ, ν) trichotomy is a...

We investigate an initial boundary value problem for a second-order hyperbolic equation with only integral conditions. We show the existence, uniqueness, and continuous dependence of a strongly generalized solution. The proof is based on an energy inequality established in a nonclassical function space, and on the density of the range of the operator associated to the abstract formulation of th...

In this paper, after recalling the definition of generalized anticipated backward stochastic differential equations (generalized anticipated BSDEs for short) and the existence and uniqueness theorem for their solutions, we show there is a duality between them and stochastic differential delay equations. We then provide a continuous dependence property for their solutions with respect to the par...

In this note we give fundamental existence, uniqueness, and continuous dependence results (well-posedness) for a variational formulation of a class of damped second order partial diierential equations with unbounded input or control coee-cients. Included as special cases in this class are structures with piezoceramic actua-tors. We then consider approximation techniques leading to computational...

This paper deals with weak solution in weighted Sobolev spaces, of three-point boundary value problems which combine Dirichlet and integral conditions, for linear and quasilinear parabolic equations in a domain with curved lateral boundaries. We, firstly, prove the existence, uniqueness, and continuous dependence of the solution for the linear equation. Next, analogous results are established f...

Abstract This paper investigates the optimal control for a class of nonlocal fractional evolution equations order $\gamma \in (1,2)$ ? ? ( 1 , 2 ) in Banach spaces. An adequate definition ? -mild solutions is...