In this paper, it is analyzed a non-classical non-autonomous diffusion equation with delay. First, the well-posedness and the existence of a local solution is proved by using a fixed point theorem. Then, the existence of solutions defined globally in future is ensured. The asymptotic behaviour of solutions is analyzed within the framework of pullback attractors as it has revealed a powerful the...

function, 51 Riemann integral, 51 adjoint bilinear form, 65 almost separably valued, 53 Ascoli Theorem, 8 Banach space reflexive, 6 separable, 2 Banach-Steinhaus Theorem, 5 Bessel’s inequality, 9 biharmonic, 61 bilinear form, 55, 58 bounded, 58 strongly positive, 58 Bochner-Pettis Theorem Theorem, 53 Bounded Inverse Theorem, 6 Calculus in W 1,p(I;X) Theorem, 54 Campanato ’63 Theorem, 86 Campana...

In this paper we continue the research begun in [CA-2]. Some new results are shown and proven, like structure theorem for n-dimensional almost periodic functions by using Bochner Transform. Also, Haraux [Har] condition case, some topological theorems similar to Ascoli theorems. Furthermore, answer a question formulated Prof. Fischer [Fis], study an average integrals of functions.

n 1 (0) ! R and ⌦ open ⇢ B, define A(u; ⌦) = Area (graph u| ⌦) = Z ⌦ p 1 + |Du| 2 , where, by 'area', we mean n-dimensional Hausdor↵ measure. The notation A(u) simply means Area (graph u). It can be established (firstly for C 1 functions using integration by parts and duality and then by approximating a Lipschitz function in the L 1 norm by a sequence of C 1 functions) that for any Lipschitz u ...

In this paper, we provide several Arzelà–Ascoli-type results on the space of all continuous functions from a Tychonoff X into fuzzy sets Rn, (FUSCB(Rn),Hend), which are upper semi-continuous and have bounded support endowed with endograph metric. Namely, obtain positive when is considered to be kr-space C(X,(FUSCB(Rn),Hend)) compact open topology, as well assume that pseudocompact equipped unif...

Gradient schemes is a framework that enables the unified convergence analysis of many numerical methods for elliptic and parabolic partial differential equations: conforming and non-conforming Finite Element, Mixed Finite Element and Finite Volume methods. We show here that this framework can be applied to a family of degenerate non-linear parabolic equations (which contain in particular the Ri...

In this paper we investigate stochastic dynamics and invariant measures for 3D Lagrangian-averaged Navier–Stokes (LANS) equations driven by infinite delay additive noise. We first use Galerkin approximations, a priori estimates the standard Gronwall lemma to show well-posedness corresponding random equation, whose solution operators generate dynamical system. Next, asymptotic compactness system...