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## Abstract

The paper deals with a posteriori analysis of the spectral element discretization of a non linear heat equation. The discretization is based on Euler’s backward scheme in time and spectral discretization in space. Residual error indicators related to the discretization in time and in space are defined. We prove that those indicators are upper and lower bounded by the error estimation.

## Abstract

We consider nonlinear sub-elliptic systems with VMO-coefficients for the case 1 < p < 2 under controllable growth conditions, as well as natural growth conditions, respectively, in the Heisenberg group. On the basis of a generalization of the technique of 𝓐-harmonic approximation introduced by Duzaar-Grotowski-Kronz, and an appropriate Sobolev-Poincaré type inequality established in the Heisenberg group, we prove partial Hölder continuity results for vector-valued solutions of discontinuous sub-elliptic problems. The primary model covered by our analysis is the non-degenerate sub-elliptic p-Laplacian system with VMO-coefficients, involving sub-quadratic growth terms.

## Abstract

We introduce a family of pairings between a bounded divergence-measure vector field and a function u of bounded variation, depending on the choice of the pointwise representative of u. We prove that these pairings inherit from the standard one, introduced in [G. Anzellotti, Pairings between measures and bounded functions and compensated compactness, Ann. Mat. Pura Appl. (4) 135 1983, 293–318], [G.-Q. Chen and H. Frid, Divergence-measure fields and hyperbolic conservation laws, Arch. Ration. Mech. Anal. 147 1999, 2, 89–118], all the main properties and features (e.g. coarea, Leibniz, and Gauss–Green formulas). We also characterize the pairings making the corresponding functionals semicontinuous with respect to the strict convergence in $BV$. We remark that the standard pairing in general does not share this property.

## Abstract

The aim of this study was to present a simple method for finding the asymptotic relations for products of elements of some positive real sequences. The main reason to carry out this study was the result obtained by Alzer and Sandor concerning an estimation of a sequence of the product of the first k primes.

## Abstract

In this article, we obtain the general solution and prove the Hyers-Ulam stability of the following quadratic-multiplicative functional equation:

$ϕ(st−uv)+ϕ(sv+tu)=[ϕ(s)+ϕ(u)][ϕ(t)+ϕ(v)]$
by using the direct method and the fixed point method.

## Abstract

Consider an ordered Banach space and $f,g$ two self-operators defined on the interior of its positive cone. In this article, we prove that the equation $f(X)=g(X)$ has a positive solution, whenever f is strictly $α$-concave g-monotone or strictly $(−α)$-convex g-antitone with g super-homogeneous and surjective. As applications, we show the existence of positive definite solutions to new classes of nonlinear matrix equations.

## Abstract

In this paper, we prove the existence and regularity of solutions of the homogeneous Dirichlet initial-boundary value problem for a class of degenerate elliptic equations with lower order terms. The results we obtained here, extend some existing ones of [, , ] in some sense.

## Abstract

Using monotonicity methods and some variational argument we consider nonlinear problems which involve monotone potential mappings satisfying condition (S) and their strongly continuous perturbations. We investigate when functional whose minimum is obtained by a direct method of the calculus of variations satisfies the Palais-Smale condition, relate minimizing sequence and Galerkin approximaitons when both exist, then provide structure conditions on the derivative of the action functional under which bounded Palais-Smale sequences are convergent. Finally, we make some comment concerning the convergence of Palais-Smale sequence obtained in the mountain pass theorem due to Rabier.

## Abstract

This paper is concerned with the positivity of solutions to the Cauchy problem for linear and nonlinear parabolic equations with the biharmonic operator as fourth order elliptic principal part. Generally, Cauchy problems for parabolic equations of fourth order have no positivity preserving property due to the change of sign of the fundamental solution. One has eventual local positivity for positive initial data, but on short time scales, one will in general have also regions of negativity.

The first goal of this paper is to find sufficient conditions on initial data which ensure the existence of solutions to the Cauchy problem for the linear biharmonic heat equation which are positive for all times and in the whole space.

The second goal is to apply these results to show existence of globally positive solutions to the Cauchy problem for a semilinear biharmonic parabolic equation.

## Abstract

This paper is concerned with the following nonlinear Hamiltonian elliptic system with gradient term

$−Δu+b→(x)⋅∇u+V(x)u=Hv(x,u,v)inRN,−Δv−b→(x)⋅∇v+V(x)v=Hu(x,u,v)inRN.$

Compared with some existing issues, the most interesting feature of this paper is that we assume that the nonlinearity satisfies a local super-quadratic condition, which is weaker than the usual global super-quadratic condition. This case allows the nonlinearity to be super-quadratic on some domains and asymptotically quadratic on other domains. Furthermore, by using variational method, we obtain new existence results of ground state solutions and infinitely many geometrically distinct solutions under local super-quadratic condition. Since we are without more global information on the nonlinearity, in the proofs we apply a perturbation approach and some special techniques.