Un problema de difusión para una ecuación del tipo Ni-Serrin con término de convección

The purpose of this article is to study the existence of weak solutions for a class of nonlinear nonlocal Dirichlet parabolic problem involving the p(x)-Kirchhoff type triharmonic operator with a nonlinear source. We apply degree theory to operators of the type T + S + C, where T is maximal monotone, S is bounded pseudomonotone, and C is compact with D(T) ⊆ D(C) and satisfies a sublinearity condition, to establish our result within the context of Sobolev spaces with variable exponents.

In this article, a finite element Galerkin method is applied to a general class of nonlinear and nonlocal parabolic problems. Based on an exponential weight function, new a priori bounds which are valid for uniform in time are derived. As a result, existence of an attractor is proved for the problem with nonhomogeneous right hand side which is independent of time. In particular, when the forcing function is zero or decays exponentially, it is shown that solution has exponential decay property which improves even earlier results in one dimensional problems. For the semidiscrete method, global existence of a unique discrete solution is derived and it is shown that the discrete problem has an attractor. Moreover, optimal error estimates are derived in both and ‐norms with later estimate is a new result in this context. For completely discrete scheme, backward Euler method with its linearized version is discussed and existence of a unique discrete solution is established. Further, optimal estimates in ‐norm are proved for fully discrete schemes. Finally, several numerical experiments are conducted to confirm our theoretical findings. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 1232–1264, 2016

We are studying a nonlinear nonlocal Dirichlet parabolic problem associated with the following p(x)−laplacian equation ∂u∂t−a(∫Ω| ∇u |p(x)p(x))Δp(x)u=f(x,t) We apply Galerkin’s approximation technique, some uniform estimates and a compactness arguments to prove the existence result. Then we investigate the uniqueness of solutions, finally we give an answer to the asymptotic behavior question.

In this paper, an expanded mixed finite element method with lowest order Raviart Thomas elements is developed and analyzed for a class of nonlinear and nonlocal parabolic problems. After obtaining some regularity results for the exact solution, a priori error estimates for the semidiscrete problem are established. Based on a linearized backward Euler method, a complete discrete scheme is proposed and a variant of Brouwer’s fixed point theorem is used to derive an existence of a fully discrete solution. Further, a priori error estimates for the fully discrete scheme are established. Finally, numerical experiments are conducted to confirm our theoretical findings.

This paper studies the existence of solutions for Robin problems involving p ( x )-Laplacian-like operators which arise from capillarity phenomena. When we only consider the convective term, due to the lack of a variational structure, the well-known variational methods are not applicable. Using Galerkin method together with Brouwer’s fixed point theorem, we obtain the existence of finite-dimensional approximate solution and generalized solution. On the other hand, utilizing local linking theorem without Ambrosetti-Rabinowitz ((A-R) for short) condition, we obtain the existence of a nontrivial solution under some conditions. The main difficulties and innovations of the present article are that we consider the convective term, the weaker assumptions on the nonlinear term, and p ( x )-Laplacian-like operators with Robin boundary condition.

This paper is devoted to the study of existence and multiplicity results for a class of elliptic problems driven by a q ( x ) -Laplacian-like operator and a q ( x ) -biharmonic operator in the presence of a Hardy-type singular potential. The problem is formulated within the framework of variable exponent Lebesgue–Sobolev spaces, which naturally accommodate the non-standard growth conditions of the operators involved. By employing variational methods, notably the Mountain Pass Theorem and its Z 2 -symmetric version, we establish the existence of at least one nontrivial weak solution as well as the existence of infinitely many solutions under suitable symmetry assumptions. A central feature of the analysis is the restriction imposed on the parameter μ associated with the Hardy-type term, which ensures the coercivity of the energy functional and the validity of the Palais–Smale condition. The results obtained extend and complement existing contributions on variable exponent biharmonic problems with singular nonlinearities and provide a foundation for further investigations, including regularity analysis and possible extensions to fractional or nonlocal q ( x ) -biharmonic operators.

Monotone Iterative Method for a System of Nonlocal Initial-Boundary Parabolic Problems

Let X be a real locally uniformly convex reflexive Banach space with locally uniformly convex dual space X⁎. Let T:X⊇D(T)→2X⁎ be maximal monotone, S:X→2X⁎ be bounded and of type (S+), and C:D(C)→X⁎ be compact with D(T)⊆D(C) such that C lies in Γστ (i.e., there exist σ≥0 and τ≥0 such that Cx≤τx+σ for all x∈D(C)). A new topological degree theory is developed for operators of the type T+S+C. The theory is essential because no degree theory and/or existence result is available to address solvability of operator inclusions involving operators of the type T+S+C, where C is not defined everywhere. Consequently, new existence theorems are provided. The existence theorem due to Asfaw and Kartsatos is improved. The theory is applied to prove existence of weak solution (s) for a nonlinear parabolic problem in appropriate Sobolev spaces.

We discuss the existence and uniqueness of the weak solution of the following nonlinear parabolic problem: [Formula: see text] which involves a quasilinear elliptic operator of Leray–Lions type with variable exponents. Next, we discuss the global behavior of solutions and in particular the convergence to a stationary solution as [Formula: see text].

In this paper, we investigate a nonlinear diffusion-convection problem with measure data, involving a general anisotropic operator with variable exponent and a maximal monotone graph. Utilizing Yosida’s regularization, we apply an approximation technique to formulate a regularized problem. We then employ the theory of maximal monotone operators in Banach spaces to demonstrate the existence of at least one solution for the approximate problem. Subsequently, we show that the sequence of solutions to the approximate problem converges, in the limit, to a renormalized and/or entropic solution of the original problem. Finally, we establish the uniqueness of the solution under certain additional assumptions regarding the convection term.

In this paper, we study the existence of a weak solution for a class of Neumann boundary value problems for equations involving the p ⁢ ( x ) p(x) -Laplacian-like operator. Using a topological degree theory for a class of demicontinuous operators of generalized ( S + ) (S_{+}) -type and the theory of the variable exponent Sobolev spaces, we establish the existence of a weak solution of this problem. Our results extend and generalize several corresponding results from the existing literature.

A topological degree theory for constrained problems with compact perturbations and application to nonlinear parabolic problem

The aim of this paper is to study a class of nonlocal nonlinear parabolic boundary value problems. First the existence, uniqueness, and continuous dependence of the solution upon the data are demonstrated, and then finite difference methods, backward Euler and Crank–Nicolson schemes are studied. It is proved that both numerical schemes are stable and convergent to the real solution. The results of some numerical examples are presented, which demonstrate the efficiency and rapid convergence of the methods.

In this paper, we prove the existence of at least one periodic solution for some nonlinear parabolic boundary value problems associated with Leray–Lions’s operators with variable exponents under the hypothesis of existence of well-ordered sub- and supersolutions.

The aim of the paper is to prove a theorem about a weak impulsive nonlinear parabolic differential inequality together with weak impulsive nonlocal nonlinear inequalities. A weak maximum principle for an impulsive nonlinear parabolic differential inequality together with weak impulsive nonlocal nonlinear inequalities and an uniqueness criterion for the existence of the classical solution of an impulsive nonlocal nonlinear parabolic differential problem are obtained as a consequence of the theorem about the weak impulsive nonlinear parabolic differential inequality together with weak impulsive nonlocal nonlinear inequalities.