MDPI - Publisher of Open Access Journals

14 pages, 275 KiB

Open AccessArticle

Strong Stability for a Viscoelastic Transmission Problem Under a Nonlocal Boundary Control

by Noureddine Touati Brahim, Abderrahmane Beniani, Abderrazak Chaoui, Zayd Hajjej, Perikles Papadopoulos and Khaled Zennir

Axioms 2024, 13(10), 714; https://doi.org/10.3390/axioms13100714 - 16 Oct 2024

Viewed by 975

Abstract

The purpose of this paper is to consider a transmission problem of a viscoelastic wave with nonlocal boundary control. It should be noted that the present paper is based on the previous C. G. Gal and M. Warma works, together with H. Atoui and A. Benaissa. Namely, they focused on a transmission problem consisting of a semilinear parabolic equation in a general non-smooth setting with an emphasis on rough interfaces and nonlinear dynamic (possibly, nonlocal) boundary conditions along the interface, where a transmission problem in the presence of a boundary control condition of a nonlocal type was investigated in these papers. Owing to the semigroup theory, we prove the question of well-posedness. For the very rare cases, we combined between the frequency domain approach and the Borichev–Tomilov theorem to establish strong stability results. Full article

(This article belongs to the Special Issue New Perspectives in Applied Mathematics with Nonlinear Equations and Dynamical Systems)

20 pages, 327 KiB

Open AccessArticle

Existence and Nonexistence of Positive Solutions for Semilinear Elliptic Equations Involving Hardy–Sobolev Critical Exponents

by Lin-Lin Wang and Yong-Hong Fan

Mathematics 2024, 12(11), 1616; https://doi.org/10.3390/math12111616 - 21 May 2024

Cited by 1 | Viewed by 1021

Abstract

The following semi-linear elliptic equations involving Hardy–Sobolev critical exponents [...] Read more.

The following semi-linear elliptic equations involving Hardy–Sobolev critical exponents

- Δ u - μ \frac{u}{{|x|}^{2}} = \frac{{|u|}^{2^{*} (s) - 2}}{{|x|}^{s}} u + g (x, u), x \in Ω ∖ \{0\}, u = 0, x \in \partial Ω

have been investigated, where

Ω

is an open-bounded domain in

R^{N} (N \geq 3)

, with a smooth boundary

\partial Ω

0 \in Ω, 0 \leq μ < \bar{μ} : = {(\frac{N - 2}{2})}^{2}, 0 \leq s < 2

, and

2^{*} (s) = 2 (N - s) / (N - 2)

is the Hardy–Sobolev critical exponent. This problem comes from the study of standing waves in the anisotropic Schrödinger equation; it is very important in the fields of hydrodynamics, glaciology, quantum field theory, and statistical mechanics. Under some deterministic conditions on

g

, by a detailed estimation of the extremum function and using mountain pass lemma with

{(P S)}_{c}

conditions, we obtained that: (a) If

μ \leq \bar{μ} - 1

, and

λ < λ_{1} (μ),

then the above problem has at least a positive solution in

H_{0}^{1} (Ω)

; (b) If

\bar{μ} - 1 < μ < \bar{μ}

, then when

λ_{*} (μ) < λ < λ_{1} (μ)

, the above problem has at least a positive solution in

H_{0}^{1} (Ω)

; (c) if

\bar{μ} - 1 < μ < \bar{μ}

and

Ω = B (0, R)

, then the above problem has no positive solution for

λ \leq λ_{*} (μ) .

These results are extensions of E. Jannelli’s research (

g (x, u) = λ u

). Full article

(This article belongs to the Section C1: Difference and Differential Equations)

33 pages, 481 KiB

Open AccessArticle

Galerkin Finite Element Approximation of a Stochastic Semilinear Fractional Wave Equation Driven by Fractionally Integrated Additive Noise

by Bernard A. Egwu and Yubin Yan

Foundations 2023, 3(2), 290-322; https://doi.org/10.3390/foundations3020023 - 29 May 2023

Viewed by 5500

Abstract

We investigate the application of the Galerkin finite element method to approximate a stochastic semilinear space–time fractional wave equation. The equation is driven by integrated additive noise, and the time fractional order

α \in (1, 2)

. The existence of [...] Read more.

α \in (1, 2)

. The existence of a unique solution of the problem is proved by using the Banach fixed point theorem, and the spatial and temporal regularities of the solution are established. The noise is approximated with the piecewise constant function in time in order to obtain a stochastic regularized semilinear space–time wave equation which is then approximated using the Galerkin finite element method. The optimal error estimates are proved based on the various smoothing properties of the Mittag–Leffler functions. Numerical examples are provided to demonstrate the consistency between the theoretical findings and the obtained numerical results. Full article

(This article belongs to the Special Issue Recent Advances in Fractional Differential Equations and Inclusions II)

► Show Figures

Figure 1

17 pages, 335 KiB

Open AccessArticle

A One-Dimensional Time-Fractional Damped Wave Equation with a Convection Term

by Ibtisam Aldawish, Mohamed Jleli and Bessem Samet

Symmetry 2023, 15(5), 1071; https://doi.org/10.3390/sym15051071 - 12 May 2023

Viewed by 1215

Abstract

We investigate a semilinear time-fractional damped wave equation in one dimension, posed in a bounded interval. The considered equation involves a convection term and singular potentials on one extremity of the interval. A Dirichlet boundary condition depending on the time-variable is imposed. Using nonlinear capacity estimates, we establish sufficient conditions for the nonexistence of weak solutions to the considered problem. In particular, when the boundary condition is independent of time, we show the existence of a Fujita-type critical exponent. Full article

(This article belongs to the Special Issue Symmetries in Evolution Equations and Applications)

17 pages, 333 KiB

Open AccessArticle

Fractional Evolution Equations with Infinite Time Delay in Abstract Phase Space

by Ahmed Salem, Kholoud N. Alharbi and Hashim M. Alshehri

Mathematics 2022, 10(8), 1332; https://doi.org/10.3390/math10081332 - 17 Apr 2022

Cited by 15 | Viewed by 2142

Abstract

In the presented research, the uniqueness and existence of a mild solution for a fractional system of semilinear evolution equations with infinite delay and an infinitesimal generator operator are demonstrated. The generalized Liouville–Caputo derivative of non-integer-order

1 < α \leq 2

and the [...] Read more.

1 < α \leq 2

and the parameter

0 < ρ < 1

are used to establish our model. The

ρ

-Laplace transform and strongly continuous cosine and sine families of uniformly bounded linear operators are adapted to obtain the mild solution. The Leray–Schauder alternative theorem and Banach contraction principle are used to demonstrate the mild solution’s existence and uniqueness in abstract phase space. The results are applied to the fractional wave equation. Full article

24 pages, 433 KiB

Open AccessArticle

Integrating Semilinear Wave Problems with Time-Dependent Boundary Values Using Arbitrarily High-Order Splitting Methods

by Isaías Alonso-Mallo and Ana M. Portillo

Mathematics 2021, 9(10), 1113; https://doi.org/10.3390/math9101113 - 14 May 2021

Cited by 1 | Viewed by 2059

Abstract

The initial boundary-value problem associated to a semilinear wave equation with time-dependent boundary values was approximated by using the method of lines. Time integration is achieved by means of an explicit time method obtained from an arbitrarily high-order splitting scheme. We propose a technique to incorporate the boundary values that is more accurate than the one obtained in the standard way, which is clearly seen in the numerical experiments. We prove the consistency and convergence, with the same order of the splitting method, of the full discretization carried out with this technique. Although we performed mathematical analysis under the hypothesis that the source term was Lipschitz-continuous, numerical experiments show that this technique works in more general cases. Full article

(This article belongs to the Special Issue Numerical Methods for Evolutionary Problems)

► Show Figures

Figure 1

28 pages, 1480 KiB

Open AccessReview

Line Integral Solution of Hamiltonian PDEs

by Luigi Brugnano, Gianluca Frasca-Caccia and Felice Iavernaro

Mathematics 2019, 7(3), 275; https://doi.org/10.3390/math7030275 - 18 Mar 2019

Cited by 14 | Viewed by 3628

Abstract

In this paper, we report on recent findings in the numerical solution of Hamiltonian Partial Differential Equations (PDEs) by using energy-conserving line integral methods in the Hamiltonian Boundary Value Methods (HBVMs) class. In particular, we consider the semilinear wave equation, the nonlinear Schrödinger [...] Read more.

(This article belongs to the Special Issue Geometric Numerical Integration)

► Show Figures

Figure 1

Search Results (7)

Further Information

Guidelines

MDPI Initiatives

Follow MDPI

Saved Queries

Search Filter Reset All

Years

Feature Papers

Subjects

Journals

Article Types

Countries / Regions

Search Results (7)

Further Information

Guidelines

MDPI Initiatives

Follow MDPI