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1
Content available remote General decay for a nonlinear pseudo-parabolic equation with viscoelastic term
Vu Ngo Tran, Dung Dao Bao, Dung Huynh Thi Hoang
Demonstratio Mathematica
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2022
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Vol. 55, nr 1
737--751
EN
This work is concerned with a multi-dimensional viscoelastic pseudo-parabolic equation with critical Sobolev exponent. First, with some suitable conditions, we prove that the weak solution exists globally. Next, we show that the stability of the system holds for a much larger class of kernels than the ones considered in previous literature. More precisely, we consider the kernel g:[0,∞)⟶(0,∞) satisfying g′(t)⩽−ξ(t)G(g(t)) , where ξ and G are functions satisfying some specific properties.
2
Content available remote Asymptotic stability of solutions for a diffusive epidemic model
Bouaziz Khelifa, Douaifia Redouane, Abdelmalek Salem
Demonstratio Mathematica
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2022
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Vol. 55, nr 1
553--573
EN
The aim of this paper is to study the existence and the asymptotic stability of solutions for an epidemiologically emerging reaction-diffusion model. We show that the model has two types of equilibrium points to resolve the proposed system for a fairly broad class of nonlinearity that describes the transmission of an infectious disease between individuals. The model is analyzed by using the basic reproductive number R0 . Finally, we present the numerical examples simulations that clarifies and confirms the results of the study throughout the paper.
3
Content available Remarks on damped Schrodinger equation of Choquard type
Chergui Lassaad
Opuscula Mathematica
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2021
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Vol. 41, no. 4
465--488
EN
This paper is devoted to the Schrodinger-Choquard equation with linear damping. Global existence and scattering are proved depending on the size of the damping coefficient.
4
Content available remote Global existence and dynamic structure of solutions for damped wave equation involving the fractional Laplacian
Bidi Younes, Beniani Abderrahmane, Zennir Khaled, Himadan Ahmed
Demonstratio Mathematica
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2021
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Vol. 54, nr 1
245--258
EN
We consider strong damped wave equation involving the fractional Laplacian with nonlinear source. The results of global solution under necessary conditions on the critical exponent are established. The existence is proved by using the Galerkin approximations combined with the potential well theory. Moreover, we showed new decay estimates of global solution.
5
Content available remote Existence and uniqueness of the weak solution for Keller-Segel model coupled with Boussinesq equations
Slimani Ali, Bouzettouta Lamine, Guesmia Amar
Demonstratio Mathematica
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2021
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Vol. 54, nr 1
558--575
EN
Keller-Segel chemotaxis model is described by a system of nonlinear partial differential equations: a convection diffusion equation for the cell density coupled with a reaction-diffusion equation for chemoattractant concentration. In this work, we study the phenomenon of Keller-Segel model coupled with Boussinesq equations. The main objective of this work is to study the global existence and uniqueness and boundedness of the weak solution for the problem, which is carried out by the Galerkin method.
6
Content available Global existence and blow up of solution for semi-linear hyperbolic equation with the product of logarithmic and power-type nonlinearity
Lian Wei, Ahmed Md Salik, Xu Runzhang
Opuscula Mathematica
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2020
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Vol. 40, no. 1
111--130
EN
In this paper we consider the semilinear wave equation with the multiplication of logarithmic and polynomial nonlinearities. We establish the global existence and finite time blow up of solutions at three different energy levels (E(0) < d, E(0) = d and E(0) > 0) using potential well method. The results in this article shed some light on using potential wells to classify the solutions of the semilinear wave equation with the product of polynomial and logarithmic nonlinearity.
7
Content available Existence and decay of finite energy solutions for semilinear dissipative wave equations in time-dependent domains
Nakao Mitsuhiro
Opuscula Mathematica
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2020
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Vol. 40, no. 6
725--736
EN
We consider the initial-boundary value problem for semilinear dissipative wave equations in noncylindrical domain [formula]. We are interested in finite energy solution. We derive an exponential decay of the energy in the case Ω (t) is bounded in [formula] and the estimate [formula] in the case Ω (t) is unbounded. Existence and uniqueness of finite energy solution are also proved.
8
Content available Global well-posedness of a class of fourth-order strongly damped nonlinear wave equations
Yanbing Yang, Ahmed Md Salik, Lanlan Qin, Runzhang Xu
Opuscula Mathematica
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2019
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Vol. 39, no. 2
297--313
EN
Global well-posedness and finite time blow up issues for some strongly damped nonlinear wave equation are investigated in the present paper. For subcritical initial energy by employing the concavity method we show a finite time blow up result of the solution. And for critical initial energy we present the global existence, asymptotic behavior and finite time blow up of the solution in the framework of the potential well. Further for supercritical initial energy we give a sufficient condition on the initial data such that the solution blows up in finite time.
9
Content available remote The analytical solution of Van der Pol and Lienard differential equations within conformable fractional operator by retarded integral inequalities
Usta Fuat, Sarıkaya Mehmet Zeki
Demonstratio Mathematica
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2019
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Vol. 52, nr 1
204--212
EN
In this study we introduced and tested retarded conformable fractional integral inequalities utilizing non-integer order derivatives and integrals. In line with this purpose, we used the Katugampola type conformable fractional calculus which has several practical properties. These inequalities generalize some famous integral inequalities which provide explicit bounds on unknown functions. The results provided here had been implemented to the global existence of solutions to the conformable fractional differential equations with time delay.
10
On some nonlinear hyperbolic p(x,t)-Laplacian equations
Ahmedatt T., Touzani A., Aberqi A., Yazough C.
Journal of Applied Analysis
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2018
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Vol. 24, nr 1
55--69
EN
This paper is devoted to study the global existence of solutions of the hyperbolic Dirichlet equation Utt=Lu+f(x,t) in ΩT=Ω×(0,T), where L is a nonlinear operator and ϕ(x,t,⋅), f(x,t) and the exponents of the nonlinearities p(x,t) and μ(x,t) are given functions.
11
Content available Global existence and asymptotic behavior for a nonlinear degenerate SIS model
Ziane T. A.
Opuscula Mathematica
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2013
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Vol. 33, no. 4
615--630
EN
In this paper we investigate the global existence and asymptotic behavior of a reaction diffusion system with degenerate diffusion arising in the modeling and the spatial spread of an epidemic disease.
12
Content available remote Global solution of reaction diffusion system with non diagonal matrix
Moumeni A., Derradji L. S.
Demonstratio Mathematica
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2012
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Vol. 45, nr 1
81-93
EN
The purpose of this paper is to prove the global existence in time of solutions for the coupled reaction-diffusion system: (…) with triangular matrix of diffusion coefficients. By combining the Lyapunov functional method with the regularizing effect, we show that global solutions exist. Our investigation applied for a wide class of the nonlinear terms f and g.
13
Content available remote On integrated Volterra integrodifferential equation of higher order
Pachpatte B. G.
Fasciculi Mathematici
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2008
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Nr 39
47-61
EN
In this paper we study the existence and other properties of solutions of a certain iterated Volterra integrodifferential equation of higher order. The tools employed in the analysis are based on application of the Leray-Schauder alternative and a certain integral inequality which provides explicit bound on the unknown function.
14
Content available remote Global existence for a class of reaction-diffusion systems
Kouachi S., Youkana A.
Bulletin of the Polish Academy of Sciences. Mathematics
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2001
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Vol. 49, no 3
303-308
EN
The purpose of this paper is to give sufficient conditions guaranteeing global existence, uniqueness and uniform boundedness of solutions for a class of reaction-diffusion systems.
15
Content available Uniform stabilization of the quasi-linear Kirchhoff wave equation with a nonlinear boundary feedback
Lasiecka I.
Control and Cybernetics
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2000
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Vol. 29, no 1
179-197
EN
An n-dimensional quasi-linear wave equation defined on bounded domain Omega with Neumann boundary conditions imposed on the boundary Gamma and with a nonlinear boundary feedback acting on a portion of the boundary [Gamma sup 1 is a subset of Gamma] is considered. Global existence, uniqueness and uniform decay rates are established for the model, under the assumption that the H[sup 1](Omega) x L[sub 2](Omega) norms of the initial data are sufficiently small. The result presented in this paper extends these obtained recently in Lasiecka and Ong (1999), where the Dirichlet boundary conditions are imposed on the uncontrolled portion of the boundary Gamma[sub o] = Gamma \ [closure of a set Gamma sub 1], and the two portions of the boundary are assumed disjoint, i.e. [... ]. The goal of this paper is to remove this restriction. This is achieved by considering the "pure" Neumann problem subject to convexity assumption imposed on Gamma[sub o]. \@eng\\
16
Content available Strong decay for one-dimensional wave equations with nonmonotone boundary damping
Pierre M., Vancostenoble J.
Control and Cybernetics
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2000
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Vol. 29, no 2
473-484
EN
This paper is a contribution to the following question : consider the classical wave equation damped by a nonlinear feedback control which is only assumed to decrease the energy. Then, do solutions to the perturbed system still exist for all time? Does strong stability occur in the sense that the energy tends to zero as time tends to infinity? We prove here that the answer to both questions is positive in the specific case of the one-dimensional wave equation damped by boundary controls which are functions of the observed velocity. The main point is that no monotonicity assumption is made on the damping term.
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