Salah Badraoui - Academia.edu (original) (raw)

Papers by Salah Badraoui

Global existence, positivity, uniform boundedness and extinction results of solutions to a system... more Global existence, positivity, uniform boundedness and extinction results of solutions to a system of reaction-diffusion equations on unbounded domain modeling two species on a predator–prey relationship is considered.

This article concerns the generation of analytic semigroups by an operator matrix in the space L2... more This article concerns the generation of analytic semigroups by an operator matrix in the space L2(Ω) × L2(Ω). We assume that one of the diagonal elements is strongly elliptic and the other is weakly elliptic, while the sum of the non-diagonal elements is weakly elliptic.

Recent Developments in the Solution of Nonlinear Differential Equations, 2021

We prove in this work the existence of a unique global nonnegative classical solution to the clas... more We prove in this work the existence of a unique global nonnegative classical solution to the class of reaction–diffusion systemsuttx=aΔutx−guvm,vttx=dΔvtx+λtxguvm,

Applicationes Mathematicae, 1999

We are concerned with the boundedness and large time behaviour of the solution for a system of re... more We are concerned with the boundedness and large time behaviour of the solution for a system of reaction-diffusion equations modelling complex consecutive reactions on a bounded domain under homogeneous Neumann boundary conditions. Using the techniques of E. Conway, D. Hoff and J. Smoller [3] we also show that the bounded solution converges to a constant function as t → ∞. Finally, we investigate the rate of this convergence.

Mathematical Methods in the Applied Sciences, 2016

We prove in this paper a generalized result with a unified proof of global existence in time of c... more We prove in this paper a generalized result with a unified proof of global existence in time of classical solutions to a class of a reaction diffusion system with triangular diffusion matrix on a bounded domain in Rn . The system in question is ut=aΔu − f(x,t,u,v), vt=cΔu + dΔv + ρf(x,t,u,v), x∈Ω⊂Rn(n≥1) , t > 0 with f(x,t,0,η) = 0 and f(x,t,ξ,η)≤Kφ(ξ)eση, for all x∈Ω, t > 0, ξ≥0, η≥0; where ρ, K and σ are real positive constants. Copyright © 2016 John Wiley & Sons, Ltd.

Electronic Journal of Differential Equations, 2006

This article concerns the behavior at mpinftymp infty mpinfty of solutions to a reaction-diffusion system wit... more This article concerns the behavior at mpinftymp infty mpinfty of solutions to a reaction-diffusion system with a cross diffusion matrix on unbounded domains. We show that the solutions satisfy the free diffusion system for all positive time whenever the initial distribution has limits at mpinftymp inftympinfty.

This article concerns the behavior at ∓∞ of solutions to a reactiondiffusion system with a cross ... more This article concerns the behavior at ∓∞ of solutions to a reactiondiffusion system with a cross diffusion matrix on unbounded domains. We show that the solutions satisfy the free diffusion system for all positive time whenever the initial distribution has limits at ∓∞.

Mediterranean Journal of Mathematics, 2012

This paper concerns with a class of reaction-diffusion systems with triangular diffusion matrix o... more This paper concerns with a class of reaction-diffusion systems with triangular diffusion matrix on the unbounded domain R n. The system with diagonal diffusion matrix has been studied by J. D. Avrin and F. Rothe in [4]. We prove two new results about uniform boundedness to solutions of such class of reaction-diffusion systems in BU C(R n), the space of bounded uniformly continuous functions from R n to R.

Boundary Value Problems, 2010

We study the following reaction-diffusion system with a cross-diffusion matrix and fractional der... more We study the following reaction-diffusion system with a cross-diffusion matrix and fractional derivatives u t a 1 Δu a 2 Δv − c 1 −Δ α1 u − c 2 −Δ α2 v 1 ω f 1 x, t in Ω× 0, t * , v t b 1 Δu b 2 Δv − d 1 −Δ β1 u − d 2 −Δ β2 v 1 ω f 2 x, t in Ω× 0, t * , u v 0 on ∂Ω× 0, t * , u x, 0 u 0 x , v x, 0 v 0 x in x ∈ Ω, where Ω ⊂ R N N ≥ 1 is a smooth bounded domain, u 0 , v 0 ∈ L 2 Ω , the diffusion matrix M a1 a2 b1 b2 has semisimple and positive eigenvalues 0 < ρ 1 ≤ ρ 2 , 0 < α 1 , α 2 , β 1 , β 2 < 1, ω ⊂ Ω is an open nonempty set, and 1 ω is the characteristic function of ω. Specifically, we prove that under some conditions over the coefficients a i , b i , c i , d i i 1, 2 , the semigroup generated by the linear operator of the system is exponentially stable, and under other conditions we prove that for all t * > 0 the system is approximately controllable on 0, t * .

Electronic Journal of Qualitative Theory of Differential Equations, 2020

Global existence, positivity, uniform boundedness and extinction results of solutions to a system... more Global existence, positivity, uniform boundedness and extinction results of solutions to a system of reaction-diffusion equations on unbounded domain modeling two species on a predator-prey relationship is considered.

Global existence, positivity, uniform boundedness and extinction results of solutions to a system... more Global existence, positivity, uniform boundedness and extinction results of solutions to a system of reaction-diffusion equations on unbounded domain modeling two species on a predator–prey relationship is considered.

This article concerns the generation of analytic semigroups by an operator matrix in the space L2... more This article concerns the generation of analytic semigroups by an operator matrix in the space L2(Ω) × L2(Ω). We assume that one of the diagonal elements is strongly elliptic and the other is weakly elliptic, while the sum of the non-diagonal elements is weakly elliptic.

Recent Developments in the Solution of Nonlinear Differential Equations, 2021

We prove in this work the existence of a unique global nonnegative classical solution to the clas... more We prove in this work the existence of a unique global nonnegative classical solution to the class of reaction–diffusion systemsuttx=aΔutx−guvm,vttx=dΔvtx+λtxguvm,

Applicationes Mathematicae, 1999

We are concerned with the boundedness and large time behaviour of the solution for a system of re... more We are concerned with the boundedness and large time behaviour of the solution for a system of reaction-diffusion equations modelling complex consecutive reactions on a bounded domain under homogeneous Neumann boundary conditions. Using the techniques of E. Conway, D. Hoff and J. Smoller [3] we also show that the bounded solution converges to a constant function as t → ∞. Finally, we investigate the rate of this convergence.

Mathematical Methods in the Applied Sciences, 2016

We prove in this paper a generalized result with a unified proof of global existence in time of c... more We prove in this paper a generalized result with a unified proof of global existence in time of classical solutions to a class of a reaction diffusion system with triangular diffusion matrix on a bounded domain in Rn . The system in question is ut=aΔu − f(x,t,u,v), vt=cΔu + dΔv + ρf(x,t,u,v), x∈Ω⊂Rn(n≥1) , t > 0 with f(x,t,0,η) = 0 and f(x,t,ξ,η)≤Kφ(ξ)eση, for all x∈Ω, t > 0, ξ≥0, η≥0; where ρ, K and σ are real positive constants. Copyright © 2016 John Wiley & Sons, Ltd.

Electronic Journal of Differential Equations, 2006

This article concerns the behavior at mpinftymp infty mpinfty of solutions to a reaction-diffusion system wit... more This article concerns the behavior at mpinftymp infty mpinfty of solutions to a reaction-diffusion system with a cross diffusion matrix on unbounded domains. We show that the solutions satisfy the free diffusion system for all positive time whenever the initial distribution has limits at mpinftymp inftympinfty.

This article concerns the behavior at ∓∞ of solutions to a reactiondiffusion system with a cross ... more This article concerns the behavior at ∓∞ of solutions to a reactiondiffusion system with a cross diffusion matrix on unbounded domains. We show that the solutions satisfy the free diffusion system for all positive time whenever the initial distribution has limits at ∓∞.

Mediterranean Journal of Mathematics, 2012

This paper concerns with a class of reaction-diffusion systems with triangular diffusion matrix o... more This paper concerns with a class of reaction-diffusion systems with triangular diffusion matrix on the unbounded domain R n. The system with diagonal diffusion matrix has been studied by J. D. Avrin and F. Rothe in [4]. We prove two new results about uniform boundedness to solutions of such class of reaction-diffusion systems in BU C(R n), the space of bounded uniformly continuous functions from R n to R.

Boundary Value Problems, 2010

We study the following reaction-diffusion system with a cross-diffusion matrix and fractional der... more We study the following reaction-diffusion system with a cross-diffusion matrix and fractional derivatives u t a 1 Δu a 2 Δv − c 1 −Δ α1 u − c 2 −Δ α2 v 1 ω f 1 x, t in Ω× 0, t * , v t b 1 Δu b 2 Δv − d 1 −Δ β1 u − d 2 −Δ β2 v 1 ω f 2 x, t in Ω× 0, t * , u v 0 on ∂Ω× 0, t * , u x, 0 u 0 x , v x, 0 v 0 x in x ∈ Ω, where Ω ⊂ R N N ≥ 1 is a smooth bounded domain, u 0 , v 0 ∈ L 2 Ω , the diffusion matrix M a1 a2 b1 b2 has semisimple and positive eigenvalues 0 < ρ 1 ≤ ρ 2 , 0 < α 1 , α 2 , β 1 , β 2 < 1, ω ⊂ Ω is an open nonempty set, and 1 ω is the characteristic function of ω. Specifically, we prove that under some conditions over the coefficients a i , b i , c i , d i i 1, 2 , the semigroup generated by the linear operator of the system is exponentially stable, and under other conditions we prove that for all t * > 0 the system is approximately controllable on 0, t * .

Electronic Journal of Qualitative Theory of Differential Equations, 2020

Global existence, positivity, uniform boundedness and extinction results of solutions to a system... more Global existence, positivity, uniform boundedness and extinction results of solutions to a system of reaction-diffusion equations on unbounded domain modeling two species on a predator-prey relationship is considered.