Akkoyunlu, Ebubekir

This paper deals with the following Neumann initial-boundary value problem (Formula presented.) where (Formula presented.) (Formula presented.) is a bounded domain with smooth boundary, (Formula presented.), (Formula presented.), (Formula presented.), (Formula presented.), and (Formula presented.). The initial data satisfy (Formula presented.) where (Formula presented.), (Formula presented.), and (Formula presented.)  on (Formula presented.). We rigorously prove that the problem admits a unique global classical solution which is uniformly bounded in time if one of the following conditions holds: (Formula presented.) or (Formula presented.).

İncele

Ayazoglu, Rabil; Akkoyunlu, Ebubekir; Naghizadeh, Zohreh

We consider an initial boundary value problem for a class of anisotropic parabolic Ni-Serrin type equations with nonstandard nonlinearity in a bounded smooth domain with homogeneous Dirichlet boundary condition. Because the nonlinear perturbation leads to difficulties (it does not have a definite sign) in obtaining a priori estimates in the energy method, we had to modify the Tartar method significantly. Under suitable assumptions, we obtain the global existence, decay and extinction of solutions.

AtıfWoS1·OpenAlex1

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Akkoyunlu, Ebubekir

This paper is concerned with the attraction-repulsion chemotaxis system involving logistic source: ut = ∆u − χ∇ · (u∇υ) + ξ∇ · (u∇ω) + f(u), ρυt = ∆υ − α1υ + β1u, ρωt = ∆ω − α2ω + β2u under homogeneous Neumann boundary conditions with nonnegative initial data (u0, υ0, ω0) (W1,∞ (Ω))3,the parameters χ, ξ,α1, α2, β1, β2 > 0, ρ ≥ 0 subject to the non-flux boundary conditions in a bounded domain Ω c RN (N ≥ 3) with smooth boundary and f(u) ≤ au − µu2 with f(0) ≥ 0 and a ≥ 0, µ > 0 for all u > 0. Based on the maximal Sobolev regularity and semigroup technique, it is proved that the system admits a globally bounded classical solution provided that χ + ξ <µ/2 and there exists a constant β* > 0 is sufficiently small for all β1, β2 < β*.

İncele

Akkoyunlu, Ebubekir

One of the fundamental processes in ecology is the interaction between predator and prey. Predator-prey interactions refer to the relative changes in population density of two species as they share the same environment and one species preys on the other. There are many studies global existence or blow-up of solutions on the predator-prey model. Our this paper related to the predator-prey model with nonlinear indirect chemotaxis mechanism under homogeneous Neumann boundary conditions. We establish the global existence and boundedness of classical solutions of our problem by using parabolic regularity theory. Namely, firstly we show that and boundedness in for some, then we obtain the-bound of and by using Alikakos-Moser iteration. Thus, it is proved that the model has a unique global classical solution under suitable conditions on the parameters in a smooth bounded domain.

AtıfScopus1·OpenAlex1

İncele

Ayazoglu, Rabil; Kadakal, Mahir; Akkoyunlu, Ebubekir

We consider the parabolic-elliptic chemotaxis system with the exponents depending on the spatial variables logistic source and nonlinear signal production: ut = ∆u−χ∇·(u∇υ)+f(x, u), (x, t) ∈ Ω×(0, T), 0 = ∆υ−υ+uγ in a bounded domain Ω ⊂ RN (N ≥ 1) with smooth boundary, subject to nonnegative initial data and homogeneous Neumann boundary conditions, where χ > 0, γ ≥ 1 and ∂ν∂ denotes the outward normal derivative on ∂Ω. The logistic function f fulfilling f(x, s) ≤ ηs− µsα(x)+1, η ≥ 0, µ > 0 for all s > 0 with f(x, 0) ≥ 0 ∀x ∈ Ω, where α : Ω → [1, ∞) is a measurable function. It is proved that if 1 ≤ α(x) < ∞ for all x ∈ Ω such that essinfx∈Ω α(x) > γ or essinfx∈Ω α(x) = γ with µ > χ, then there exists a nonnegative classical solution (u, υ) that is global-in-time and bounded. In addition, under the particular conditions γ = 1 and f(x, s) = µ(s − sα(x)+1), if µ is sufficiently large, the global bounded solution (u, υ) satisfies ∥u(·, t) − 1∥L∞(Ω) + ∥υ (·, t) − 1∥L∞(Ω) ≤ Ce− Nk+2 t for all t > 0 with k = min n χ42 12o , C > 0. The global-in-time existence and uniform-in-time boundedness of solutions are established under specific parameter conditions, which improves the known results.

AtıfScopus11·WoS9·OpenAlex11

İncele

Ayazoglu, Rabil; Akkoyunlu, Ebubekir

This paper is concerned with the blow-up time of the solutions to an integro-differential problem of parabolic type with variable growth if blow-up occurs. By using the differential inequality technique, we obtain lower bounds for the blow-up time and some global existence results under some conditions to variable exponent of reaction, memory kernel, and initial value.

AtıfScopus1·WoS1

İncele

Ayazoglu Mashiyev, Rabil; Akkoyunlu, Ebubekir

In this paper, we study a class of p(·)-Laplace equation including nonstandard growth nonlinearity in a bounded smooth domain with homogeneous Dirichlet boundary condition. We establish the conditions of non-extinction and extinction are studied of global weak solutions in finite time for any initial data u0 . Moreover, we show the global existence results for N ≥ 1 with constant p for any initial data u0 .

AtıfScopus2

İncele

Ayazoğlu, Rabil; Akbulut, Sezgin; Akkoyunlu, Ebubekir

This paper is concerned with the existence and multiplicity of solutions of a Dirichlet problem for p(.) - Kirchhoff-type equation (Formula Presented). Using the mountain pass theorem, fountain theorem, dual fountain theorem and the theory of the variable exponent Sobolev spaces, under appropriate assumptions on f and M, we obtain results on existence and multiplicity of solutions

AtıfScopus9·WoS9·OpenAlex9

İncele

Ayazoglu, Rabil; Akkoyunlu, Ebubekir

This paper deals with the higher dimension quasilinear parabolic–parabolic chemotaxis model involving a source term of logistic type ut= ∇ · (ϕ(u) ∇ u) - ∇ · (ψ(u) ∇ υ) + g(x, u) , τυt= Δ υ- υ+ u, in(x,t)∈Ω×(0,T), subject to nonnegative initial data and homogeneous Neumann boundary condition, where Ω is a smooth and bounded domain in RN, N≥ 1 and ψ, ϕ, g are smooth, positive functions satisfying νsq≤ ψ≤ χsq, ϕ≥ σsp, p, q∈ R, ν, χ, σ> 0 when s≥ s> 1 , g(x, s) ≤ ηsk(x)- μsm(x) for s> 0 , η≥ 0 , μ> 0 constants and g(x, 0) ≥ 0 , x∈ Ω , where k, m are measurable functions with 0≤k-:=essinfx∈Ωk(x)≤k(x)≤m+:=esssupk(x)x∈Ω<+∞, 1<m-:=essinfx∈Ωm(x)≤m(x)≤m+:=esssupm(x)x∈Ω<+∞. We extend the constant exponents k= { 0 , 1 } , m> 1 which in logistic source term g(s) ≤ ηsk- μsm for s> 0 , η≥ 0 , μ> 0 as variable exponents k(·) ≥ 0 , m(·) > 1 with k+< m-. It is proved that if q= m-- 1 (critical case) with μ properly large that μ > μ for some μ> 0 , then there exists a classical solution which is global in time and bounded. Furthermore, if q< m-- 1 , we prove that the classical solutions to the above system are uniformly in-time-bounded without restriction on μ.

AtıfScopus10·WoS8·OpenAlex10

İncele

Ayazoglu, Rabil; Akkoyunlu, Ebubekir; Aydin, Tuba Agirman

This paper is concerned with the blow-up of the solutions to a semilinear heat equation with a reaction given by parametric variable sources. Some conditions to parameters and exponents of sources are given to obtain lower–upper bounds for the time of blow-up and some global existence results.

AtıfScopus3·WoS2·OpenAlex2

İncele

Ayazoglu, Rabil; Akbulut, Sezgin; Akkoyunlu, Ebubekir

In this paper, we prove the existence of multiple solutions for the nonhomogeneous Schrödinger-Kirchhoff-type problem involving the p(.)-Laplacian (Formula presented.) where b≥0 is a constant, N ≥2, Δp(.)u:=div(|∇u|p(.)−2∇u) is the p(.)-Laplacian operator, (Formula presented.) is Lipschitz continuous, (Formula presented.) is a coercive type potential, (Formula presented.) and (Formula presented.) functions verifying suitable conditions. We propose different assumptions on the nonlinear term (Formula presented.) to yield bounded Palais-Smale sequences and then prove that the special sequences we found converge to critical points, respectively. The solutions are obtained by the Mountain Pass Theorem, Ekeland variational principle, and Krasnoselskii genus theory.

AtıfScopus1·WoS1

İncele

Ayazoglu, Rabil; Akbulut, Sezgin; Akkoyunlu, Ebubekir

In this paper, the Kantorovich operators Kn, n ϵ N are shown to be uniformly bounded in variable exponent Lebesgue spaces on the closed interval [0, 1]. Also an upper estimate is obtained for the difference Kn (f) − f for functions f of regularity of order 1 and 2 measured in variable exponent Lebesgue spaces, which is of interest on its own and can be applied to other problems related to the Kantorovich operators.

AtıfScopus3·WoS3·OpenAlex3

İncele

Akkoyunlu, Ebubekir; Ayazoglu, Rabil

In the present paper, we investigate the existence of multiple solutions for the nonhomogeneous fractional p-Kirchhoff equation M(∬R2N|u(x)-u(y)|p|x-y|N+psdxdy+∫RNV(x)|u|pdx)×((-Δ)psu+V(x)|u|p-2u)=f(x,u)inRN,where (-Δ)ps is the fractional p-Laplacian operator, 0 < s< 1 < p< ∞ with sp< N, M:R0+→R0+ is a nonnegative, continuous and increasing Kirchhoff function, the nonlinearity f: RN× R→ R is a Carathéodory function that obeys some conditions which will be stated later and V∈ C(RN, R+) is a non-negative potential function. We first establish the Bartsch–Pankov–Wang type compact embedding theorem for the fractional Sobolev spaces. Then multiplicity results are obtained by using the variational method, (S+) mapping theory and Krasnoselskii’s genus theory.

AtıfScopus4·WoS4

İncele