Ayazoglu, Rabil; Salmanova, Kamala A.

This paper deals with the higher dimension quasilinear parabolic-parabolic chemotaxis model involving a source term of logistic type (formula presented), subject to the homogeneous Neumann boundary conditions in a Ω ⊂ RN (N ≥ 1) with smooth boundary. It is shown that for the attractionrepulsion case with χ2 ≤ 0, the global boundedness of solutions can be ensured by µ1, µ2 > 0 without any other assumptions, due to the contribution of the logistic sources included in addition to the repulsion mechanism. While for the attraction-attraction case with χ2 > 0, the global boundedness of solutions has to require logistic coefficients µ1, µ2 > 0 such that µ2 properly large.

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İncele

Ayazoglu, Rabil

We study the quasilinear attraction-repulsion chemotaxis system of parabolic-elliptic type ut = ∆u−χ∇·(u∇υ)+ξ∇·(u∇ω) with nonlinear production 0 = ∆υ −βυ +αus, 0 = ∆ω −δω +γur, subject to the homogeneous Neumann boundary conditions in a bounded domain Ω ⊂ ℝN (N ≥ 3) with smooth boundary. It is proved that for every α, β, δ, γ, χ, ξ > 0 and s ≥ N2,r > N−2/N s, there exists ξ∗ > 0 such that if ξ > ξ∗ and any sufficiently regular initial datum u0(x) ≥ 0, then the model has a unique global classical solution (u, υ, ω), which is bounded in Ω × (0, ∞).

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 .

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İncele

Ayazoglu, Rabil; Şule Şener, Sidika; Aydın, Tuba Ağırman

The present paper is concerned with the resonant problem −div (a (x, ∇u)) = λ1|u|p −2 u + f (x, u) − g(x) in Ω, where Ω is a bounded domain with smooth boundary in RN (N ≥ 2), p ∈ (1, ∞) and div(a (x, ∇u)) is a general elliptic operators in divergence form. By assuming a Landesman-Lazer type condition and using a variational method based on the Minimum Principle, we show the existence of a weak solution in the Sobolev space W0 1,p (Ω).

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