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Öğe Boundedness of solutions to a quasilinear parabolic-parabolic chemotaxis model with variable logistic source(Springer Int Publ Ag, 2022) Ayazoglu, Rabil; Akkoyunlu, EbubekirThis paper deals with the higher dimension quasilinear parabolic-parabolic chemotaxis model involving a source term of logistic type u(t) = del . (phi(u)del u) - V . (psi(u)del upsilon) + g(x, u), tau upsilon(t) = del upsilon - upsilon + u, in (x, t) is an element of Omega x (0, T), subject to nonnegative initial data and homogeneous Neumann boundary condition, where Omega is a smooth and bounded domain in R-N, N >= 1 and psi, phi, g are smooth, positive functions satisfying nu s(q) <= psi <= chi s(q), phi >= sigma s(p), p,q is an element of R, nu, chi, sigma > 0 when s >= s(0) >1, g(x, s) <= eta s(k(x))-mu s(rn)(x) for s > 0, eta >= 0, mu > 0 constants and g(x, 0) >= 0, x is an element of Omega, where k, m are measurable functions with 0 <= k(-) := ess(x is an element of Omega )infk (x) <= k(x) <= m(+) := ess(x is an element of Omega)sup k(x) < +infinity, 1 < m(-) := ess(x is an element of Omega)infm (x) <= m(x) <= m(+) := ess(x is an element of Omega)sup m(x)< +infinity. We extend the constant exponents k = {0, 1} , m > 1 which in logistic source term g(s) < eta s(k) - mu s(m) for s > 0, eta >= 0 , > 0 as variable exponents k(.) >= 0, m(.) > 1 with k(+) < m(-) . It is proved that if q = m(-) -1 (critical case) with mu properly large that mu > mu(0) for some mu(0) > 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 mu.Öğe Bounds for the blow-up time a class of integro-differential problem of parabolic type with variable reaction term(Acad Sciences, 2023) Ayazoglu, Rabil; Akkoyunlu, EbubekirThis 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.Öğe DYNAMICS IN A PARABOLIC-ELLIPTIC CHEMOTAXIS SYSTEM WITH LOGISTIC SOURCE INVOLVING EXPONENTS DEPENDING ON THE SPATIAL VARIABLES(Amer Inst Mathematical Sciences-Aims, 2024) Ayazoglu, Rabil; Kadakal, Mahir; Akkoyunlu, EbubekirWe consider the parabolic-elliptic chemotaxis system with the exponents depending on the spatial variables logistic source and nonlinear signal production: ut = Delta u-chi del (u del upsilon)+f (x, u), (x, t) is an element of Omega x (0, T), 0 = Delta upsilon - upsilon +u(gamma) in a bounded domain Omega subset of R-N (N > 1) with smooth boundary, subject to non negative initial data and homogeneous Neumann boundary conditions, where chi > 0, gamma >= 1 and partial derivative/partial derivative nu denotes the outward normal derivative on partial derivative Omega. The logistic function f fulfilling f (x, s) <= eta s - mu s(alpha(x)+1), eta >= 0, mu > 0 for all s > 0 with f (x, 0) >= 0 for all x is an element of Omega, where alpha : Omega -> [1, infinity) is a measurable function. It is proved that if 1 <= alpha (x) < infinity for all x is an element of Omega such that ess inf(x is an element of Omega) alpha (x) > gamma or ess inf(x is an element of Omega) alpha (x) = gamma with mu > chi, then there exists a nonnegative classical solution (u, upsilon) that is global-in-time and bounded. In addition, under the particular conditions gamma = 1 and f (x, s) = mu (s - s(alpha(x)+1)), if mu is sufficiently large, the global bounded solution (u, upsilon) satisfies IIu (, t) - 1II(L)infinity(Omega) + II upsilon (, t) - 1II(L)infinity(Omega) <= Ce (- k/N+2t) for all t > 0 with k = min{ chi 2/4 , 1/2 } , 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.Öğe Existence and multiplicity of solutions for p(.)-Kirchhoff-type equations(Tubitak Scientific & Technological Research Council Turkey, 2022) AyazoClu, Rabil; Akbulut, Sezgin; Akkoyunlu, EbubekirThis paper is concerned with the existence and multiplicity of solutions of a Dirichlet problem for p(.)- Kirchhoff-type equation {M(integral(Omega)vertical bar del vertical bar p(x)/p(x)dx) (-Delta(p(x))u) = f(x,u), in Omega, u = 0, on partial derivative Omega 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.Öğe Existence of multiple solutions of Schrodinger-Kirchhoff-type equations involving the p(.) -Laplacian in RN(Wiley, 2020) Ayazoglu (Mashiyev), Rabil; Akbulut, Sezgin; Akkoyunlu, EbubekirIn this paper, we prove the existence of multiple solutions for the nonhomogeneous Schrodinger-Kirchhoff-type problem involving the p(.)-Laplacian {-(1+b integral(N)(R)1/p(x)vertical bar del u vertical bar(p(x)) dx) Delta(p(x))u+V(x)vertical bar u vertical bar p((x)-2) u=f(x,u) + g(x) in R-N, u is an element of W-1,W-p(.)(R-N), where b >= 0 is a constant, N >= 2, Delta(p)(.)u := div(vertical bar del u vertical bar p((.)-2)del u) is the p(.)-Laplacian operator, p : R-N -> R is Lipschitz continuous, V : R-N -> R is a coercive type potential, integral : R-N x R -> R and g : R-N -> R functions verifying suitable conditions. We propose different assumptions on the nonlinear term f : R-N x R -> R 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.Öğe Existence of solutions for anisotropic parabolic Ni-Serrin type equations originated from a capillary phenomena with nonstandard growth nonlinearity(Taylor & Francis Ltd, 2024) Ayazoglu, Rabil; Akkoyunlu, Ebubekir; Naghizadeh, ZohrehWe 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.Öğe Extinction properties of solutions for a parabolic equation with a parametric variable exponent nonlinearity(Institute of Mathematics and Mechanics, National Academy of Sciences of Azerbaijan, 2022) Ayazoglu Mashiyev, Rabil; Akkoyunlu, EbubekirIn 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 . © 2022, Institute of Mathematics and Mechanics, National Academy of Sciences of Azerbaijan. All rights reserved.Öğe Infinitely many solutions for the stationary fractional p-Kirchhoff problems in RN(Springer India, 2019) Akkoyunlu, Ebubekir; Ayazoglu, RabilIn the present paper, we investigate the existence of multiple solutions for the nonhomogeneous fractional p-Kirchhoff equation M(integral integral R2N vertical bar u(x) - u(y)vertical bar(p)/vertical bar x - y vertical bar(N+ps)dxdy + integral V-RN(x)vertical bar u vertical bar p dx) x((-Delta)(p)(s) u + V(x) vertical bar u vertical bar(p-2) u) = f (x, u) in R-N, where (-Delta)(p)(s) is the fractional p-Laplacian operator, 0 < s < 1 < p < infinity with sp < N, M : R-0(+) -> R-0(+) is a nonnegative, continuous and increasing Kirchhoff function, the nonlinearity f : R-N x R -> R is a Caratheodory function that obeys some conditions which will be stated later and V is an element of C(R-N, 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.Öğe On global existence and bounds for the blow-up time in a semilinear heat equation involving parametric variable sources(Acad Sciences, 2021) Ayazoglu (Mashiyev), Rabil; Akkoyunlu, Ebubekir; Aydin, Tuba AgirmanThis 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.Öğe Uniform Boundedness of Kantorovich Operators in Variable Exponent Lebesgue Spaces(Univ Nis, Fac Sci Math, 2019) Ayazoglu (Mashiyev), Rabil; Akbulut, Sezgin; Akkoyunlu, EbubekirIn this paper, the Kantorovich operators K-n, n is an element of 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 K-n(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.
