TY - JOUR AB - Using the (Formula presented.) -equivariant degree, we develop a global Hopf bifurcation theory for system of differential equations with multiple threshold-type state-dependent delays whose prototype is the human respiratory system with multiple blood transport time delays. To establish a theoretic framework for modeling practices of periodic breathing, we further investigate the periodic oscillations of carbon dioxide concentrations in the lung, brain, and tissue compartments and conduct a local and global Hopf bifurcation analysis for the model when varying the commensurate scale of the multiple delays in a transformed system. Such a global Hopf bifurcation will indicate the onset and persistence of the periodic oscillations. AU - Efendiyev, M.A. AU - Hu, Q.* C1 - 68861 C2 - 53686 CY - 111 River St, Hoboken 07030-5774, Nj Usa SP - 1065-1094 TI - Global Hopf bifurcation for differential equations with multiple threshold-type state-dependent delays. JO - Math. Meth. Appl. Sci. VL - 47 IS - 2 PB - Wiley PY - 2024 SN - 0170-4214 ER - TY - JOUR AB - In the article, we establish the global well-posedness in (Formula presented.) of the integro-differential equation in the case of anomalous diffusion when the one-dimensional negative Laplace operator is raised to a fractional power in the presence of the transport term. The model is relevant to the cell population dynamics in the mathematical biology. Our proof relies on a fixed point technique. AU - Efendiyev, M.A. AU - Vougalter, V.* C1 - 71855 C2 - 56240 CY - 111 River St, Hoboken 07030-5774, Nj Usa TI - On the well-posedness of some model arising in the mathematical biology. JO - Math. Meth. Appl. Sci. PB - Wiley PY - 2024 SN - 0170-4214 ER - TY - JOUR AB - We analyze existence and asymptotic behavior of a system of semilinear diffusion-reaction equations that arises in the modeling of the mitochondrial swelling process. The model itself expands previous work in which the mitochondria were assumed to be stationary, whereas now their movement is modeled by linear diffusion. While in the previous model certain formal structural conditions were required for the rate functions describing the swelling process, we show that these are not required in the extended model. Numerical simulations are included to visualize the solutions of the new model and to compare them with the solutions of the previous model. AU - Efendiyev, M.A. AU - Ôtani, M.* AU - Eberl, H.J.* C1 - 52822 C2 - 44184 CY - Hoboken SP - 2162-2177 TI - Analysis of a PDE model of the swelling of mitochondria accounting for spatial movement. JO - Math. Meth. Appl. Sci. VL - 41 IS - 5 PB - Wiley PY - 2018 SN - 0170-4214 ER - TY - JOUR AB - A cell growth model for a size-structured cell population with a stochastic growth rate for size and division into two daughter cells of unequal size is studied in this paper. The model entails an initial boundary value problem that involves a second-order parabolic partial differential equation with two nonlocal terms, the presence of which is a consequence of asymmetry in the cell division. The solution techniques for solving such problems are rare due to the nonlocal terms. In this paper, we solve the initial boundary value problem for arbitrary initial distributions. We obtain a separable solution, as well as the general solution to the partial differential equation, and show that the solutions converge to the separable solution for large time. As in the symmetric division case, the dispersion term does not affect the rate of convergence to the separable solution. AU - Efendiyev, M.A. AU - van Brunt, B.* AU - Zaidi, A.A.* AU - Shah, T.H.* C1 - 54534 C2 - 45569 SP - 8059-8069 TI - Asymmetric cell division with stochastic growth rate. Dedicated to the memory of the late Spartak Agamirzayev. JO - Math. Meth. Appl. Sci. VL - 41 IS - 17 PY - 2018 SN - 0170-4214 ER - TY - JOUR AB - In this paper we solve an initial-boundary value problem that involves a pde with a nonlocal term. The problem comes from a cell division model where the growth is assumed to be stochastic. The deterministic version of this problem yields a first-order pde; the stochastic version yields a second-order parabolic pde. There are no general methods for solving such problems even for the simplest cases owing to the nonlocal term. Although a solution method was devised for the simplest version of the first-order case, the analysis does not readily extend to the second-order case. We develop a method for solving the second-order case and obtain the exact solution in a form that allows us to study the long time asymptotic behaviour of solutions and the impact of the dispersion term. We establish the existence of a large time attracting solution towards which solutions converge exponentially in time. The dispersion term does not appear in the exponential rate of convergence. AU - Efendiyev, M.A. AU - van Brunt, B.* AU - Wake, G.C.* AU - Zaidi, A.A.* C1 - 52588 C2 - 44034 TI - A functional partial differential equation arising in a cell growth model with dispersion. JO - Math. Meth. Appl. Sci. PY - 2017 SN - 0170-4214 ER - TY - JOUR AB - The Breakthrough Starshot Initiative is suggested to develop the concept of propelling a nanoscale spacecraft by the radiation pressure of an intense laser beam. In this project, the nanocraft is a gram-scale robotic spacecraft comprising two main parts: StarChip and Lightsail. To achieve the goal of the project, it is necessary to solve a number of scientific problems. One of these tasks is to make sure that the nanocraft position and orientation inside the intense laser beam column are stable. The nanocraft driven by intense laser beam pressure acting on its Lightsail is sensitive to the torques and lateral forces reacting on the surface of the sail. These forces influence the orientation and lateral displacement of the spacecraft, thus affecting its dynamics. If unstable, the nanocraft might be expelled from the area of laser beam. In choosing the models for nanocraft stability studies, we are using several assumptions: (i) configuration of nanocraft is treated as a rigid body; (ii) flat or concave shape of circular sail; and (iii) mirror reflection of laser beam from surface of the Lightsail. We found conditions of position stability for spherical and conical shapes of the sail. The simplest stable configurations require the StarChip to be removed from the sail to make the distance to the center of mass of the nanocraft bigger than the curvature radius of the sail. Stability criteria do not require the spinning of the nanocraft. A flat sail is never stable. AU - Popova, E.* AU - Efendiyev, M.A. AU - Gabitov, I.* C1 - 50242 C2 - 42160 SP - 1346-1354 TI - On the stability of a space vehicle riding on an intense laser beam. JO - Math. Meth. Appl. Sci. VL - 40 IS - 4 PY - 2017 SN - 0170-4214 ER - TY - JOUR AB - Bacterial biofilms are microbial depositions on immersed surfaces. Their mathematical description leads to degenerate diffusion-reaction equations with two non-Fickian effects: (i) a porous medium equation like degeneracy where the biomass density vanishes and (ii) a super-diffusion singularity if the biomass density reaches its threshold density. In the case of multispecies interactions, several such equations are coupled, both in the reaction terms and in the nonlinear diffusion operator. In this paper, we generalize previous work on existence and uniqueness of solutions of this type of models and give a general, relatively easy to apply criterion for well-posedness. The use of the criterion is illustrated in several examples from the biofilm modeling literature. AU - Sonner, S.* AU - Efendiyev, M.A. AU - Eberl, H.J.* C1 - 45327 C2 - 37288 SP - 3753–3775 TI - On the well-posedness of mathematical models for multicomponent biofilms. JO - Math. Meth. Appl. Sci. VL - 38 IS - 17 PY - 2015 SN - 0170-4214 ER - TY - JOUR AB - In this paper, we give a detailed study of the global attractors for porous medium equations in a heterogeneous medium. Not only the existence but also the infinite dimensionality of the global attractors is obtained by showing that their ϵ-Kolmogorov entropy behaves as a polynomial of the variable 1 ∕ ϵ as ϵ tends to zero, which is not observed for non-degenerate parabolic equations. The upper and lower bounds for the Kolmogorov ϵ-entropy of infinite-dimensional attractors are also obtained. We believe that the method developed in this paper has a general nature and can be applied to other classes of degenerate evolution equations. AU - Efendiyev, M.A. C1 - 8604 C2 - 30202 SP - 1987-1996 TI - Infinite dimensional attractors for porous medium equations in heterogeneous medium. JO - Math. Meth. Appl. Sci. VL - 35 IS - 16 PB - Wiley-Blackwell PY - 2012 SN - 0170-4214 ER - TY - JOUR AB - In this paper, we prove the global in time existence for weak solutions to a Landau-Lifschitz system with magnetostriction arising from the ferromagnetism theory. We describe also the omega-limit set of a solution. AU - Carbou, G.* AU - Efendiyev, M.A. AU - Fabrie, P.* C1 - 4762 C2 - 28902 SP - 1274-1288 TI - Global weak solutions for the Landau-Lifschitz equation with magnetostriction. JO - Math. Meth. Appl. Sci. VL - 34 IS - 10 PB - Wiley-Blackwell PY - 2011 SN - 0170-4214 ER - TY - JOUR AB - We study in this article the long-time behavior of solutions of fourth-order parabolic equations in R-3. In particular, we prove that under appropriate assumptions on the nonlinear interaction function and on the external forces, these equations possess infinite-dimensional exponential attractors whose Kolmogorov's epsilon-entropy satisfies an estimate of the same type as that obtained previously for the epsilon-entropy of the global attractor. AU - Efendiyev, M.A. C1 - 5603 C2 - 28531 SP - 939-949 TI - Infinite-dimensional exponential attractors for fourth-order nonlinear parabolic equations in unbounded domains. JO - Math. Meth. Appl. Sci. VL - 34 IS - 8 PB - Wiley-Blackwell PY - 2011 SN - 0170-4214 ER - TY - JOUR AB - We analyze a system of reaction-diffusion equations that models quorum-sensing in a growing biofilm. The model comprises two nonlinear diffusion effects: a porous medium-type degeneracy and super diffusion. We prove the well-posedness of the model. In particular, we present for the first time a uniqueness result for this type of problem. Moreover, we illustrate the behavior of model solutions in numerical simulations. AU - Sonner, S. AU - Efendiyev, M.A. AU - Eberl, H.J.* C1 - 6844 C2 - 29348 SP - 1667-1684 TI - On the well-posedness of a mathematical model of quorum-sensing in patchy biofilm communities. JO - Math. Meth. Appl. Sci. VL - 34 IS - 13 PB - Wiley-Blackwell PY - 2011 SN - 0170-4214 ER - TY - JOUR AB - We consider the following doubly nonlinear parabolic equation in a bounded domain Omega subset of R-3: f (x, partial derivative(t)u) = Delta(x)u - g (x, u) where the nonlinearity f is allowed to have a degeneracy with respect to partial derivative(t)u of the form partial derivative(t)u vertical bar partial derivative(t)u vertical bar(p) at some points x is an element of Omega. Under some natural assumptions on the nonlinearities f and g, we prove the existence and uniqueness of a solution of that problem and establish the finite-dimensionality of global and exponential attractors of the semigroup associated with this equation in the appropriate phase space. AU - Efendiyev, M.A. AU - Zelik, S.* C1 - 511 C2 - 26980 SP - 1638-1668 TI - Finite-dimensional attractors and exponential attractors for degenerate doubly nonlinear equations. JO - Math. Meth. Appl. Sci. VL - 32 IS - 13 PB - Wiley-Blackwell PY - 2009 SN - 0170-4214 ER - TY - JOUR AB - We consider a chemotaxis-growth model which takes into account diffusion, chemotaxis, production of chemical substance, and growth. We present estimates from above and below of the fractal dimension dim M of the exponential attractor M in terms of the coefficients of the system. Comparisons are made between the sizes of the global and exponential attractors. Numerical simulations are presented which confirm the analytical results obtained AU - Efendiyev, M.A. AU - Kläre, M.* AU - Lasser, R. C1 - 1713 C2 - 28316 SP - 579-594 TI - Dimension estimate of the exponential attractor for the chemotaxis-growth system. JO - Math. Meth. Appl. Sci. VL - 30 IS - 5 PB - Wiley-Blackwell PY - 2007 SN - 0170-4214 ER -