arXiv subjects

Sema Yayla

Publications and source records attributed to Sema Yayla.

4 recordsLinked to original sources

arXiv2026

Long-time dynamics of a bulk-surface convective Cahn--Hilliard system: Pullback attractors and convergence to equilibrium

We study the long-time dynamics of a bulk-surface convective Cahn--Hilliard system describing phase separation processes with bulk-surface interaction. The presence of convection terms leads to a non-autonomous dynamical system and prevents the associated free energy from being a Lyapunov functional, which makes the analysis of the asymptotic behavior considerably more challenging. First, we establish an instantaneous regularization property for weak solutions. Next, interpreting the evolution as a continuous two-parameter process, we prove the existence of a minimal pullback attractor. Finally, under suitable decay assumptions on the velocity fields, we show that every solution converges as $t\to\infty$ to a single steady state. The proof of this convergence relies on the {\L}ojasiewicz--Simon inequality combined with customized decay estimates that compensate for the lack of a monotone energy functional.

math.AP↗

arXiv2021

Long-time dynamics of the Cahn--Hilliard equation with kinetic rate dependent dynamic boundary conditions

We consider a Cahn--Hilliard model with kinetic rate dependent dynamic boundary conditions that was introduced by Knopf, Lam, Liu and Metzger (ESAIM Math. Model. Numer. Anal., 2021) and will thus be called the KLLM model. In the aforementioned paper, it was shown that solutions of the KLLM model converge to solutions of the GMS model proposed by Goldstein, Miranville and Schimperna (Physica D, 2011) as the kinetic rate tends to infinity. We first collect the weak well-posedness results for both models and we establish some further essential properties of the weak solutions. Afterwards, we investigate the long-time behavior of the KLLM model. We first prove the existence of a global attractor as well as convergence to a single stationary point. Then, we show that the global attractor of the GMS model is stable with respect to perturbations of the kinetic rate. Eventually, we construct exponential attractors for both models, and we show that the exponential attractor associated with the GMS model is robust against kinetic rate perturbations.

math.AP↗

arXiv2020

Long-time Dynamics for a Cahn-Hilliard Tumor Growth Model with Chemotaxis

Mathematical models that describe the tumor growth process have been formulated by several authors in order to understand how cancer develops and to develop new treatment approaches. In this study, it is aimed to investigate the long-time behavior of the two-phase diffuse-interface model, which was proposed in [11] to model a tumor tissue as a mixture of cancerous and healthy cells. Up to now, studies on the long-time behavior of this model have neglected chemotaxis and active transport, which have a significant effect on tumor growth. In this research, our main aim is to study this model with chemotaxis and active transport. We prove an asymptotic compactness result for the weak solutions of the problem in the whole phase-space $H^1 (Ω) \times L^2 (Ω)$. We establish the existence of the global attractor in a phase space denoted via mass conservation. We also prove that the global attractor equals to the unstable manifold emanating from the set of stationary points. Moreover, we obtain that the global attractor has a finite fractal dimension.

math.AP↗

arXiv2017

Long-time dynamics of the strongly damped semilinear plate equation in $\mathbb{R}^{n}$

We investigate the initial-value problem for the semilinear plate equation containing localized strong damping, localized weak damping and nonlocal nonlinearity. We prove that if nonnegative damping coefficients are strictly positive almost everywhere in the exterior of some ball and the sum of these coefficients is positive a.e. in $% \mathbb{R} ^{n}$, then the semigroup generated by the considered problem possesses a global attractor in $H^{2}\left( \mathbb{R} ^{n}\right) \times L^{2}\left( \mathbb{R} ^{n}\right) $. We also establish boundedness of this attractor in $ H^{3}\left( \mathbb{R} ^{n}\right) \times H^{2}\left( \mathbb{R} ^{n}\right) $.

math.AP↗