Acta Applicandae Mathematicae最新文献

This paper is devoted to a charge transfer model with cross-diffusion under Neumann boundary conditions. We investigate how the cross-diffusion could destabilize the stable periodic solutions bifurcating from the unique positive equilibrium point. By the implicit function theorem and Floquet theory, we obtain some conditions on the self-diffusion and cross-diffusion coefficients under which the stable periodic solutions can become unstable. New irregular Turing patterns then generate by the destabilization of stable spatially homogeneous periodic solutions. We also present some numerical simulations to further support the results of theoretical analysis.

In this paper, we study the variational inequality over the solution set of the split common solution problem with multiple output sets for monotone operator equations in real Hilbert spaces. We propose a new approach for solving this problem without depending on the norm of the transfer mappings. Some of its applications for solving the variational inequality over the solution set of the split common fixed point problem with multiple output sets and related problems are also mentioned. Finally, we give numerical examples to illustrate the performance of the proposed algorithm and compare it with several existing previous algorithms.

Motivated by the incompressible limit of a cell density model, we propose a free boundary tumor growth model where the pressure satisfies an obstacle problem on an evolving domain (Omega (t)), and the coincidence set (Lambda (t)) captures the emerging necrotic core. We contribute to the analytical characterization of the solution structure in the following two aspects. By deriving a semi-analytical solution and studying its dynamical behavior, we obtain quantitative transitional properties of the solution separating phases in the development of necrotic cores and establish its long time limit with the traveling wave solutions. Also, we prove the existence of traveling wave solutions incorporating non-zero outer densities outside the tumor bulk, provided that the size of the outer density is below a threshold.

Tumour growth is a complex process influenced by various factors, including cell proliferation, migration, and chemotaxis. In this study, a biphasic chemotaxis model for tumour growth is considered, and the effect of chemotherapy on the growth process is investigated. We use optimal control theory to derive the optimized treatment strategy that minimises the tumour size while minimising the toxicity associated with chemotherapy. Moreover, the existence, uniqueness, and strong solution estimates for the biphasic chemotaxis model subsystem in one dimension are derived. These results are achieved through semigroup theory and the truncation method. In addition, the research provides evidence of the existence of an optimal pair through the utilization of the minimising sequence technique. It also demonstrates the differentiability of the mapping from control variable to state variable and establishes the first-order necessary optimality condition. Lastly, a sequence of numerical simulations are presented to showcase the impact of chemotherapy and the influence of parameters in restraining tumour growth when applied in an optimized manner. Our results show that optimal control can provide a more effective and personalised treatment for cancer patients, and the approach can be extended to other tumour growth models.

This work focusses on the well–posedness and the exponential stability of a simplified porous elastic system. By omitting the second time derivative term of the function (varphi ) in the porous elastic system, we make the system free of the damaging consequences of the second spectrum. We consider an elastic dissipation and prove the well–posedness by the use of Hille–Yosida therorem and some arguments of elliptic equations. Next, we use the multiplier method and establish an exponential decay result without any restriction on coefficients of the system.

We consider the propagation of surface water waves described by the Boussinesq system. Following (Molinet et al. in Nonlinearity 34:744–775, 2021), we introduce a regularized Boussinesq system obtained by adding a non-local pseudo-differential operator define by (widehat{g_{lambda }[zeta ]}=|k|^{lambda }hat{zeta }_{k}) with (lambda in ]0,2]). In this paper, we display a twofold approach: first, we study theoretically the existence of an asymptotic expansion for the solution to the Cauchy problem associated to this regularized Boussinesq system with respect to the regularizing parameter (epsilon ). Then, we compute numerically the function coefficients of the expansion (in (epsilon )) and verify numerically the validity of this expansion up to order 2. We also check the numerical (L^{2}) stability of the numerical algorithm.

In this paper, we propose a diffusive prey-predator model with saturated hunting cooperation and predator-taxis. We first establish the global classical solvability and boundedness, and provide some sufficient conditions to assure the existence of a unique positive homogeneous steady state and the global uniform asymptotic stability of the predator-free homogeneous steady state. Secondly, we study the pattern formation mechanism and reveal that pattern formation is driven by the joint effect of predator-taxis, hunting cooperation, and slow diffusivity of predators. Moreover, we find that a strong predator-taxis can annihilate the spatiotemporal patterns, but a weak predator-taxis supports the pattern formation when diffusion-driven instability is present in the model without predator-taxis. However, if diffusion-driven instability is absent, predator-taxis cannot destabilize the unique positive spatially homogeneous steady state. Additionally, we highlight that spatially heterogeneous steady states do not exist when the diffusion coefficient ratio of predators to prey is sufficiently large under specific parametric conditions. To explore the various types of spatially heterogeneous steady states, we derive amplitude equations based on the weakly nonlinear analysis theory. Finally, numerical simulations, including the hexagonal pattern, stripe pattern, a mixed pattern combining hexagons and stripes, and the square pattern, are presented to illustrate the theoretical results.

We first propose a new error bound for the linear complementarity problems when the involved matrices are generalized Nekrasov matrices, which generalizes the recent result obtained by Li et al. (Numer. Algorithms 74:997–1009, 2017). Then we present two new error bounds for the linear complementarity problems when the involved matrices are Nekrasov matrices. Numerical examples are given to illustrate the effectiveness of the proposed results.

Obesity-related factors have been associated with beta cell dysfunction, potentially leading to Type 2 diabetes. To address this issue, we developed a comprehensive obesity-based diabetes model incorporating fat cells, glucose, insulin, and beta cells. We established the model’s global existence, non-negativity, and boundedness. Additionally, we introduced a delay to examine the effects of impaired insulin production resulting from beta-cell dysfunction. Bifurcation analyses were conducted for delay and non-delay models, exploring the model’s dynamic transitions through backward and forward Hopf bifurcations. Utilizing the maximal Pontryagin principle, we formulated and evaluated an optimal control problem to mitigate diabetic complications by reducing the prevalence of overweight individuals and halting disease progression. Comparative graphical outputs were generated to demonstrate the beneficial effects of glucose-regulating medication and regular exercise in managing diabetes.

The chemotaxis system

is considered in a smoothly bounded domain (Omega subset mathbb{R}^{n}) ((n in mathbb{N})), where (chi > 0), (p > 1), (lambda ge 0), (mu > 0), (kappa > 1), and (h = v) or (h = frac{1}{|Omega |} int _{Omega } u). It is firstly proved that if (n = 1) and (p > 1) is arbitrary, or (n ge 2) and (p in (1, frac{n}{n-1})), then for all continuous initial data a corresponding no-flux type initial-boundary value problem for ((ast )) admits a globally defined and bounded weak solution. Secondly, it is shown that if (n ge 2), (Omega = B_{R}(0) subset mathbb{R}^{n}) is a ball with some (R > 0), (p > frac{n}{n-1}) and (kappa > 1) is small enough, then one can find a nonnegative radially symmetric function (u_{0}) and a weak solution of ((ast )) with initial datum (u_{0}) which blows up in finite time.