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Abstract: Abstract 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. PubDate: 2024-06-07

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Abstract: Abstract 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. PubDate: 2024-06-05

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Abstract: Abstract 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. PubDate: 2024-06-03

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Abstract: Abstract 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. PubDate: 2024-06-03

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Abstract: Abstract 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. PubDate: 2024-05-23

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Abstract: Abstract 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. PubDate: 2024-05-22

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Abstract: Abstract 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. PubDate: 2024-05-16

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Abstract: Abstract The chemotaxis system ∗ $$\begin{aligned} \textstyle\begin{cases} u_{t}=\Delta u - \chi \nabla \cdot (u \nabla v ^{p-2}\nabla v) + \lambda u - \mu u^{\kappa }, \\ 0=\Delta v + u - h(u,v) \end{cases}\displaystyle \end{aligned}$$ 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. PubDate: 2024-05-16

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Abstract: Abstract In this investigation, we delve into the dynamics of an ecoepidemic model, considering the intertwined influences of fear, refuge-seeking behavior, and alternative food sources for predators with selective predation. We extend our model to incorporate the impact of fluctuating environmental noise on system dynamics. The deterministic model undergoes thorough scrutiny to ensure the positivity and boundedness of solutions, with equilibria derived and their stability properties meticulously examined. Furthermore, we explore the potential for Hopf bifurcation within the system dynamics. In the stochastic counterpart, we prioritize discussions on the existence of a globally positive solution. Through simulations, we unveil the stabilizing effect of the fear factor on susceptible prey reproduction, juxtaposed against the destabilizing roles of prey refuge behavior and disease prevalence intensity. Notably, when disease prevalence intensity is too low, the infection can be eradicated from the ecosystem. Our deterministic analysis reveals a complex interplay of factors: the system destabilizes initially but then stabilizes as the fear factor suppressing disease prevalence intensifies, or as predators exhibit a stronger preference for infected prey over susceptible ones, or as predators are provided with more alternative food sources. Moreover, for the stochastic system, the oscillations tend to cluster around the coexistence equilibrium of the corresponding deterministic model when white noise intensity is low. However, with increasing white noise intensity, oscillation amplitudes escalate. Critically, very high levels of white noise can lead to the eradication of infection from the ecosystem. PubDate: 2024-05-14

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Abstract: Abstract We study the long-time behaviour of a run and tumble model which is a kinetic-transport equation describing bacterial movement under the effect of a chemical stimulus. The experiments suggest that the non-uniform tumbling kernels are physically relevant ones as opposed to the uniform tumbling kernel which is widely considered in the literature to reduce the complexity of the mathematical analysis. We consider two cases: (i) the tumbling kernel depends on the angle between pre- and post-tumbling velocities, (ii) the velocity space is unbounded and the post-tumbling velocities follow the Maxwellian velocity distribution. We prove that the probability density distribution of bacteria converges to an equilibrium distribution with explicit (exponential for (i) and algebraic for (ii)) convergence rates, for any probability measure initial data. To the best of our knowledge, our results are the first results concerning the long-time behaviour of run and tumble equations with non-uniform tumbling kernels. PubDate: 2024-05-14

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Abstract: Abstract Random Batch Methods (RBM) for mean-field interacting particle systems enable the reduction of the quadratic computational cost associated with particle interactions to a near-linear cost. The essence of these algorithms lies in the random partitioning of the particle ensemble into smaller batches at each time step. The interaction of each particle within these batches is then evolved until the subsequent time step. This approach effectively decreases the computational cost by an order of magnitude while increasing the amount of fluctuations due to the random partitioning. In this work, we propose a variance reduction technique for RBM applied to nonlocal PDEs of Fokker-Planck type based on a control variate strategy. The core idea is to construct a surrogate model that can be computed on the full set of particles at a linear cost while maintaining enough correlations with the original particle dynamics. Examples from models of collective behavior in opinion spreading and swarming dynamics demonstrate the great potential of the present approach. PubDate: 2024-05-09

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Abstract: Abstract The derivation of an approximate Class–I model for nonisothermal multicomponent systems of fluids, as the high-friction limit of a Class–II model is justified, by validating the Chapman–Enskog expansion performed from the Class–II model towards the Class–I model. The analysis proceeds by comparing two thermomechanical theories via relative entropy. PubDate: 2024-05-08

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Abstract: Abstract This work proposes a protocol for Fermionic Hamiltonian learning. For the Hubbard model defined on a bounded-degree graph, the Heisenberg-limited scaling is achieved while allowing for state preparation and measurement errors. To achieve \(\epsilon \) -accurate estimation for all parameters, only \(\tilde{\mathcal{O}}(\epsilon ^{-1})\) total evolution time is needed, and the constant factor is independent of the system size. Moreover, our method only involves simple one or two-site Fermionic manipulations, which is desirable for experiment implementation. PubDate: 2024-04-30

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Abstract: This paper concerns the polynomial decay of the dissipative wave equation subject to Kinetic boundary condition and non-neglected density in the square. After reformulating this problem into an abstract Cauchy problem, we show the existence and uniqueness of the solution. Then, by analyzing a family of eigenvalues of the corresponding operator, we prove that the rate of energy decay decreases in a polynomial way. PubDate: 2024-04-22

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Abstract: Abstract This paper is dedicated to studying the steady state problem of a population-toxicant model with negative toxicant-taxis, subject to homogeneous Neumann boundary conditions. The model captures the phenomenon in which the population migrates away from regions with high toxicant density towards areas with lower toxicant concentration. This paper establishes sufficient conditions for the non-existence and existence of non-constant positive steady state solutions. The results indicate that in the case of a small toxicant input rate, a strong toxicant-taxis mechanism promotes population persistence and engenders spatially heterogeneous coexistence (see, Theorem 2.3). Moreover, when the toxicant input rate is relatively high, the results unequivocally demonstrate that the combination of a strong toxicant-taxis mechanism and a high natural growth rate of the population fosters population persistence, which is also characterized by spatial heterogeneity (see, Theorem 2.4). PubDate: 2024-04-12

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Abstract: Abstract In this paper, we consider a system of two degenerate wave equations coupled through the velocities, only one of them being controlled. We assume that the coupling parameter is sufficiently small and we focus on null controllability problem. To this aim, using multiplier techniques and careful energy estimates, we first establish an indirect observability estimate for the corresponding adjoint system. Then, by applying the Hilbert Uniqueness Method, we show that the indirect boundary controllability of the original system holds for a sufficiently large time. PubDate: 2024-04-12

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Abstract: Abstract We consider in this paper numerical approximation and simulation of a two-species Keller-Segel model. The model enjoys an energy dissipation law, mass conservation and bound or positivity preserving for the population density of two species. We construct a class of very efficient numerical schemes based on the generalized scalar auxiliary variable with relaxation which preserve unconditionally the essential properties of the model at the discrete level. We conduct a sequence of numerical tests to validate the properties of these schemes, and to study the blow-up phenomena of the model in a three-dimensional domain in parabolic-elliptic form and parabolic-parabolic form. PubDate: 2024-04-10

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Abstract: Abstract We deal with the following predator-prey model involving nonlinear indirect chemotaxis mechanism $$ \left \{ \textstyle\begin{array}{l@{\quad }l} u_{t}=\Delta u+\xi \nabla \cdot (u \nabla w)+a_{1}u(1-u^{r_{1}-1}-b_{1}v), \ &\ \ x\in \Omega , \ t>0, \\ v_{t}=\Delta v-\chi \nabla \cdot (v \nabla w)+a_{2}v(1-v^{r_{2}-1}+b_{2}u), \ &\ \ x\in \Omega , \ t>0, \\ w_{t}=\Delta w-w+z^{\gamma }, \ &\ \ x\in \Omega , \ t>0, \\ 0=\Delta z-z+u^{\alpha }+v^{\beta }, \ &\ \ x\in \Omega , \ t>0 , \end{array}\displaystyle \right . $$ under homogeneous Neumann boundary conditions in a bounded and smooth domain \(\Omega \subset \mathbb{R}^{n}\) ( \(n\geq 1\) ), where the parameters \(\xi ,\chi ,a_{1},a_{2},b_{1},b_{2},\alpha ,\beta ,\gamma >0\) . It has been shown that if \(r_{1}>1\) , \(r_{2}>2\) and \(\gamma (\alpha +\beta )<\frac{2}{n}\) , then there exist some suitable initial data such that the system has a global classical solution \((u,v,w,z)\) , which is bounded in \(\Omega \times (0,\infty )\) . Compared to the previous contributions, in this work, the boundedness criteria are only determined by the power exponents \(r_{1}\) , \(r_{2}\) , \(\alpha \) , \(\beta \) , \(\gamma \) and spatial dimension \(n\) instead of the coefficients of the system and the sizes of initial data. PubDate: 2024-04-10

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Abstract: Abstract In this paper, we prove a new blowup criterion for the strong solution to the Cauchy problem of three-dimensional micropolar fluid equation with vacuum. Specifically, we establish a blowup criterion in terms of \(L_{t}^{\infty }L_{x}^{q}\) of the density, where \(1< q<\infty \) , and it is independent on the velocity of rotation of the microscopic particles. PubDate: 2024-04-04 DOI: 10.1007/s10440-024-00642-5

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Abstract: Abstract The present study investigates an algorithm numerically for finding the solution of partial differential equation with differences involved singular perturbation parameter(SPPDE) on non-uniform grid. Taylor series expansion provides a close approximation of the delay and advance terms in the convection-diffusion terms. After the approximations in shift containing terms, we applied the Crank-Nicolson application on uniform grid in the vertical direction. Subsequently, the resultant system is employed by the method of tension spline on a piece-wise uniform grid. Empirical evidence has shown that the suggested approach exhibits second-order characteristics in both the spatial and temporal dimensions. The effectiveness of derived scheme demonstrated through the solution of examples and the results are compared with existed methods. In the conclusion section, we will discuss the effect of shift parameters behavior for various singular perturbation parameter. PubDate: 2024-04-03 DOI: 10.1007/s10440-024-00645-2