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This study considers the American VIX option pricing problem under the models of mixing the 3/2, and 1/2 classes of volatility processes. The pricing formulas for the perpetual American VIX call are provided for the generalized 1/2 volatility model, the generalized 3/2 volatility model and the model mixing these two processes. For these models, we derive approximate formulas for the finite-lived American VIX call option by the quadratic approximation approach. Then, we illustrate the influence of the varied parameters on the free boundaries and the corresponding continuation regions through numerical analysis. Finally, in the real application, we obtain the price for the finite-lived American VIX call option using the CBOE volatility index. [ABSTRACT FROM AUTHOR]

The typical treatment for tumor involves a combination of radiation therapy and chemotherapy. In this study, we explore a mathematical model based on proliferation and diffusion, considering the impacts of both radiotherapeutic and chemotherapeutic treatments. The mathematical model is formulated as a system of partial differential equations, accompanied by initial, boundary, and free boundary conditions. Existence and uniqueness of this mathematical problem are obtained by developing an iteration algorithm in this paper. [ABSTRACT FROM AUTHOR]

In this paper, we address the utility maximization problem of an infinitely lived agent who has the option to increase their income. The agent can increase their income at any time, but doing so incurs a wealth cost proportional to the amount of the increase. To prevent the agent from infinitely increasing their income and borrowing against future income, we additionally consider a non-negative wealth constraint that prohibits borrowing based on future income. This utility maximization problem is a mixture of stochastic control, where the agent chooses consumption and investment, and singular control, where the agent chooses a non-decreasing income process. To solve this non-trivial and challenging problem, we derive the Hamilton–Jacobi–Bellman (HJB) equation with a gradient constraint using the dynamic programming principle (DPP). Then, using the guess-and-verify method and a linearization technique, we obtain a closed-form solution to the HJB equation and, based on this, find the optimal strategy. [ABSTRACT FROM AUTHOR]

In this paper, we analyze the free boundary problem in three dimensions describing the growth of tumor cords. The model consists of a reaction-diffusion equation describing the concentration $ \sigma $ of nutrients and an elliptic equation describing the distribution of the internal pressure $ p $. The model is defined in a bounded domain in $ \mathbb{R}^{3} $ whose boundary consists of two disjoint closed curves, the known interior part and the unknown exterior part. The concentration of nutrients outside the tumor region is denoted by $ \bar{\sigma} $. We shall show that there is a positive integer $ n^{**} $ and a sequence $ \bar{\sigma}_{n} $ such that for each $ \bar{\sigma}_{n}(n>n^{**}) $, symmetry-breaking stationary solutions bifurcate from the annular stationary solution in the longitude direction. [ABSTRACT FROM AUTHOR]

We consider the incompressible viscous surface wave problem in the setting that the fluid domain is a horizontal infinite layer in 3 D. The fluid dynamics is governed by the gravity-driven incompressible Navier–Stokes equations, and the effect of surface tension is ignored on the upper free surface. We prove the optimal time-decay rate of the low-order energy of the solution with minimal derivative count 3, which implies that the Lipschitz norm of the velocity decays at the rate (1 + t) − 1. This together with a time-weighted estimate for the highest order spatial derivatives of the free surface function leads to the boundedness of the high-order energy, which improves the result of Wang [9]. [ABSTRACT FROM AUTHOR]

During embryonic development of the retina of the eye, astrocytes, a type of glial cell, migrate over the retinal surface and form a dynamic mesh. This mesh then serves as scaffolding for blood vessels to form the retinal vasculature network that supplies oxygen and nutrients to the inner portion of the retina. Astrocyte spreading proceeds in a radially symmetric manner over the retinal surface. Additionally, astrocytes mature from astrocyte precursor cells (APCs) to immature perinatal astrocytes (IPAs) during this embryonic stage. We extend a previously-developed continuum model that describes tension-driven migration and oxygen and growth factor influenced proliferation and differentiation. Comparing numerical simulations to experimental data, we identify model equation components that can be removed via model reduction using approximate Bayesian computation (ABC). Our results verify experimental studies indicating that the choroid oxygen supply plays a negligible role in promoting differentiation of APCs into IPAs and in promoting IPA proliferation, and the hyaloid artery oxygen supply and APC apoptosis play negligible roles in astrocyte spreading and differentiation. [ABSTRACT FROM AUTHOR]

In [3] a mathematical model of the initiation and development of atherosclerosis involving LDL and HDL cholesterol, macrophages, and foam cells was introduced. The model is a highly nonlinear and coupled system of PDEs with a free boundary – the interface between the plaque and the blood flow. We establish infinite branches of symmetry-breaking stationary solutions that bifurcate from the stationary annular solution in the combined longitude-latitude direction. After establishing various estimates for our PDE system, the Crandall-Rabinowitz theorem is applied to prove our main bifurcation theorem. [ABSTRACT FROM AUTHOR]

This paper deals with the free boundary problem for a double‐layered tumor filled with quiescent cells and proliferating cells, where time delay τ>0$$ \tau >0 $$ in cell proliferation is taken into account. These two types of living cells exhibit different metabolic responses and consume nutrients σ$$ \sigma $$ at different rates λ1$$ {\lambda}_1 $$ and λ2$$ {\lambda}_2 $$ (λ1⩽λ2$$ {\lambda}_1\leqslant {\lambda}_2 $$). Time delay happens between the time at which a cell commences mitosis and the time at which the daughter cells are produced. The problem is reduced to a delay differential equation on the tumor radius R(t)$$ R(t) $$ over time, and the difficulty arises from the jump discontinuity of the consumption rate function. We give rigorous analysis on this new model and study the dynamical behavior of the global solutions for any initial φ(t)$$ \varphi (t) $$. [ABSTRACT FROM AUTHOR]

This paper concerns with the one dimensional compressible isentropic Navier–Stokes equations with a free boundary separating fluid and vacuum when the viscosity coefficient depends on the density. Precisely, the pressure P and the viscosity coefficient μ are assumed to be proportional to ρ γ and ρ θ respectively, where ρ is the density, and γ and θ are constants. We establish the unique solvability in the framework of global classical solutions for this problem when γ ≥ θ > 1. Since the previous results on this topic are limited to the case when θ ∈ (0 , 1 ] , the result in this paper fills in the gap for θ > 1. Note that the key estimate is to show that the density has a positive lower bound and the new ingredient of the proof relies on the study of the quasilinear parabolic equation for the viscosity coefficient by reducing the nonlocal terms in order to apply the comparison principle. [ABSTRACT FROM AUTHOR]

In this paper, we study a free boundary problem modeling the growth of 3-dimensional tumor cords. Since tumor cells grow freely in both the longitudinal and cross-sectional directions of blood vessels, the investigation of symmetry-breaking phenomena in both directions is biologically very reasonable. This forces the possible bifurcation value γ m , n to be dependent on two variables m and n. Some monotonicity properties of the possible bifurcation value μ n or μ j obtained in Friedman and Hu (2008) [1] and He and Xing (2023) [2] no longer hold here, which brings a great challenge to the bifurcation analysis. The novelty of this paper lies in determining the order of γ m , n for m 2 + n 2 . Together with periodicity and symmetry, we propose an effective method to avoid the need for the monotonicity of γ m , n. We give symmetry-breaking bifurcation results for every γ m , n > 0. [ABSTRACT FROM AUTHOR]

We study an optimal control problem for a stochastic model of tumour growth with drug application. This model consists of three stochastic hyperbolic equations describing the evolution of tumour cells. It also includes two stochastic parabolic equations describing the diffusions of nutrient and drug concentrations. Since all systems are subject to many uncertainties, we have added stochastic terms to the deterministic model to consider the random perturbations. Then, we have added control variables to the model according to the medical concepts to control the concentrations of drug and nutrient. In the optimal control problem, we have defined the stochastic and deterministic cost functions and we have proved the problems have unique optimal controls. For deriving the necessary conditions for optimal control variables, the stochastic adjoint equations are derived. We have proved the stochastic model of tumour growth and the stochastic adjoint equations have unique solutions. For proving the theoretical results, we have used a change of variable which changes the stochastic model and adjoint equations (a.s.) to deterministic equations. Then we have employed the techniques used for deterministic ones to prove the existence and uniqueness of optimal control. [ABSTRACT FROM AUTHOR]

This paper is concerned with the spreading behavior of a two-species strong-weak competition system with two free boundaries. The model may describe how a strong competing species invades into the habitat of a native weak competing species. The asymptotic spreading speed of invading fronts has been determined by making use of semi-wave systems in Du et al. (J Math Pures Appl 107:253–287, 2017). Here we give a sharp estimate for the asymptotic spreading speed of invading fronts. Moreover, we prove that the solution of the free boundary problem evolves eventually into a semi-wave solution when the spreading happens, while the solution of the free boundary problem exponentially converges to a semi-trivial solution of such system when the vanishing happens. [ABSTRACT FROM AUTHOR]

This paper deals with the free boundary problem for a double‐layered tumor filled with quiescent cells and proliferating cells, where time delay τ>0$$ \tau >0 $$ in cell proliferation is taken into account. These two types of living cells exhibit different metabolic responses and consume nutrients σ$$ \sigma $$ at different rates λ1$$ {\lambda}_1 $$ and λ2$$ {\lambda}_2 $$ ( λ1⩽λ2$$ {\lambda}_1\leqslant {\lambda}_2 $$). Time delay happens between the time at which a cell commences mitosis and the time at which the daughter cells are produced. The problem is reduced to a delay differential equation on the tumor radius R(t)$$ R(t) $$ over time, and the difficulty arises from the jump discontinuity of the consumption rate function. We give rigorous analysis on this new model and study the dynamical behavior of the global solutions for any initial φ(t)$$ \varphi (t) $$. [ABSTRACT FROM AUTHOR]

In a tumor, the ratio between nutrient diffusion time and tissue growth time, ε = T diffusion / T growth , is small and set to zero as a quasi-steady state approximation in models [2,19]. Under this assumption, the periodic solution and stability were studied there. In this paper, we shall establish the existence and uniqueness of the periodic solution in the case ε > 0. Furthermore, the periodic solution is globally stable: it is a global attractor in the class of radially symmetric initial data. [ABSTRACT FROM AUTHOR]

In this paper, we prove the existence of a solution of a novel free boundary problem for reaction-diffusion systems describing growth of biological tissues due to cell influx and proliferation. For this aim, we transform it into a problem with fixed boundary, through a change of variables. The new problem thus obtained has space and time dependent coefficients with nonlinear terms. We then prove the existence of a solution for the corresponding linear problem, and deduce the existence of a solution for the nonlinear problem using the fixed point theorem. Finally, we return to the problem with free boundary to conclude the existence of its solution. [ABSTRACT FROM AUTHOR]

In this paper, we apply Markov-modulated models to value continuous-installment options of the European style with a partial differential equation approach. Under regime-switching models and the opportunity for continuing or stopping to pay installments, the valuation problem can be formulated as coupled partial differential equations (CPDE) with free boundary features, which in many ways is similar to the free boundary problem for vanilla American options due to the possibility of early exercise. In this paper, to value the continuous-installment options under the proposed model with a numerical approach, we first express the truncated CPDE as a linear complementarity problem (LCP), and then a finite element method is applied to solve the resulting variational inequality. We studied the existence and uniqueness of the solution and analyzed the stability of the proposed method under some appropriate assumptions, then we illustrated the error estimates on the appropriate spaces. We presented some numerical results to examine the rate of convergence and accuracy of the proposed method for the pricing problem under the regime-switching model. [ABSTRACT FROM AUTHOR]

We consider the equilibrium shape of a liquid drop on a hexagonal substrate as motivated by vapor–liquid growth of nanowires. We numerically determine the energy-minimizing liquid drop shape on a hexagonal base using the software Surface Evolver in conjunction with an efficient regridding algorithm and convergence monitoring. The drop shape depends on two nondimensional parameters, the drop volume, and the equilibrium contact angle. We show that sufficiently large drops are well approximated away from the base by a spherical cap drop with geometric parameters determined by the area of the hexagonal base. Notably, however, the drop/base contact region does not extend to the corners of the hexagonal base, even in the limit of large volume V. In particular, there is a self-similar structure to the dry corner region with a length scale proportional to V - 3 / 2 . Since steady-state growth of faceted hexagonal nanowires by vapor–liquid–solid growth requires the liquid drop to be commensurate with the underlying wire cross-section, our findings mean that steady-state growth of hexagonal wires is not strictly compatible with an equilibrium liquid drop acting as a catalyst. [ABSTRACT FROM AUTHOR]

Biological models for cardiac regeneration and remodeling, along with the effects of cytokines or chemokines during the therapy with mesenchymal stem cells after a myocardial infarction, are of crucial importance for understanding the complex underlying mechanisms. This paper presents a mathematical model composed of three coupled partial differential equations that describes the dynamics of stem cells, nutrients and chemokines, highlighting the fundamental role of the chemokines during the myocardial tissue regeneration process. The system is solved numerically using mimetic difference operators and the MOLE library for MATLAB. The results show the tissue regeneration process in the necrotic part closest to the cell implantation area. [ABSTRACT FROM AUTHOR]

A celebrated theorem of Serrin asserts that one overdetermined condition on the boundary is enough to obtain radial symmetry in the so-called one-phase overdetermined torsion problem. It is also known that imposing just one overdetermined condition on the boundary is not enough to obtain radial symmetry in the corresponding multi-phase overdetermined problem. In this paper we show that, in order to obtain radial symmetry in the two-phase overdetermined torsion problem, two overdetermined conditions are needed. Moreover, it is noteworthy that this pattern does not extend to multi-phase problems with three or more layers, for which we show the existence of nonradial configurations satisfying countably infinitely many overdetermined conditions on the outer boundary. [ABSTRACT FROM AUTHOR]

We study a free boundary problem on the evolution of a radial tumor with triple-layered structure. Based on the nutrient thresholds $ \sigma_Q $ and $ \sigma_D $, cells in the tumor spheroid are divided into three types, i.e. proliferating cells, quiescent cells and necrotic cells, hence it forms a layered structure from tumor's surface to its center. The novelty of this paper is that the growth of the triple-layered tumor is first taken into rigorous analysis, and the difficulty arises from the two unknown moving interfaces $ \eta $ and $ \rho $ between different cell layers inside the tumor. We show existence and uniqueness of the stationary solution to the system for different external nutrient supply $ \bar \sigma $. It is proved that there exist two constants $ \sigma^* $ and $ \sigma_* $ with the relation $ \sigma^*> \sigma_*>\tilde \sigma $, such that if $ \bar \sigma> \sigma^* $, the dormant tumor forms a triple-layered structure; if $ \sigma_*<\bar \sigma \le \sigma^* $, the dormant tumor has double layers with proliferating cells and quiescent cells; if $ \tilde \sigma <\bar \sigma\le \sigma_* $, the dormant tumor is filled with only proliferating cells and appears as a single layer; and if $ \bar \sigma\le\tilde \sigma $, the dormant tumor vanishes. The asymptotic stability of the stationary solution above is also studied. We can show the tendency of the evolutionary tumor towards its dormant state by using only $ \bar \sigma $ and the initial radius $ R_0 $ as well. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the existence, boundedness, and integrability of minimizers of heterogeneous, two-phase free boundary problems Jγ(u) = ∫Ω (f(x, ∇u) + λ+(u+)γ + λ−(u−)γ + gu) dx → min under non-standard growth conditions. Included in such problems are heterogeneous jets and cavities of Prandtl-Batchelor type with γ = 0, chemical reaction problems with 0 < γ < 1, and obstacle type problems with γ = 1, respectively. [ABSTRACT FROM AUTHOR]

In this paper, a mathematical model for a solid spherically symmetric vascular tumor growth with nutrient periodic supply and time delays is studied. Compared to the apoptosis process of tumor cells, there is a time delay in the process of tumor cell division. The cells inside the tumor obtain nutrient σ(r,t) through blood vessels, and the tumor attracts blood vessels at a rate proportional to α(t). So, the boundary value condition ∂σ ∂r + α(t)(σ − ψ(t)) = 0,r = R(t),t > 0, holds on the boundary, where the function ψ(t) is the concentration of nutrient externally supplied to the tumor. Considering that the nutrients provided by the outside world are often periodic, the research in this paper assumes that ψ(t) is a periodic function. Sufficient conditions for the global stability of zero steady state are presented. Under certain conditions, we prove that there exists at least one periodic solution to the model. The results are illustrated by computer simulations. [ABSTRACT FROM AUTHOR]

In this study, we present a mathematical model for plasmid spread in a growing biofilm, formulated as a nonlocal system of partial differential equations in a 1-D free boundary domain. Plasmids are mobile genetic elements able to transfer to different phylotypes, posing a global health problem when they carry antibiotic resistance factors. We model gene transfer regulation influenced by nearby potential receptors to account for recipient-sensing. We also introduce a promotion function to account for trace metal effects on conjugation, based on literature data. The model qualitatively matches experimental results, showing that contaminants like toxic metals and antibiotics promote plasmid persistence by favoring plasmid carriers and stimulating conjugation. Even at higher contaminant concentrations inhibiting conjugation, plasmid spread persists by strongly inhibiting plasmid-free cells. The model also replicates higher plasmid density in biofilm’s most active regions. [ABSTRACT FROM AUTHOR]

Atherosclerosis, as a chronic inflammatory disease, has been a threat to human health. How to diagnose and prevent this disease has long been the focus of medical and biomathematics study. In this paper, we investigate a free boundary model of an atherosclerotic plaque formation in the presence of reverse cholesterol transport, which includes low-density lipoprotein, high-density lipoprotein, endothelial stimulating cytokines, pro-inflammatory and anti-inflammatory macrophages, as well as foam cells. For this model, we use the Holling type-III response function instead of the usual Holling type-II to describe the change of the concentration of each substance. We first introduce the auxiliary function ξ (r) to prove the existence and uniqueness of small radially symmetric stationary plaque for appropriate L 0 and H 0 by using the contraction mapping principle and maximum principle, and then establish a condition to ensure their asymptotic stability behavior by expanding the corresponding steady-state solution. [ABSTRACT FROM AUTHOR]

We consider the long-time behaviour of a West Nile virus (WNv) model consisting of a reaction–diffusion system with free boundaries. Such a model describes the spreading of WNv with the free boundary representing the expanding front of the infected region, which is a time-dependent interval $[g(t), h(t)]$ in the model (Lin and Zhu, Spatial spreading model and dynamics of West Nile virus in birds and mosquitoes with free boundary. J. Math. Biol. 75, 1381–1409, 2017). The asymptotic spreading speed of the front has been determined in Wang et al. (Spreading speed for a West Nile virus model with free boundary. J. Math. Biol. 79, 433–466, 2019) by making use of the associated semi-wave solution, namely $\lim _{t\to \infty } h(t)/t=\lim _{t\to \infty }[\!-g(t)/t]=c_\nu$ , with $c_\nu$ the speed of the semi-wave solution. In this paper, by employing new techniques, we significantly improve the estimate in Wang et al. (Spreading speed for a West Nile virus model with free boundary. J. Math. Biol. 79, 433–466, 2019): we show that $h(t)-c_\nu t$ and $g(t)+c_\nu t$ converge to some constants as $t\to \infty$ , and the solution of the model converges to the semi-wave solution. The results also apply to a wide class of analogous Ross–MacDonold epidemic models. [ABSTRACT FROM AUTHOR]

One of the fundamental questions in population dynamics concerns the criterion for the persistence or extinction of a biological species subjected to their habitat changes. In this communication, to understand more clearly the impact of climate change on the global dynamics of mosquitoes proposed in Bao (JMB 76:841-875, 2018), we consider a reaction-diffusion free boundary model with conditional dispersal in a heterogeneous environment. Our main interest is to study long-time dynamics of solutions assuming that the environment is globally unfavorable determined by a spectral condition at infinity. The mathematical models to describe the dynamics of a population facing climate change have arisen many challenges in science and application and our result makes a theoretically substantial contribution besides the previous works (Bao in JMB 76:841-875, 2018; Monobe in JDE 261:6144-6177, 2016; Shen in JMB 84:30-42, 2022; Shen in JDE 269:6236-6268, 2020; Vo in JDE 259:4947-4988, 2015) for the study of the impact of the climate change with the conditional dispersal and free boundary. [ABSTRACT FROM AUTHOR]

In this paper we derive the best constant for the following L^{\infty }-type Gagliardo-Nirenberg interpolation inequality \begin{equation*} \|u\|_{L^{\infty }}\leq C_{q,\infty,p} \|u\|^{1-\theta }_{L^{q+1}}\|\nabla u\|^{\theta }_{L^p},\quad \theta =\frac {pd}{dp+(p-d)(q+1)}, \end{equation*} where parameters q and p satisfy the conditions p>d\geq 1, q\geq 0. The best constant C_{q,\infty,p} is given by \begin{equation*} C_{q,\infty,p}=\theta ^{-\frac {\theta }{p}}(1-\theta)^{\frac {\theta }{p}}M_c^{-\frac {\theta }{d}},\quad M_c≔\int _{\mathbb {R}^d}u_{c,\infty }^{q+1} dx, \end{equation*} where u_{c,\infty } is the unique radial non-increasing solution to a generalized Lane-Emden equation. The case of equality holds when u=Au_{c,\infty }(\lambda (x-x_0)) for any real numbers A, \lambda >0 and x_{0}\in \mathbb {R}^d. In fact, the generalized Lane-Emden equation in \mathbb {R}^d contains a delta function as a source and it is a Thomas-Fermi type equation. For q=0 or d=1, u_{c,\infty } have closed form solutions expressed in terms of the incomplete Beta functions. Moreover, we show that u_{c,m}\to u_{c,\infty } and C_{q,m,p}\to C_{q,\infty,p} as m\to +\infty for d=1, where u_{c,m} and C_{q,m,p} are the function achieving equality and the best constant of L^m-type Gagliardo-Nirenberg interpolation inequality, respectively. [ABSTRACT FROM AUTHOR]

Similarity solutions for the two‐phase Rubinstein binary‐alloy solidification problem in a semi‐infinite material are developed. These new explicit solutions are obtained by considering two cases: A heat flux or a convective boundary conditions at the fixed face, and the necessary and sufficient conditions on data are also given in order to have an instantaneous solidification process. We also show that all solutions for the binary‐alloy solidification problem are equivalent under some restrictions for data. Moreover, this implies that the coefficient that characterizes the solidification front for the Rubinstein solution must verify an inequality as a function of all thermal and boundary conditions. [ABSTRACT FROM AUTHOR]

This paper is concerned with a free boundary problem with resource-dependent motility in a weak heterogeneous environment. The existence and uniqueness of global solutions are discussed first. Next, we establish long-time behaviors of solutions which is a spreading-- vanishing dichotomy.Moreover,we obtain sharp criteria on spreading and vanishing by investigating the associated linearized eigenvalue problem. The theoretical analyses reveal an important biological phenomenon. (i) Resource abundance reduces the motility, providing advantages for individuals to persist in a habitat. (ii) Resource shortage enhances the motility, forcing individuals to expand outward to survive in an environment. Consequently, resource-dependent motility is more beneficial to the survival of species compared with random dispersal. [ABSTRACT FROM AUTHOR]

In this paper, a free boundary model of tumor growth with drug application including the heterogeneity or different types of tumor cells (caused by mutations and different values of drug and nutrient concentrations inside the tumor) is studied. Heterogeneity is included in the model by a variable −1≤y≤1$$ -1\le y\le 1 $$. It is assumed that converting from mutation y1$$ {y}_1 $$ to mutation y2$$ {y}_2 $$ happens with probability P(y1,y2)$$ P\left({y}_1,{y}_2\right) $$. A Caputo time fractional‐order hyperbolic equation describes the evolution of tumor cells depending on y$$ y $$. It also includes two Caputo time fractional‐order parabolic equations describing the diffusions of nutrients (e.g., oxygen and glucose) and drug concentrations. Instead of integer‐order time derivatives, the fractional ones are considered. In this study, it is aimed to employ the least squares curve fitting method to fit the order of fractional derivatives, coefficient, and rates of the model. For this purpose, using the mathematical model, we have considered the boundary densities of the tumor cells of different types and near‐boundary concentrations of drug and nutrient as the functions of the unknown orders, coefficients and rates (unknown variables). Due to the complexity of the problem, we have obtained the functions numerically. For this, using a change of variable, we have changed the free boundary problem to a problem with fixed domain. Thus, Riemann–Liouville fractional‐order integrals are added to the problem. In the spatial domain, the problem is discretized using the collocation method. In the temporal domain, the fractional derivatives and integrals (with order α$$ \alpha $$) are approximated in mesh points (with step size t∗$$ {t}&#x0005E;{\ast } $$) applying a method with error O((t∗)2−α)$$ O\left({\left({t}&#x0005E;{\ast}\right)}&#x0005E;{2-\alpha}\right) $$. Then, the unknown variables are obtained by fitting the functions for unknown variables to the data. In order to obtain the variables, a quadratic objective function is considered. The discretized function is substituted into the objective function, then the objective function is minimized using the trust‐region reflective algorithm. Finally, some numerical examples are presented to verify the efficiency of the method. In the examples, we have added noises generated from Gaussian distributions to the data and the effects of noise on the fitted coefficients, and rates are illustrated using some figures and tables. The noisy data are also plotted to have a clear vision of the effects of noises on the data. It is shown that despite the noise, the prediction of the radius of the tumor is acceptable. [ABSTRACT FROM AUTHOR]

This paper is devoted to study the population dynamics of a single species in a one-dimensional environment which is modeled by a reaction–diffusion–advection equation with free boundary condition. We find three critical values c 0 , 2 and β ∗ for the advection coefficient - β with β ∗ > 2 > c 0 > 0 , which play key roles in the dynamics, and prove that a spreading-vanishing dichotomy result holds when - 2 < β ⩽ c 0 ; a small spreading-vanishing dichotomy result holds when c 0 < β < 2 ; a virtual spreading-transition-vanishing trichotomy result holds when 2 ⩽ β < β ∗ ; only vanishing happens when β ⩾ β ∗ ; a virtual vanishing-transition-vanishing trichotomy result holds when β ⩽ - 2 . When spreading or small spreading or virtual spreading happens for a solution, we make use of the traveling semi-wave solutions to give a estimate for the asymptotic spreading speed and asymptotic profile of the right front. [ABSTRACT FROM AUTHOR]

This paper is concerned with the spreading properties for a reaction-diffusion equation with free boundary condition. We obtained a complete description of the long-time dynamical behavior of this problem. By introducing a parameter $ \sigma $ in the initial data, we revealed a threshold value $ \sigma^* $ such that spreading happens when $ \sigma > \sigma^* $ and vanishing happens when $ \sigma\leq \sigma^* $. There exists a unique $ L^* > 0 $ independent of the initial data such that $ \sigma^* = 0 $ if and only if the length of initial occupying interval is no smaller than $ 2L^* $. These theoretical results may have important implications for prediction and prevention of biological invasions.

This paper was concerned with a free boundary problem modeling the growth of tumor cord with a time delay in cell proliferation, in which the cell location was incorporated, the domain was bounded in $ \mathbb{R}^2 $, and its boundary included two disjoint closed curves, one fixed and the other moving and a priori unknown. A parameter $ \mu $ represents the aggressiveness of the tumor. We proved that there exists a unique radially symmetric stationary solution for sufficiently small time delay, and this stationary solution is linearly stable under the nonradially symmetric perturbations for any $ \mu > 0 $. Moreover, adding the time delay in the model leads to a larger stationary tumor. If the tumor aggressiveness parameter is bigger, the time delay has a greater effect on the size of the stationary tumor, but it has no effect on the stability of the stationary solution.

In this paper, we address the utility maximization problem of an infinitely lived agent who has the option to increase their income. The agent can increase their income at any time, but doing so incurs a wealth cost proportional to the amount of the increase. To prevent the agent from infinitely increasing their income and borrowing against future income, we additionally consider a non-negative wealth constraint that prohibits borrowing based on future income. This utility maximization problem is a mixture of stochastic control, where the agent chooses consumption and investment, and singular control, where the agent chooses a non-decreasing income process. To solve this non-trivial and challenging problem, we derive the Hamilton–Jacobi–Bellman (HJB) equation with a gradient constraint using the dynamic programming principle (DPP). Then, using the guess-and-verify method and a linearization technique, we obtain a closed-form solution to the HJB equation and, based on this, find the optimal strategy.

The aim of this paper is to show the global wellposedness of the Navier-Stokes equations, including surface tension and gravity, with a free surface in an unbounded domain such as bottomless ocean. In addition, it is proved that the solution decays polynomially as time t tends to infinity. To show these results, we first use the Hanzawa transformation in order to reduce the problem in a time-dependent domain Ω t ⊂ R 3 , t > 0 , to a problem in the lower half-space R − 3. We then establish some time-weighted estimate of solutions, in an L p -in-time and L q -in-space setting, for the linearized problem around the trivial steady state with the help of L r - L s time decay estimates of semigroup. Next, the time-weighted estimate, combined with the contraction mapping principle, shows that the transformed problem in R − 3 admits a global-in-time solution in the L p - L q setting and that the solution decays polynomially as time t tends to infinity under the assumption that p , q satisfy the conditions: 2 < p < ∞ , 3 < q < 16 / 5 , and (2 / p) + (3 / q) < 1. Finally, we apply the inverse transformation of Hanzawa's one to the solution in R − 3 to prove our main results mentioned above for the original problem in Ω t. Here we want to emphasize that it is not allowed to take p = q in the above assumption about p , q , which means that the different exponents p , q of L p - L q setting play an essential role in our approach. [ABSTRACT FROM AUTHOR]

Atherosclerosis, a chronic inflammatory disease that originates from a plaque that forms in the artery, is a leading cause of disability and premature death in the world. In this paper, we consider a free boundary model of early atherosclerotic plaque formation under the influence of reverse cholesterol transport (RCT), which includes low-density lipoprotein (LDL), high-density lipoprotein (HDL), endothelial stimulating cytokines, macrophages and foam cells. For this model, we first reduce the free boundary to a fixed boundary by using the Hanzawa transformation, and then prove the existence and uniqueness of the solution in suitable Hölder spaces by using the theory of analytic semigroups. Our main novelty is to reduce the regional regularity requirement for the corresponding problem from $ m \geq 4 $ to $ m \geq 3 $ with respect to (J. Math. Anal. Appl., 2022, Paper No. 125606). [ABSTRACT FROM AUTHOR]

We study an optimal investment and consumption problem with heterogeneous consumption of basic and luxury goods, together with the choice of time for retirement. The optimal heterogeneous consumption strategies for a class of nonhomothetic utility maximizer are shown to consume only basic goods when the wealth is small, to consume basic goods and make savings when the wealth is intermediate, and to consume almost all in luxury goods when the wealth is large. The optimal retirement policy is shown to be both universal, in the sense that all individuals should retire at the same level of marginal utility that is determined only by income, labor cost, discount factor as well as market parameters, and not universal, in the sense that all individuals can achieve the same marginal utility with different utility and wealth. We also show that individuals prefer to retire as time goes by if the marginal labor cost increases faster than that of income. This paper studies an optimal investment and consumption problem with heterogeneous consumption of basic and luxury goods, together with the choice of time for retirement. The utility for luxury goods is not necessarily a concave function. The optimal heterogeneous consumption strategies for a class of nonhomothetic utility maximizer are shown to consume only basic goods when the wealth is small, to consume basic goods and make savings when the wealth is intermediate, and to consume almost all in luxury goods when the wealth is large. The optimal retirement policy is shown to be both universal, in the sense that all individuals should retire at the same level of marginal utility that is determined only by income, labor cost, discount factor and market parameters, and not universal, in the sense that all individuals can achieve the same marginal utility with different utility and wealth. It is also shown that individuals prefer to retire as time goes by if the marginal labor cost increases faster than that of income. The main tools used in analyzing the problem are from a partial differential equation and stochastic control theory including variational inequality and dual transformation. We finally conduct the simulation analysis for the featured model parameters to investigate practical and economic implications by providing their figures. Funding: This work was supported by Hong Kong Research Grants Council General Research Fund [Grants 15202421 and 15202817], the National Research Foundation of Korea [Grant 2021R1C1C1004647], the PolyU-SDU Joint Research Center on Financial Mathematics, the CAS AMSS-PolyU Joint Laboratory of Applied Mathematics, and Hong Kong Polytechnic University, the National Natural Science Foundation of China [Grant 11971409], and the Engineering and Physical Sciences Research Council (UK) [Grant EP/V008331/1]. Supplemental Material: The online appendix is available at https://doi.org/10.1287/opre.2022.2328. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the notion of variational free boundary problem. Namely, we say that a free boundary problem is variational if its solutions can be characterized as the critical points of some shape functional. Moreover, we extend the notion of nondegeneracy of a critical point to this setting. As a result, we provide a unified functional-analytical framework that allows us to construct families of solutions to variational free boundary problems whenever the shape functional is nondegenerate at some given solution. As a clarifying example, we apply this machinery to construct families of nontrivial solutions to the two-phase Serrin's overdetermined problem in both the degenerate and nondegenerate case. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence, uniqueness, and stability of solutions to the pricing problem of European continuous-installment options under the regime-switching model based on a numerical approach. For this, we consider a two-state continuous-time Markov chain for the regime-switching model and a one-dimensional finite element method for the numerical scheme. Under our proposed model and the installment option feature for the option holder to continue paying installments until maturity and receive payoff or to stop installments and terminate the contract, the valuation problem has been formulated as coupled partial differential equations (CPDE) with free boundaries. For this problem, we obtained some appropriate assumptions for the model parameters to prove that the pricing problem has unique solutions under the regime-switching model. We also illustrated some proven theorems to show the stability of solutions with a numerical approach. Finally, some numerical examples are considered to show the performance of the obtained theoretical results through numerical implementations. [ABSTRACT FROM AUTHOR]

We consider the Stefan problem of nonlinear diffusion equation $ u_t = \Delta u+f(u) $ for $ t>0 $ and $ x\in\Omega(t)(\subset\mathbb{R}^N) $ with positive bistable nonlinearity $ f $. We first prove that for any initial data, the long-time behavior of the solution is classified into four cases: vanishing, small spreading, big spreading and transition. In particular, we show that when transition occurs for the solution $ u $, there exists an $ x_0\in\mathbb{R}^N $ such that $ u(t,\,\cdot\,) $ converges as $ t\to\infty $ to an equilibrium solution which is radially symmetric and radially decreasing with center $ x_0 $.We next give some results about large-time estimates of the expanding speed of $ \Omega(t) $ for small and big spreading cases. As in our previous paper [12], it can be expected that, under a certain condition, every big spreading solution accompanies a propagating terrace. We have succeeded in understanding the large-time behavior of such a solution with terrace in terms of its level set. [ABSTRACT FROM AUTHOR]

In this paper, we prove the existence of smooth initial data for the two-dimensional free boundary incompressible viscous magnetohydrodynamics (MHD) equations, for which the interface remains regular but collapses into a splash singularity (self-intersects in at least one point) in finite time. The existence of the splash singularities is guaranteed by a local existence theorem, in which we need suitable spaces for the modified magnetic field together with the modification of the velocity and pressure such that the modified initial velocity is zero, and a stability result which allows us to construct a class of initial velocities and domains for an arbitrary initial magnetic field. It turns out that the presence of the magnetic field does not prevent the viscous fluid from forming splash singularities for certain smooth initial data. [ABSTRACT FROM AUTHOR]

This paper is devoted to the study of a mathematical model of atherosclerosis in one-dimensional geometry with a free boundary. The motion of the boundary is attributable to the concentration of cells in the intima and their interaction in the subendothelial space in addition to their influx through the boundary. A mathematical model that describes the main inflammatory processes in atherosclerosis is proposed, then, by considering some simplifications, a reduced model is obtained. Using a change of variables, the reduced model is converted to a fixed boundary model with space- and time-dependent coefficients and nonlinear terms. We study the existence of solution for the fixed boundary model starting with a model with linear terms then by applying the fixed point theorem. The wave solution is as well investigated along with numerical simulations. Then, we return to the reduced model, prove the existence of solution and present numerical results. Finally, we generalize the results to the complete model initially proposed. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence and interior W 2 , p -regularity of the solution, and the regularity of the free boundary ∂ ⁡ { u > ϕ } to the obstacle problem of the porous medium equation, u t = Δ ⁢ u m ( m > 1 ) with the obstacle function ϕ. The penalization method is applied to have the existence and interior regularity. To deal with the interaction between two free boundaries ∂ ⁡ { u > ϕ } and ∂ ⁡ { u > 0 } , we consider two cases on the initial data which make the free boundary ∂ ⁡ { u > ϕ } separate from the free boundary ∂ ⁡ { u > 0 } . Then the problem is converted into the obstacle problem for a fully nonlinear operator. Hence, the C 1 -regularity of the free boundary ∂ ⁡ { u > ϕ } is obtained by the regularity theory of a class of obstacle problems for the general fully nonlinear operator. [ABSTRACT FROM AUTHOR]

We consider a model of congestion dynamics with chemotaxis, where the density of cells follows the chemical signal it generates, while observing an incompressibility constraint (incompressible parabolic-elliptic Patlak-Keller-Segel model). We show that when the chemical diffuses slowly and attracts the cells strongly, then the dynamics of the congested cells is well approximated by a surface-tension driven free boundary problem. More precisely, we rigorously establish the convergence of the solution to the characteristic function of a set whose evolution is determined by the classical Hele-Shaw free boundary problem with surface tension. The problem is set in a bounded domain, which leads to an interesting analysis on the limiting boundary conditions. Namely, we prove that the assumption of Robin boundary conditions for the chemical potential leads to a contact angle condition for the free interface (in particular Neumann boundary conditions lead to an orthogonal contact angle condition, while Dirichlet boundary conditions lead to a tangential contact angle condition). [ABSTRACT FROM AUTHOR]

This paper is concerned with the spreading properties for a reaction-diffusion equation with free boundary condition. We obtained a complete description of the long-time dynamical behavior of this problem. By introducing a parameter σ in the initial data, we revealed a threshold value σ ∗ such that spreading happens when σ > σ ∗ and vanishing happens when σ ≤ σ ∗ . There exists a unique L ∗ > 0 independent of the initial data such that σ ∗ = 0 if and only if the length of initial occupying interval is no smaller than 2 L ∗ . These theoretical results may have important implications for prediction and prevention of biological invasions. This paper is concerned with the spreading properties for a reaction-diffusion equation with free boundary condition. We obtained a complete description of the long-time dynamical behavior of this problem. By introducing a parameter in the initial data, we revealed a threshold value such that spreading happens when and vanishing happens when . There exists a unique independent of the initial data such that if and only if the length of initial occupying interval is no smaller than . These theoretical results may have important implications for prediction and prevention of biological invasions. [ABSTRACT FROM AUTHOR]

The coefficient inverse problem for a degenerate parabolic equation is studied in a free boundary domain. The degeneration of the equation is caused by time dependent function at the higher order derivative of unknown function. It is assumed that the coefficient at the minor derivative of the equation is a polynomial of the first order for the space variable with two unknown time depended functions. The conditions of existence and uniqueness of the classical solution to such inverse problem are established for the weak degeneration case at the Dirichlet boundary conditions and the values of heat moments as overdetermination conditions. [ABSTRACT FROM AUTHOR]

We establish local interior Lipschitz continuity of the solutions of a class of free boundary elliptic problems assuming the coefficients of the equation of Dini mean oscillation in at least one direction. The novelty in this regularity result lies in the fact that it allows discontinuous coefficients in all but one variable. [ABSTRACT FROM AUTHOR]

In this paper, we study the symmetry properties of nondegenerate critical points of shape functionals using the implicit function theorem. We show that, if a shape functional is invariant with respect to some one-parameter group of rotations, then its nondegenerate critical points (bounded open sets with smooth enough boundary) share the same symmetries. We also consider the case where the shape functional exhibits translational invariance in addition to just rotational invariance. Finally, we study the applications of this result to the theory of one/two-phase overdetermined problems of Serrin-type. En passant, we give a simple proof of the fact that, under suitable smoothness assumptions, the ball is the only nondegenerate critical point of the Lagrangian associated to the maximization problem for the torsional rigidity under a volume constraint. We remark that the proof does not rely on either the method of moving planes or rearrangement techniques. [ABSTRACT FROM AUTHOR]