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In this paper, we consider the three-dimensional free boundary problem of incompressible and compressible neo-Hookean viscoelastic fluid equations in an infinite strip without surface tension, provided that the initial data is sufficiently close to the equilibrium state. By reformulating the problems in Lagrangian coordinates, we can get the stabilizing effect of elasticity. In both cases, we utilize the elliptic estimates to improve the estimates. Moreover, for the compressible case, we find there is an extra ODE structure that can improve the regularity of the free boundary, thus we can have the global well-posedness. To prove the global well-posedness for the incompressible case, we employ two-tier energy method introduced in [11] [12] [13] to compensate for the inferior structure. [ABSTRACT FROM AUTHOR]

To study the influence of the moving front of the infected interval and the spatial movement of individuals on the spreading or vanishing of infectious disease, we consider a nonlocal susceptible–infected–susceptible (SIS) reaction–diffusion model with media coverage, hospital bed numbers and free boundaries. The principal eigenvalue of the integral operator is defined, and the impacts of the diffusion rate of infected individuals and interval length on the principal eigenvalue are analyzed. Furthermore, sufficient conditions for spreading and vanishing of the disease are derived. Our results show that large media coverage and hospital bed numbers are beneficial to the prevention and control of disease. The difference between the model with nonlocal diffusion and that with local diffusion is also discussed and nonlocal diffusion leads to more possibilities. [ABSTRACT FROM AUTHOR]

In this paper, we consider the following free boundary problem: \[ (P)\begin{cases} \Delta u = \lambda \phi(x)\sum_{i=1}^n H(u-\mu_i) & \quad \mbox{in}\ \Omega=\Omega_2\setminus \bar{\Omega}_1, \\ u =0 & \quad \mbox{on}\ \partial \Omega_2, \\ u =M & \quad \mbox{on}\ \partial \Omega_1. \end{cases}\] (P) { Δu = λϕ (x) ∑ i = 1 n H (u − μ i) in Ω = Ω 2 ∖ Ω ¯ 1 , u = 0 on ∂ Ω 2 , u = M on ∂ Ω 1. The domain Ω is a convex ring where $ \Omega _1 $ Ω 1 and $ \Omega _2 $ Ω 2 are bounded convex domains in $ \mathbb {R}^N $ R N , $ N\geq 2 $ N ≥ 2 , H is the Heaviside step function, $ \lambda, M,\mu _i $ λ , M , μ i $ (i=1,..,n) $ (i = 1 ,.. , n) are given positive real parameters and ϕ is a given function. We show that under suitable conditions, there exists a solution to problem (P) with a convex level set. We derive also an interesting result for the convexity of the set which is delimited by the free boundary in obstacle problem when the nonlinearity is discontinuous. At last, a detailed analysis is given by a construction of function ϕ which shows that there are more solutions. [ABSTRACT FROM AUTHOR]

In this paper, we examine the effect of a new free boundary condition on the propagation dynamics of the nonlocal diffusion model considered in [9] , which describes the spreading of a species with density u (t , x) and population range [ g (t) , h (t) ] ⊂ R. The existing free boundary condition can be written as { h ′ (t) = μ ∫ g (t) h (t) u (t , x) W J (h (t) − x) d x , g ′ (t) = − μ ∫ g (t) h (t) u (t , x) W J (x − g (t)) d x , where W J (x) = ∫ x + ∞ J (y) d y , and J is the kernel function of the nonlocal diffusion operator in the model. In the new free boundary condition, we replace W J by a general nonnegative locally Lipschitz continuous function W with W (0) > 0 , independent of J. This represents a very different assumption that the movement of the range boundary of the species is independent of its dispersal strategy, as in [20]. Our analysis shows that the dynamics of the model with the new free boundary condition resembles that of the old model except in the case that J is thin-tailed and ∫ 0 ∞ W (x) d x = ∞ , where new propagation phenomena appear. [ABSTRACT FROM AUTHOR]

This paper concerns a nonlocal diffusion problem with a free boundary. We first give accurate estimates on the longtime behaviors of solution by constructing suitably upper and lower solutions. In particular, for two important kinds of kernel functions, one of which is compactly supported and the other behaves like |x|−γ with γ ∈ (1, 2] near infinity, some sharp estimates on the longtime behaviors and rates of accelerated spreading are obtained. Then the limiting behaviors of the solution pair of a semi-wave problem and asymptotic dynamics of a nonlocal diffusion problem on half space are given, respectively. Finally, we investigate the limiting profiles of this free boundary problem when the expanding coefficient of free boundary converges to 0 and ∞, respectively. [ABSTRACT FROM AUTHOR]

Atherosclerosis is a chronic inflammatory disease that poses a serious threat to human health. It starts with the buildup of plaque in the artery wall, which results from the accumulation of pro‐inflammatory factors and other substances. In this paper, we propose a mathematical model of early atherosclerosis with a free boundary and time delay. The time delay represents the time required for macrophages to transit to foam cells through cholesterol accumulation. We obtain an explicit solution and analyze the stability of the model and the effect of the time delay on plaque size. We show that in the form of perturbation cos(nθ)$$ \cos \left(n\theta \right) $$ (where n$$ n $$ represents the mode of angle), when n=0$$ n=0 $$ or 1, the steady‐state solution (M∗,p∗,r∗)$$ \left({M}_{\ast },{p}_{\ast },{r}_{\ast}\right) $$ is linearly stable; when n≥2$$ n\ge 2 $$, there exists a critical parameter L∗$$ {L}_{\ast } $$ such that the steady‐state solution is linearly stable for LL∗$$ L>{L}_{\ast } $$. Moreover, we find that smaller plaque are associated with the presence of time delay. [ABSTRACT FROM AUTHOR]

This paper concerns the free-surface problem of incompressible ideal MHD fluids with velocity damping in a two-dimensional (2D) horizontally periodic slab domain impressed by a uniform horizontal magnetic field. We prove that the problem is globally well-posed for the small initial data and the solution decays exponentially in time to the equilibrium, where both the Taylor sign condition and the surface tension are not required in this paper due to the stabilizing effect of horizontal magnetic field. [ABSTRACT FROM AUTHOR]

This paper involves a diffusive epidemic model whose domain has one free boundary with the Stefan boundary condition, and one fixed boundary subject to the usual homogeneous Dirichlet or Neumann condition. By using the standard upper and lower solutions method and the regularity theory, we first study some related steady-state problems which help us obtain the exact longtime behaviors of solution component (u, v). Then we prove there exists the unique classical solution whose longtime behaviors are governed by a spreading-vanishing dichotomy. Lastly, the criteria determining when spreading or vanishing happens are given with respect to the basic reproduction number R 0 , the initial habitat [ 0 , h 0 ] , the expanding rates μ 1 and μ 2 as well as the initial function (u 0 , v 0) . The criteria reveal the effect of the cooperative behaviors of agents and humans on spreading and vanishing. [ABSTRACT FROM AUTHOR]

We propose a notion of stability for capillary hypersurfaces with constant higher order mean curvature and we generalize some results of the classical stability theory for constant mean curvature capillary hypersurfaces. [ABSTRACT FROM AUTHOR]

In this paper, we study a nonlocal diffusion problem with free boundaries in a spatially heterogeneous environment. Since both the diffusion and reaction terms include nonlocal operators that do not satisfy the local Lipschitz continuity condition, many traditional methods are not directly applicable. We introduce new techniques to address these challenges. We first establish the global existence and uniqueness of the solution. By studying the principal eigenvalue of the problem, we obtain the spreading-vanishing dichotomy and the corresponding criteria. Under a suitable condition, we determine the long-term behavior of the solution when spreading occurs. [ABSTRACT FROM AUTHOR]

In this paper, we study an optimal dividend payout problem of a firm facing with both diffusion risk and Poisson risk. Mathematically, we need to solve a parabolic variational inequality involving an integro-differential operator with gradient constraint. Assuming the Poisson intensity is less than a given constant $ \lambda_0 $, we prove the existence, uniqueness, and monotonicity of a classical solution to the variational inequality without putting any constraint on the loss distribution function. The properties such as existence and strict monotonicity and the upper bound of the free boundary are also obtained. [ABSTRACT FROM AUTHOR]

Adverse selection is a version of the principal-agent problem that includes monopolist nonlinear pricing, where a monopolist with known costs seeks a profit-maximizing price menu facing a population of potential consumers whose preferences are known only in the aggregate. For multidimensional spaces of agents and products, Rochet and Choné (Econometrica, 1998) reformulated this problem as a concave maximization over the set of convex functions, by assuming agent preferences combine bilinearity in the product and agent parameters with a quasilinear sensitivity to prices. We characterize solutions to this problem by identifying a dual minimization problem. This duality allows us to reduce the solution of the square example of Rochet–Choné to a novel free boundary problem, giving the first analytical description of an overlooked market segment. [ABSTRACT FROM AUTHOR]

This is a sequel work on the existence of the solution to the free boundary problem on injection of fluid from a slot into a uniform stream with two free boundaries by Stojanovic (IMA J Appl Math 41:237–253, 1988). However, the uniqueness of the solution to the two-phase fluids problem with two free boundaries remains unresolved. In this paper, we will establish the asymptotic behavior of the flow in the upstream and prove the uniqueness of the solution to this problem. [ABSTRACT FROM AUTHOR]

This paper investigates the longtime behavior of solutions to a temporally discrete diffusion equation with a fixed boundary and a free boundary respectively in one space dimension. Such equation can be equivalent in any case to an integro-difference equation, another important time discrete equation that provides powerful tools for the study of dispersal phenomena. In this paper, we first discuss the global dynamics of the equation in a fixed bounded domain. With a Stefan type free boundary, we then give a new well-posedness proof and the regular spreading-vanishing dichotomy for the corresponding problem. Moreover, a modified comparison principle for the time discrete free boundary problem is proved in an effort to provide the sufficient conditions for dichotomy. It is the first attempt to study the temporally discrete diffusive phenomenon with a free boundary. [ABSTRACT FROM AUTHOR]

Given is a bounded domain Ω ⊂ ℝ n {\Omega\subset\mathbb{R}^{n}} , and a vector-valued function defined on ∂ ⁡ Ω {\partial\Omega} (depicting temperature distributions from different sources), our objective is to study the mathematical model of a physical problem of enclosing ∂ ⁡ Ω {\partial\Omega} with a specific volume of insulating material to reduce heat loss in a stationary scenario. Mathematically, this task involves identifying a vector-valued function 퐮 = ( u 1 , … , u m ) {\mathbf{u}=(u^{1},\dots,u^{m})} ( m ≥ 1 {m\geq 1} ) that represents the temperature within Ω and gives rise to a free boundary, somehow reminiscent of, but not equivalent to, the Bernoulli free boundary problem. [ABSTRACT FROM AUTHOR]

This paper is concerned with affine solutions to the isentropic compressible Navier–Stokes equations with physical vacuum free boundary. Motivated by the result for Euler equations by Sideris (Arch Ration Mech Anal 225:141–176, 2017), we established the existence theories of affine solutions for the Navier–Stokes equations in ℝN(N≥2)$$ {\mathrm{\mathbb{R}}}^N\left(N\ge 2\right) $$ space under the homogeneity assumption that the pressure and the nonlinear viscosity parameters as functions of the density have a common degree of homogeneity. We derived an N×N$$ N\times N $$ second‐order system of nonlinear ODEs of the deformation gradient A(t)$$ A(t) $$ and provided an asymptotic analysis of the corresponding matrix system. The results show that both the diameter and volume of viscous fluids expand to infinity as time goes to infinity, and the algebraic rate of expansion is not bigger than that of inviscid fluids (Euler equations). In particular, the results contain the spherically symmetric case, in which the free boundary will grow linearly in time, exactly as that in inviscid fluids. Moreover, these results can be applied to the Navier–Stokes equations with constant viscosity and the Euler equations. [ABSTRACT FROM AUTHOR]

To explore the bending process and theory of the free-boundary aerodynamic film forming method for curved detectors, this study integrates practical forming structures with theoretical analysis and establishes a simulation model to investigate stress, strain, and morphological changes during bending. The analysis indicates that the shift from "projection" to "wrapping" in forming theory is due to the release of boundary degrees of freedom. The forming process can be summarized as the mold's arc characteristics, originating from the chip's corners, gradually replacing the chip's rectangular characteristics along the diagonal, resulting in corresponding stress and strain changes. The "wrapping" bending theory of this method has significant advantages over traditional methods and represents a crucial direction for achieving higher curvature in the future. However, this study found that the use of film pressure can only inhibit out-of-plane deformation to a certain extent, and the buckling phenomenon will still occur when the thinner chip is bent. It prevents the use of thinner chips in the thinning–bending method, so avoiding out-of-plane deformation during the molding process is the direction that needs to be broken in the future. [ABSTRACT FROM AUTHOR]

In this paper, we consider the free boundary problem of the radially symmetric compressible Navier–Stokes equations with viscosity coefficients of the form μ(ρ) = ρθ, λ(ρ) = (θ − 1)ρθ in R N . Under the continuous density boundary condition, we correct some errors in (Z. H. Guo and Z. P. Xin, "Analytical solutions to the compressible Navier–Stokes equations with density-dependent viscosity coefficients and free boundaries," J. Differ. Equ., vol. 253, no. 1, pp. 1–19, 2012) for N = 3, θ = γ > 1 and improve the spreading rate of the free boundary, where γ is the adiabatic exponent. Moreover, we construct an analytical solution for θ = 2 3 , N = 3 and γ > 1, and we prove that the free boundary grows linearly in time by using some new techniques. When θ = 1, under the stress free boundary condition, we construct some analytical solutions for N = 2, γ = 2 and N = 3, γ = 5 3 , respectively. [ABSTRACT FROM AUTHOR]

Atherosclerosis is a chronic inflammatory disease characterized by the gradual formation of plaques due to lipid deposition on the inner walls of blood vessels, which can lead to narrowing, blockage, or rupture of the arteries, posing a significant risk to cardiovascular health. In this study, we study the bifurcation of a highly nonlinear and coupled model describing the early growth of atherosclerotic plaques with Holling type III functional response, involving low-density lipoprotein cholesterol, high-density lipoprotein cholesterol, cytokines, pro-inflammatory macrophages, anti-inflammatory macrophages, and foam cells, with a free boundary separating the plaque from the blood flow region. We first prove the existence and uniqueness of radially symmetric steady-state solutions, and then establish a finite branch of symmetry-breaking steady-state solutions from the radially symmetric ones. This result helps us to understand the reason of plaque asymmetry in real life. [ABSTRACT FROM AUTHOR]

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Two two-dimensional free boundary problems describing the erosion of solid surfaces by the flow of inviscid fluid in the presence of trapped vortices are considered. The first problem tackles an initially flat, infinite fluid-solid interface with uniform flow at infinity and a vortex in equilibrium above the surface. The second involves flow around a finite body with a trailing Föppl-type vortex pair. The conformal invariance of the complex potential permits both problems to be formulated as a Polubarinova–Galin (PG) type equation in which the time-dependent eroding surface in the physical z-plane is mapped to the fixed boundary of the ζ -disk. The Hamiltonian governing the equilibrium position of the vortex (or vortex pair in the second problem) is also found from the same map. In each problem, the PG equation giving the conformal map is found numerically and the time-dependent evolution of the interface and vortex location is determined. Different models governing the erosion of the interface are investigated in which the normal velocity of the boundary depends on some given function of the fluid flow velocity at the boundary. Typically, in the infinite surface case, erosion leads to the formation of a symmetric valley beneath the vortex which, in turn, moves downward toward the interface. A finite body undergoes erosion which is asymmetric in the flow direction leading to a flattening of the lee surface of the body so displaying some similarity to the experiments and associated viscous theory of Ristroph et al, Moore et al (Proc Natl Acad Sci 109(48):19606–19609, 2012, Phys Fluids 25(11):116602, 2013). [ABSTRACT FROM AUTHOR]

This paper concerns subsonic jet flows from two-dimensional finitely long divergent nozzles with straight solid walls, which are governed by a free boundary problem for a quasilinear elliptic equation. It is assumed that the angle of the nozzle and the location of the inlet are fixed, while the length of the nozzle is free. For a given surrounding pressure and a given incoming mass flux, it is shown that there is a critical number not greater than π for the angle of the nozzle such that there exists a unique subsonic jet flow if the angle of the nozzle is less than the critical number. If this critical number is less than π, then there is not a subsonic jet flow when the angle of the nozzle takes this critical number; furthermore, as the angle of the nozzle tends to this critical number, either the length of the nozzle tends to zero, or a sonic point will occur at the inlet. Moreover, it is shown that the subsonic jet flow tends to a uniform horizontal flow exponentially at the downstream. As to the jet, it is smooth away from the connecting point with the wall of the nozzle, and it connects the wall of the nozzle with C1+α regularity for each exponent α ∈(0,1/2). Furthermore, the jet is strictly concave to the fluid and tends to a line parallel to the symmetrical axis exponentially. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a free boundary problem relevant in several applications, such as tumor growth models. Our problem is expressed as an elliptic equation involving discontinuous nonlinearities in a specified domain with a moving boundary. We establish the existence and uniqueness of solutions and provide a qualitative analysis of the free boundaries generated by the nonlinear term (inner boundaries). Furthermore, we analyze the dynamics of the outer region boundary. The final result demonstrates that under certain conditions, our problem is solvable in the neighborhood of a radial solution. [ABSTRACT FROM AUTHOR]

This paper is concerned with a long-standing optimal dividend payout problem subject to the so-called ratcheting constraint, that is, the dividend payout rate shall be nondecreasing over time and is thus self-path-dependent. The surplus process is modeled by a drifted Brownian motion process and the aim is to find the optimal dividend ratcheting strategy to maximize the expectation of the total discounted dividend payouts until the ruin time. Due to the self-path-dependent control constraint, the standard control theory cannot be directly applied to tackle the problem. The related Hamilton--Jacobi--Bellman (HJB) equation is a new type of variational inequality. In the literature, it is only shown to have a viscosity solution, which is not strong enough to guarantee the existence of an optimal dividend ratcheting strategy. This paper proposes a novel partial differential equation method to study the HJB equation. We not only prove the existence and uniqueness of the solution in some stronger functional space, but also prove the strict monotonicity, boundedness, and C∞-smoothness of the dividend ratcheting free boundary. Based on these results, we eventually derive an optimal dividend ratcheting strategy, and thus solve the open problem completely. Economically speaking, we find that if the surplus volatility is above an explicit threshold, then one should pay dividends at the maximum rate, regardless of the surplus level. Otherwise, by contrast, the optimal dividend ratcheting strategy relies on the surplus level and one should only ratchet up the dividend payout rate when the surplus level touches the dividend ratcheting free boundary. Moreover, our numerical results suggest that one should invest in those companies with stable dividend payout strategies since their income rates should be higher and volatility rates smaller. [ABSTRACT FROM AUTHOR]

The motion of viscous incompressible magnetohydrodynamics (MHD) is considered in a domain that is bounded by a free surface. The motion interacts through the free surface with an electromagnetic field located in a domain exterior to the free surface and bounded by a given fixed surface. Some electromagnetic fields are prescribed on this fixed boundary. On the free surface, jumps in the magnetic and electric fields are assumed. The global existence of solutions to this problem assuming appropriate smallness conditions on the initial and boundary data is proved. [ABSTRACT FROM AUTHOR]

We study the reaction diffusion equation $ u_t-du_{xx} = f(u) $ with a monostable nonlinear function $ f(u) $ over a changing interval $ [g(t), h(t)] $, viewed as a model for the spreading of a species with population range $ [g (t), h(t)] $ and density $ u(t,x) $. The free boundaries $ x = g(t) $ and $ x = h(t) $ are not governed by the same Stefan condition as in Du and Lin [20] and other previous works; instead, they satisfy a related but different set of equations obtained from a 'preferred population density' assumption at the range boundary, which allows the population range to shrink. We obtain a rather complete understanding of the longtime dynamics of the model, which exhibits persistent propagation with a finite asymptotic propagation speed determined by a certain semi-wave solution, and the density function converges to the semi-wave profile as time goes to infinity. The asymptotic propagation speed is always smaller than that of the corresponding classical Cauchy problem where the reaction-diffusion equation is satisfied for $ x $ over the entire real line with no free boundary. Moreover, when the preferred population density used in the free boundary condition converges to 0, the solution $ u $ of our free boundary problem converges to the solution of the corresponding classical Cauchy problem, and the propagation speed also converges to that of the Cauchy problem. [ABSTRACT FROM AUTHOR]

In Chen and Liang previous work, a model, together with its well-posedness, was established for credit rating migrations with different upgrade and downgrade thresholds (i.e. a buffer zone, also called dead band in engineering). When positive dividends are introduced, the model in Chen and Liang (SIAM Financ. Math. 12, 941–966, 2021) may not be well-posed. Here, in this paper, a new model is proposed to include the realistic nonzero dividend scenarios. The key feature of the new model is that partial differential equations in Chen and Liang (SIAM Financ. Math. 12, 941–966, 2021) are replaced by variational inequalities, thereby creating a new free boundary, besides the original upgrading and downgrading free boundaries. Well-posedness of the new model, together with a few financially meaningful properties, is established. In particular, it is shown that when time to debt paying deadline is long enough, the underlying dividend paying company is always in high grade rating, that is, only when time to debt paying deadline is within a certain range, there can be seen the phenomenon of credit rating migration. [ABSTRACT FROM AUTHOR]

We consider the motion of an incompressible magnetohydrodynamics with resistivity in a domain bounded by a free surface which is coupled through the free surface with an electromagnetic field generated by a magnetic field prescribed on an exterior fixed boundary. On the free surface, transmission conditions for the electromagnetic field are imposed. As transmission condition we assume jumps of tangent components of magnetic and electric fields on the free surface. We prove local existence of solutions such that velocity and magnetic fields belong to H 2 + α , 1 + α / 2 , α > 5 / 8 . [ABSTRACT FROM AUTHOR]

We study the motion of charged liquid drop in three dimensions where the equations of motions are given by the Euler equations with free boundary with an electric field. This is a well-known problem in physics going back to the famous work by Rayleigh. Due to experiments and numerical simulations one may expect the charged drop to form conical singularities called Taylor cones, which we interpret as singularities of the flow. In this paper, we study the well-posedness of the problem and regularity of the solution. Our main theorem is a criterion which roughly states that if the flow remains C 1 , α -regular in shape and the velocity remains Lipschitz-continuous, then the flow remains smooth, i.e., C ∞ in time and space, assuming that the initial data is smooth. Our main focus is on the regularity of the shape of the drop. Indeed, due to the appearance of Taylor cones, which are singularities with Lipschitz-regularity, we expect the C 1 , α -regularity assumption to be optimal. We also quantify the C ∞ -regularity via high order energy estimates which, in particular, implies the well-posedness of the problem. [ABSTRACT FROM AUTHOR]

We study a minimization problem with free boundary, resulting in hybrid quadrature domains for the Helmholtz equation, as well as some applications to inverse scattering problems. [ABSTRACT FROM AUTHOR]

This paper investigates the possible scattering and nonscattering behavior of an anisotropic and inhomogeneous Lipschitz medium at a fixed wave number and with a single incident field. We connect the anisotropic nonscattering problem to a Bernoulli type free boundary problem. By invoking methods from the theory of free boundaries, we show that an anisotropic medium with Lipschitz but not C1,α boundary scatters every incident wave that satisfies a nondegeneracy condition. [ABSTRACT FROM AUTHOR]

In this paper, we study a free boundary problem of tumor growth with necrotic core. The model is a parabolic-hyperbolic partial differential equations, which is composed of three first-order nonlinear hyperbolic equations, a parabolic equation and an ordinary differential equation. First, we obtained the approximation model by polishing the Heaviside function, and then proved the existence and uniqueness of the solution of the approximation model. In addition, we improved the regularity of solution of the approximate problem by using the characteristic curves method, and finally proved the global existence of the weak solution of the original problem by the convergence. [ABSTRACT FROM AUTHOR]

In this paper, we study an SIRI epidemic model with relapse and free boundaries, which can describe the spreading process of diseases well. At first, the existence and uniqueness of the global solution is proven. And then we obtain some sufficient conditions for the disease spreading and vanishing. We can find an R0 such that the disease will vanish for R0 ≤ 1; otherwise, whether the disease will spread or not is determined by the initial infected region h0 and spreading capacity ho of the spreading front. Moreover, the longtime behavior of the solution is given. Finally, numerical simulations are provided to illustrate the results of the analysis. [ABSTRACT FROM AUTHOR]

It is proved by Brendle (Geom Funct Anal 22:621–626, 2012) that the equatorial disk D k has least area among k-dimensional free boundary minimal submanifolds in the Euclidean ball B n . By comparing the excess of free boundary minimal submanifolds with the excess of the associated cones over their boundary, we prove the existence of a gap for the area. [ABSTRACT FROM AUTHOR]

In this paper, we study the dynamics of the ratio-dependent type prey–predator model with different free boundaries. The two free boundaries, determined by prey and predator, respectively, implying that they may intersect with each other as time evolves, are used to describe the spreading of prey and predator. Our primary focus lies in analyzing the long-term behaviors of both predator and prey. We establish sufficient conditions for the spreading and vanishing of prey and predator. Furthermore, in cases where spread occurs, we offer estimates for the asymptotic spreading speeds of prey and predator, denoted as u and v, respectively, as well as the asymptotic speeds of the free boundaries, denoted by h and g. Our findings reveal that when the predator's speed is lower than that of the prey, it leads to a reduction in the prey's asymptotic speed. [ABSTRACT FROM AUTHOR]