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We give a comprehensive treatment of the parabolic Signorini problem based on a generalization of Almgren’s monotonicity of the frequency. This includes the proof of the optimal regularity of solutions, classification of free boundary points, the regularity of the regular set and the structure of the singular set.

This paper is dedicated to a free boundary system arising in the study of a class of shape optimization problems. The problem involves three variables: two functions u and v , and a domain ; with u and v being both positive in , vanishing simultaneously on , and satisfying an overdetermined boundary value problem involving the product of their normal derivatives on . Precisely, we consider solutions u, v ın C(B_1) of - u= f \\ and \\ - v=g =\\u>0\\=\\v>0\\, u n v n=Q B_1. Our main result is an epsilon-regularity theorem for viscosity solutions of this free boundary system. We prove a partial Harnack inequality near flat points for the couple of auxiliary functions uv and 12(u+v) . Then, we use the gained space near the free boundary to transfer the improved flatness to the original solutions. Finally, using the partial Harnack inequality, we obtain an improvement-of-flatness result, which allows to conclude that flatness implies C^1, regularity.

The paper deals with a West Nile virus (WNv) model, in which the nonlocal diffusion characterizes the long-range movement of birds and mosquitoes, the free boundaries describe their spreading fronts, and the seasonal succession accounts for the effect of the warm and cold seasons. The well-posedness of the mathematical model is established, and its long-term dynamical behaviours, which depend upon the generalized eigenvalues of the corresponding linearized differential operator, are investigated. For both spatially independent and nonlocal WNv models with seasonal successions, the generalized eigenvalues are studied and applied to determine whether the spreading or vanishing occurs. Our results extend those for the case with nonlocal diffusion but no free boundary and those for the case with free boundary but local diffusion, respectively. The generalized eigenvalues reveal that there exists positive correlation between the duration of the warm season and the risk of infection. Moreover, the initial infection length, the initial infection scale and the spreading ability to new areas all play important roles for the long time behaviors of the time dependent solutions.

In this paper we investigate a diffusive logistic model with a free boundary in one space dimension. We aim to use the dynamics of such a problem to describe the spreading of a new or invasive species, with the free boundary representing the expanding front. We prove a spreading-vanishing dichotomy for this model, namely the species either successfully spreads to all the new environment and stabilizes at a positive equilibrium state, or it fails to establish and dies out in the long run. Sharp criteria for spreading and vanishing are given. Moreover, we show that when spreading occurs, for large time, the expanding front moves at a constant speed. This spreading speed is uniquely determined by an elliptic problem induced from the original model. [PUBLICATION ABSTRACT]

We investigate the tumor boundary instability induced by nutrient consumption and supply based on a Hele-Shaw model. The model describes the geometric evolution of the tumor region and is derived from taking the incompressible limit of a cell density model, which leads to the pressure vanishing on the free boundary. We investigate the boundary behaviors under two different nutrient supply regimes, in vitro and in vivo, where in the former case the supply is adequate and in the latter case the supply is deficient. Our main conclusion is that by investigating typical solutions of the tumor-nutrient model with the asymptotic analysis with respect to the domain perturbation, the tumor boundary in the in vitro regime is shown to be stable regardless of the nutrient consumption rate. However, boundary instability occurs when the nutrient consumption rate exceeds a certain threshold in the in vivo regime, and the bifurcation threshold has a monotonic dependence on the frequency of the domain perturbation.

A mutualist model with nonlocal diffusions and a free boundary is first considered. We prove that this problem has a unique solution defined for t ≥ 0 , and its dynamics are governed by a spreading-vanishing dichotomy. Some criteria for spreading and vanishing are also given. Of particular importance is that we find that the solution of this problem has quite rich longtime behaviors, which vary with the conditions satisfied by kernel functions and are much different from those of the counterpart with local diffusion and free boundary. At last, we extend these results to the model with nonlocal diffusions and double free boundaries.

The present paper is devoted to the investigation of the long time dynamics for advective-cooperative system with free boundaries, which models the infectious diseases transmitted via digestive system such as fecal-oral diseases, cholera, hand-foot and mouth, etc,... The coupled advective terms yield significant obstacles, which require a different approach to analyze the system’s dynamics. To overcome this, we must prove the existence and the variational formula for the principal eigenvalue of a linear system with advective-cooperative, then use it to obtain the right limits as the dispersal rates and domain tend to zero or infinity. Additionally, we conduct numerical simulations to validate our theoretical results, and explore the effects of various parameters.

In this paper, a reaction–diffusion system is proposed to model the spatial spreading of West Nile virus in vector mosquitoes and host birds in North America. Transmission dynamics are based on a simplified model involving mosquitoes and birds, and the free boundary is introduced to model and explore the expanding front of the infected region. The spatial-temporal risk index R 0 F ( t ) , which involves regional characteristic and time, is defined for the simplified reaction–diffusion model with the free boundary to compare with other related threshold values, including the usual basic reproduction number R 0 . Sufficient conditions for the virus to vanish or to spread are given. Our results suggest that the virus will be in a scenario of vanishing if R 0 ≤ 1 , and will spread to the whole region if R 0 F ( t 0 ) ≥ 1 for some t 0 ≥ 0 , while if R 0 F ( 0 ) < 1 < R 0 , the spreading or vanishing of the virus depends on the initial number of infected individuals, the area of the infected region, the diffusion rate and other factors. Moreover, some remarks on the basic reproduction numbers and the spreading speeds are presented and compared.

We study the critical Ising model with free boundary conditions on finite domains in Zd with d≥4. Under the assumption, so far only proved completely for high d, that the critical infinite volume two-point function is of order |x-y|-(d-2) for large |x-y|, we prove the same is valid on large finite cubes with free boundary conditions, as long as x, y are not too close to the boundary. This confirms a numerical prediction in the physics literature by showing that the critical susceptibility in a finite domain of linear size L with free boundary conditions is of order L2 as L→∞. We also prove that the scaling limit of the near-critical (small external field) Ising magnetization field with free boundary conditions is Gaussian with the same covariance as the critical scaling limit, and thus the correlations do not decay exponentially. This is very different from the situation in low d or the expected behavior in high d with bulk boundary conditions.

We investigate the asymptotic behavior of a Kirchhoff plate equation with free boundary conditions in a bounded domain of R2. The model involves frictional internal damping, a time delay condition of fractional type and a source term. We derive a polynomial decay estimate for the associated semigroup by using the frequency domain method.