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We study the Bayesian problems of detecting a change in the drift rate of an observable diffusion process with linear and exponential penalty costs for a detection delay. The optimal times of alarms are found as the first times at which the weighted likelihood ratios hit stochastic boundaries depending on the current observations. The proof is based on the reduction of the initial problems into appropriate three-dimensional optimal stopping problems and the analysis of the associated parabolic-type free-boundary problems. We provide closed-form estimates for the value functions and the boundaries, under certain nontrivial relations between the coefficients of the observable diffusion.

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.

In this paper, we prove the local well-posedness of the free boundary problem for the incompressible Euler equations in low regularity Sobolev spaces, in which the velocity is a Lipschitz function and the free surface belongs to

In this paper we prove some results concerning inverse/free boundary type problems, below the continuous threshold, for the heat equation in the setting of parabolic regular graph domains.

Given an n -dimensional Riemannian manifold with non-negative sectional curvature and convex boundary, that is conformal to a Euclidean convex bounded domain, we show that it does not contain any compact stable free boundary minimal submanifold of dimension 2 k n-2 , provided that either the boundary is strictly convex with respect to any of the two metrics or the sectional curvature is strictly positive.

This article is concerned with the local well-posedness problem for the compressible Euler equations in gas dynamics. For this system we consider the free boundary problem which corresponds to a physical vacuum. Despite the clear physical interest in this system, the prior work on this problem is limited to Lagrangian coordinates, in high-regularity spaces. Instead, the objective of the present work is to provide a new, fully Eulerian approach to this problem, which provides a complete, Hadamard-style well-posedness theory for this problem in low-regularity Sobolev spaces. In particular, we give new proofs for existence, uniqueness, and continuous dependence on the data with sharp, scale-invariant energy estimates, and a continuation criterion.

In this paper we investigate a free boundary problem for the classical Lotka–Volterra type predator–prey model with double free boundaries in one space dimension. This system models the expanding of an invasive or new predator species in which the free boundaries represent expanding fronts of the predator species and are described by Stefan-like condition. We prove a spreading–vanishing dichotomy for this model, namely the predator species either successfully spreads to infinity as t→∞ at both fronts and survives in the new environment, or it spreads within a bounded area and dies out in the long run while the prey species stabilizes at a positive equilibrium state. The long time behavior of solution and criteria for spreading and vanishing are also obtained.

We prove the existence of a unique global-in-time weak solution of the Navier–Stokes equations that govern the motion of a compressible viscous fluid with density-dependent viscosity in two-dimensional space. The initial velocity belongs to the Sobolev space H1(R2), and the initial fluid density is α-Hölder continuous on both sides of a C1+α-regular interface with some geometrical assumption. We prove that this configuration persists over time: the initial interface is transported by the flow to an interface that maintains the same regularity as the initial one. Our result generalizes a previous known result of Hoff [Comm. Pure Appl. Math. 55 (2002), 1365–140] and Hoff and Santos [Arch. Ration. Mech. Anal. 188 (2008), 509–543] concerning the propagation of regularity for discontinuity surfaces by allowing a more general non-linear pressure law and density-dependent viscosity. Moreover, it supplements the work by Danchin, Fanelli, and Paicu [Anal. PDE 13 (2020), 275–316] with global-in-time well-posedness, even for density-dependent viscosity, and we achieve uniqueness in a large space.

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.