Related papers: Carleman estimate and unique continuation for a st…
Continuous first-order logic is used to apply model-theoretic analysis to analytic structures (e.g. Hilbert spaces, Banach spaces, probability spaces, etc.). Classical computable model theory is used to examine the algorithmic structure of…
For a semilinear elliptic equation, we prove uniqueness results in determining potentials and semilinear terms from partial Cauchy data on an arbitrary subboundary.
The goal of the present paper is to explore the long-time behavior of the growth-fragmentation equation formulated in the case of equal mitosis and variability in growth rate, under fairly general assumptions on the coefficients. The first…
We obtain a unique continuation result for the differential inequality $| (i\partial_t +\Delta)u | \leq |Vu| + | W\cdot\nabla u |$ by establishing $L^2$ Carleman estimates. Here, $V$ is a scalar function and $W$ is a vector function, which…
We prove local in time well-posedness for a class of quasilinear Hamiltonian KdV-type equations with periodic boundary conditions, more precisely we show existence, uniqueness and continuity of the solution map. We improve the previous…
This paper is devoted to the reconstruction of the time and space-dependent coefficient in an infinite cylindrical hyperbolic domain. Using a local Carleman estimate we prove the uniqueness and a H\"older stability in the determining of the…
We consider an inverse problem for the nonlinear Boltzmann equation with a time-dependent kernel in dimensions $n\ge 2$. We establish a logarithm-type stability result for the collision kernel from measurements under certain additional…
We consider stochastic growth models for populations organized in colonies and subject to uniform catastrophes. To assess population viability, we analyze scenarios in which individuals adopt dispersion strategies after catastrophic events.…
The Cauchy problem for the Schr\"odinger equations is studied with time-dependent potentials growing polynomially in the spatial direction. First the existence and the uniqueness of solutions are shown in the weighted Sobolev spaces. In…
It is proposed to monitor spatial and temporal spreads of epidemics via solution of a Coefficient Inverse Problem for a system of three coupled nonlinear parabolic equations. To solve this problem numerically, a version of the so-called…
In this paper, we consider a linear heat equation with constant coefficients and a single constant delay. Such equations are commonly used to model and study various problems arising in ecology and population biology when describing the…
In this article, we complement recent results on the convergence of the state estimate obtained by applying the discrete-time Kalman filter on a time-sampled continuous-time system. As the temporal discretization is refined, the estimate…
We consider a first-order transport equation $\ppp_tu(x,t) + (H(x)\cdot\nabla u(x,t)) + p(x)u(x,t) = F(x,t)$ for $x \in \OOO \subset \R^d$, where $\OOO$ is a bounded domain and $0<t<T$. We prove a Carleman estimate for more generous…
In this paper, a quantitative estimate of unique continuation for the stochastic heat equation with bounded potentials on the whole Euclidean space is established. This paper generalizes the earlier results in [29] and [17] from a bounded…
In this paper, we establish a quantitative weak unique continuation theorem on an annular domain for a backward degenerate parabolic equation with a degenerate interior point. Our methodology hinges on approximating the solution of the…
In this work we develop a new numerical approach for recovering a spatially dependent source component in a standard parabolic equation from partial interior measurements. We establish novel conditional Lipschitz stability and H\"{o}lder…
We study the problem of stabilization for the acoustic system with a spatially distributed damping. Without imposing any hypotheses on the structural properties of the damping term, we identify logarithmic decay of solutions with growing…
We consider very weak solutions of the Cauchy problem for the porous medium equation on Cartan-Hadamard manifolds, that are assumed to satisfy general curvature bounds and to be stochastically complete. We identify a class of initial data…
This paper analyzes a stochastic logistic difference equation under the assumption that the population distribution follows a normal distribution. Our focus is on the mathematical relationship between the average growth rate and a newly…
We introduce an arbitrary order, stabilized finite element method for solving a unique continuation problem subject to the time-harmonic elastic wave equation with variable coefficients. Based on conditional stability estimates we prove…