Unique Continuation Property Research Articles

The fractional p,-biharmonic systems: optimal Poincaré constants, unique continuation and inverse problems

This article investigates nonlocal, quasilinear generalizations of the classical biharmonic operator (-Delta )^2. These fractional p -biharmonic operators appear naturally in the variational characterization of the optimal fractional Poincaré constants in Bessel potential spaces. We study the following basic questions for anisotropic fractional p -biharmonic systems: existence and uniqueness of weak solutions to the associated interior source and exterior value problems, unique continuation properties, monotonicity relations, and inverse problems for the exterior Dirichlet-to-Neumann maps. Furthermore, we show the UCP for the fractional Laplacian in all Bessel potential spaces H^{t,p} for any tin {mathbb R}, 1 le p < infty and s in {mathbb R}_+ {setminus } {mathbb N}: If uin H^{t,p}({mathbb R}^n) satisfies (-Delta )^su=u=0 in a nonempty open set V, then uequiv 0 in {mathbb R}^n. This property of the fractional Laplacian is then used to obtain a UCP for the fractional p -biharmonic systems and plays a central role in the analysis of the associated inverse problems. Our proofs use variational methods and the Caffarelli–Silvestre extension.

Wave breaking and asymptotic analysis of solutions for a class of weakly dissipative nonlinear wave equations

We study formation of singularities and persistence properties of solutions for a class of non-local and non-linear evolution equations with an arbitrary function and a non-negative (dissipation) parameter, which includes the famous Camassa-Holm equation and other recently reported hydrodynamic models as particular members. In case such a parameter is positive, solutions emanating from Cauchy problems have time decaying norms bounded from above by the norm of the corresponding initial datum. Whenever the function is even, for a given odd initial datum with slope satisfying a certain relation involving the dissipation parameter, then the corresponding solution of the problem breaks at finite time. We can also describe scenarios for the occurrence of wave breaking for more general initial data or functions, provided that those latter satisfy certain conditions on their first derivatives. We also study asymptotic behavior and unique continuation properties for solutions of the equation.

On a Phase Field Model for RNA-Protein Dynamics

A phase field model which describes the formation of protein-RNA complexes subject to phase segregation is analyzed. A single protein, two RNA species, and two complexes are considered. Protein and RNA species are governed by coupled reaction-diffusion equations which also depend on the two complexes. The latter ones are driven by two Cahn-Hilliard equations with Flory-Huggins potential and reaction terms depending on the solution variables. The resulting nonlinear coupled system is equipped with no-flux boundary conditions and suitable initial conditions. The former ones entail some mass conservation constraints which are also due to the nature of the reaction terms. The existence of global weak solutions in a bounded (two- or) three-dimensional domain is established. In dimension two, some weighted-in-time regularity properties are shown. In particular, the complexes instantaneously get uniformly separated from the pure phases. Taking advantage of this result, a unique continuation property is proven. Among the many technical difficulties, the most significant one arises from the fact that the two complexes are initially nonexistent, so their initial conditions are zero i.e., they start from a pure phase. Thus we must solve, in particular, a system of two coupled Cahn-Hilliard equations with singular potential and nonlinear sources without the usual assumption on the initial datum, i.e., the initial phase cannot be pure. This novelty requires a new approach to estimate the chemical potential in a suitable $L^p(L^2)$-space with $p\in(1,2)$. This technique can be extended to other models like, for instance, the well-known Cahn-Hilliard-Oono equation.

An inverse problem for the fractional porous medium equation

We consider a time-independent variable coefficients fractional porous medium equation and formulate an associated inverse problem. We determine both the conductivity and the absorption coefficient from exterior partial measurements of the Dirichlet-to-Neumann map. Our approach relies on a time-integral transform technique as well as the unique continuation property of the fractional operator.

Strong Unique Continuation from the Boundary for the Spectral Fractional Laplacian

We investigate unique continuation properties and asymptotic behaviour at boundary points for solutions to a class of elliptic equations involving the spectral fractional Laplacian. An extension procedure leads us to study a degenerate or singular equation on a cylinder, with a homogeneous Dirichlet boundary condition on the lateral surface and a non-homogeneous Neumann condition on the basis. For the extended problem, by an Almgren-type monotonicity formula and a blow-up analysis, we classify the local asymptotic profiles at the edge where the transition between boundary conditions occurs. Passing to traces, an analogous blow-up result and its consequent strong unique continuation property is deduced for the nonlocal fractional equation.

An application of generalized Morrey spaces to unique continuation property of the quasilinear elliptic equations

&lt;abstract&gt;&lt;p&gt;In this paper, we study nonnegative weak solutions of the quasilinear elliptic equation $ \text{div}(A(x, u, \nabla u)) = B(x, u, \nabla u) $, in a bounded open set $ \Omega $, whose coefficients belong to a generalized Morrey space. We show that $ \log (u + \delta) $, for $ u $ a nonnegative solution and $ \delta $ an arbitrary positive real number, belongs to $ \text{BMO}(B) $, where $ B $ is an open ball contained in $ \Omega $. As a consequence, this equation has the strong unique continuation property. For the main proof, we use approximation by smooth functions to the weak solutions to handle the weak gradient of the composite function which involves the weak solutions and then apply Fefferman's inequality in generalized Morrey spaces, recently proved by Tumalun et al. &lt;sup&gt;[&lt;xref ref-type="bibr" rid="b1"&gt;1&lt;/xref&gt;]&lt;/sup&gt;.&lt;/p&gt;&lt;/abstract&gt;

Unique Continuation and Time Decay for a Higher-Order Water Wave Model

This work is devoted to prove the exponential decay for the energy of solutions of a higher order Korteweg–de Vries (KdV)–Benjamin–Bona–Mahony (BBM) equation on a periodic domain with a localized damping mechanism. Following the method in [L. Rosier and B.-Y. Zhang, J. Diff. Equ. 254 (2013) 141–178], which combines energy estimates, multipliers and compactness arguments, the problem is reduced to prove the Unique Continuation Property (UCP) for weak solutions of the model. Then, this is done by deriving Carleman estimates for a system of coupled elliptic-hyperbolic equations.

Boundary control for transport equations

&lt;p style='text-indent:20px;'&gt;This paper considers two types of boundary control problems for linear transport equations. The first one shows that transport solutions on a subdomain of a domain &lt;inline-formula&gt;&lt;tex-math id="M1"&gt;\begin{document}$ X $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; can be controlled exactly from incoming boundary conditions for &lt;inline-formula&gt;&lt;tex-math id="M2"&gt;\begin{document}$ X $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; under appropriate convexity assumptions. This is in contrast with the only approximate control one typically obtains for elliptic equations by an application of a unique continuation property, a property which we prove does not hold for transport equations. We also consider the control of an outgoing solution from incoming conditions, a transport notion similar to the Dirichlet-to-Neumann map for elliptic equations. We show that for well-chosen coefficients in the transport equation, this control may not be possible. In such situations and by (Fredholm) duality, we obtain the existence of non-trivial incoming conditions that are compatible with vanishing outgoing conditions.&lt;/p&gt;

On inverse problems for uncoupled space-time fractional operators involving time-dependent coefficients

We study the uncoupled space-time fractional operators involving time-dependent coefficients and formulate the corresponding inverse problems. Our goal is to determine the variable coefficients from the exterior partial measurements of the Dirichlet-to-Neumann map. We exploit the integration by parts formula for Riemann-Liouville and Caputo derivatives to derive the Runge approximation property for our space-time fractional operator based on the unique continuation property of the fractional Laplacian. This enables us to extend early unique determination results for space-fractional but time-local operators to the space-time fractional case.

Unique continuation results for abstract quasi-linear evolution equations in Banach spaces

Unique continuation properties for a class of evolution equations defined on Banach spaces are considered from two different point of views: the first one is based on the existence of conserved quantities, which very often translates into the conservation of some norm of the solutions of the system in a suitable Banach space. The second one is regarded to well-posed problems. Our results are then applied to some equations, most of them describing physical processes like wave propagation, hydrodynamics, and integrable systems, such as the $b-$; Fornberg-Whitham; potential and $\pi-$Camassa-Holm; generalised Boussinesq equations; and the modified Euler-Poisson system.

SOME FUNCTION SPACES AND THEIR APPLICATIONS TO ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS

In this paper we prove Fefferman's inequalities associated to potentials belonging to a generalized Morrey space or a Stummel class. We also show that the logarithm of a non-negative weak solution to a second order elliptic partial differential equation with potential in a generalized Morrey space or a Stummel class, under some assumptions, belongs to the bounded mean oscillation class. As a consequence, this elliptic partial differential equation has the strong unique continuation property. An example of an elliptic partial differential equation with potential in a Morrey space or a Stummel class which does not satisfy the strong unique continuation is presented.

On Unique Continuation for Non-local Dispersive Models

We consider unique continuation properties of solutions to a family of evolution equations. Our interest is mainly on nonlinear non-local models. This class contains the Benjamin–Ono, the Intermediate Long Wave, the Camassa–Holm, the dispersion generalized Benjamin–Ono and non-local Schrödinger equations as well as their generalizations. We shall review, discuss, expand, and comment on several results. In addition, we shall state some open questions concerning these results and their techniques.

Well-posedness and exponential stability for a Klein-Gordon system with locally distributed viscoelastic dampings in a past-history framework

Our main goal in the present work is to prove the well-posedness as well as the exponential stability for a strongly coupled Klein-Gordon system posed on a bounded domain of R2 with smooth boundary, subject to locally distributed viscoelastic effects driven by nonnegative functions a(x) and b(x) acting on Ω, where a=b=0 in a region A of Ω. Some difficulties imposed by the presence of integrals terms force us to consider a perturbed problem in order to obtain exponential decay rates. For this purpose, microlocal analysis tools combined with unique continuation property were used.

Uniqueness for the Fractional Calderón Problem with Quasilocal Perturbations

We study the fractional Schrödinger equation with quasilocal perturbations and show that the qualitative unique continuation and Runge approximation properties hold in the assumption of sufficient decay. Quantitative versions of both results are also obtained via a propagation of smallness analysis for the Caffarelli--Silvestre extension. The results are then used to show uniqueness in the inverse problem of retrieving a quasilocal perturbation from Dirichlet-to-Neumann (DN) data under suitable geometric assumptions. Our work generalizes recent results regarding the locally perturbed fractional Calderón problem.

On the space-like analyticity in the extension problem for nonlocal parabolic equations

In this note we give an elementary proof of the space-like real-analyticity of solutions to a degenerate evolution problem that arises in the study of fractional parabolic operators of the type ( ∂ t − div x ⁡ ( B ( x ) ∇ x ) ) s (\partial _t - \operatorname {div}_x(B(x)\nabla _x))^s , 0 &gt; s &gt; 1 0&gt;s&gt;1 . Our primary interest is in the so-called extension variable. We show that weak solutions that are even in such variable, are in fact real-analytic in the totality of the space variables. As an application of this result we prove the weak unique continuation property for nonlocal parabolic operators of the type above, where B ( x ) B(x) is a uniformly elliptic matrix-valued function with real-analytic entries.

Wave-breaking phenomena and persistence properties for a nonlinear dissipative Camassa–Holm equation

In present paper, we mainly consider the wave-breaking phenomena and persistence properties of a nonlinear dissipative Camassa–Holm equation. The precise blow-up criteria of the Cauchy problem are established and wave-breaking phenomenon is investigated based on the method of characteristics and the Riccati-type differential inequality. Due to the presence of high order nonlinear term, the conservation law is losed. This difficulty has been addressed by establishing the energy inequality. Finally, we establish the persistence properties and some unique continuation properties of the solutions in the weighted -spaces .

On the Calderón problem for nonlocal Schrödinger equations with homogeneous, directionally antilocal principal symbols

In this article we consider direct and inverse problems for α-stable, elliptic nonlocal operators whose kernels are possibly only supported on cones and which satisfy the structural condition of directional antilocality as introduced by Y. Ishikawa in the 80s. We consider the Dirichlet problem for these operators on the respective “domain of dependence of the operator” and in several, adapted function spaces. This formulation allows one to avoid natural “gauges” which would else have to be considered in the study of the associated inverse problems. Exploiting the directional antilocality of these operators we complement the investigation of the direct problem with infinite data and single measurement uniqueness results for the associated inverse problems. Here, due to the only directional antilocality, new geometric conditions arise on the measurement domains. We discuss both the setting of symmetric and a particular class of non-symmetric nonlocal elliptic operators, and contrast the corresponding results for the direct and inverse problems. In particular for only “one-sided operators” new phenomena emerge both in the direct and inverse problems: For instance, it is possible to study the problem in data spaces involving local and nonlocal data, the unique continuation property may not hold in general and further restrictions on the measurement set for the inverse problem arise.

The Calderón Problem for the Fractional Wave Equation: Uniqueness and Optimal Stability

We study an inverse problem for the fractional wave equation with a potential by the measurement taking on arbitrary subsets of the exterior in the space-time domain. We are interested in the issues of uniqueness and stability estimate in the determination of the potential by the exterior Dirichlet-to-Neumann map. The main tools are the qualitative and quantitative unique continuation properties for the fractional Laplacian. For the stability, we also prove that the log type stability estimate is optimal. The log type estimate shows the striking difference between the inverse problems for the fractional and classical wave equations in the stability issue. The results hold for any spatial dimension $n\in \mathbb{N}$.

Green’s Functions for First-Order Systems of Ordinary Differential Equations without the Unique Continuation Property

This paper is a contribution to the spectral theory associated with the differential equation $Ju'+qu=wf$ on the real interval $(a,b)$ when $J$ is a constant, invertible skew-Hermitian matrix and $q$ and $w$ are matrices whose entries are distributions of order zero with $q$ Hermitian and $w$ non-negative. Under these hypotheses it may not be possible to uniquely continue a solution from one point to another, thus blunting the standard tools of spectral theory. Despite this fact we are able to describe symmetric restrictions of the maximal relation associated with $Ju'+qu=wf$ and show the existence of Green's functions for self-adjoint relations even if unique continuation of solutions fails.

The fractional Schrödinger equation on compact manifolds: global controllability results

The goal of this work is to prove global controllability and stabilization properties for the fractional Schrödinger equation on d-dimensional compact Riemannian manifolds without boundary (M, g). To prove our main results we use techniques of pseudo-differential calculus on manifolds. More precisely, by using microlocal analysis, we are able to prove propagation of regularity which together with the so-called Geometric Control Condition and Unique Continuation Property help us to prove global control results for the system under consideration. As a main novelty this manuscript presents the relation between the geometric control condition and the controllability for the fractional Schrödinger equation.