Insensitizing controls for a forward stochastic heat equation

The existence of insensitizing controls for a forward stochastic heat equation is considered. To develop the duality, we obtain observability estimates for linear forward and backward coupled stochastic heat equations with general coefficients, by means of some global Carleman estimates. Furthermore, the constant in the observability inequality is estimated by an explicit function of the norm of the involved coefficients in the equation. As far as we know, our paper is the first one to address the problem of insensitizing controls for stochastic partial differential equations.     

Quantitative unique continuation for the semilinear heat equation in a convex domain

In this paper, we study certain unique continuation properties for solutions of the semilinear heat equation t_∂u−△u=g(u), with the homogeneous Dirichlet boundary condition, over Ω×(0,T_∗). Ω is a bounded, convex open subset of Rd, with a smooth boundary for the subset. The function g:R→R satisfies certain conditions. We establish some observation estimates for (u−v), where u and v are two solutions to the above-mentioned equation. The observation is made over ω×{T}, where ω is any non-empty open subset of Ω, and T is a positive number such that both u and v exist on the interval [0,T]. At least two results can be derived from these estimates: (i) if ‖(u−v)(⋅,T)_‖L²(ω)=δ, then ‖(u−v)(⋅,T)_‖L²(Ω)?Cδα where constants C>0 and α∈(0,1) can be independent of u and v in certain cases; (ii) if two solutions of the above equation hold the same value over ω×{T}, then they coincide over Ω×[0,Tm). Tm indicates the maximum number such that these two solutions exist on [0,Tm).     

Unique continuation property for anomalous slow diffusion equation

A Carleman estimate and the unique continuation of solutions for an anomalous diffusion equation with fractional time derivative of order 0 < α <1 are given. The estimate is derived through some subelliptic estimates for an operator associated to the anomalous diffusion equation using calculus of pseudo-differential operators.     

Unique continuation for the symmetric regularized long wave equation

The unique continuation property has been intensively studied for a long time due to the important role that plays in the applications. The validity of the unique continuation property for symmetric regularized long wave equation is showed in this paper. The result is established by using an appropriate Carleman‐type estimate for a partial differential operators closely related to our problem. Copyright © 2006 John Wiley & Sons, Ltd.     

Formal solutions of semilinear heat equations

We study formal power series solutions to the initial value problem for semilinear heat equation t_∂u−Δu=f(u) with polynomial nonlinearity f and prove that they belong to the formal Gevrey class G². Next we give counterexamples showing that the solution, in general, is not analytic in time at t=0.     

Carleman estimates and unique continuation property for the Kadomtsev–Petviashvili equations

We study the unique continuation property for the generalized Kadomtsev–Petviashvili (KP) equations and its regularized version. We use Carleman estimates to prove that if the solution of the KP equations vanishes in an open subset, then this solution is identically equal to zero in the horizontal component of the open subset.     

Unique continuation on a line for the Helmholtz equation

In this article, local unique continuation on a line for solutions of the Helmholtz equation is discussed. The fundamental solution of the exterior problem for the Helmholtz equation have a logarithmic singularity which behaves similar to those of the interior problem for the Laplace equation in two dimension. A Hölder-type conditional stability estimate of the proposed exterior problem for the Helmholtz equation is obtained by adopting the complex extension method in Cheng and Yamamoto [J. Cheng and M. Yamamoto, Unique continuation on a line for harmonic functions, Inverse Probl. 14 (1998), pp. 869–882]. Finally, a regularization scheme based on the collocation method is compatible with the Hölder-type stability estimate provided that the line does not intersect the boundary of the domain for both the Laplace and the Helmholtz equations.     

Unique continuation from infinity in asymptotically anti-de Sitter spacetimes II: Non-static boundaries

We generalize our unique continuation results recently established for a class of linear and nonlinear wave equations □_g?+σ? = 𝒢(?,??) on asymptotically anti-de Sitter (aAdS) spacetimes to aAdS spacetimes admitting nonstatic boundary metrics. The new Carleman estimates established in this setting constitute an essential ingredient in proving unique continuation results for the full nonlinear Einstein equations, which will be addressed in forthcoming papers. Key to the proof is a new geometrically adapted construction of foliations of pseudo-convex hypersurfaces near the conformal boundary.     

Carleman estimates and unique continuation property for the anisotropic differential-operator equations

The unique continuation theorems for the anisotropic partial differential-operator equations with variable coeffcients in Banach-valued Lp-spaces are studied.To obtain the uniform maximal regularity and the Carleman type estimates for parameter depended differential-operator equations,the suffcient conditions are founded.By using these facts,the unique continuation properties are established.In the application part,the unique continuation properties and Carleman estimates for finite or infinite systems of quasielliptic partial differential equations are studied.     

Remark on the strong unique continuation property for parabolic operators

We consider solutions , in a neighbourhood of , to a parabolic differential equation with variable coefficients depending on space and time variables. We assume that the coefficients in the principal part are Lipschitz continuous and that those in the lower order terms are bounded. We prove that, if vanishes of infinite order at , then .

     

Heisenberg群上次Laplace算子的Carleman型估计与唯一延拓性

本文在适当条件下给出了Heisenberg群上次Laplace算子的Carleman型估计, 并由此建立了唯一延拓性.     

A Liouville property and quasiconvergence for a semilinear heat equation

This paper is concerned with a supercritical semilinear diffusion equation with the power nonlinearity. Via establishing a Liouville-type property, we prove the quasiconvergence (convergence to a set of steady states) of a large class of global solutions. The method of proof relies on similarity variables and invariant manifold ideas.     

Null controllability for the heat equation with singular inverse-square potentials

We prove the null controllability of the heat equation perturbed by a singular inverse-square potential arising in quantum mechanics and combustion theory. This is done within the range of subcritical coefficients of the singular potential, provided the control acts on an annular set around the singularity. Our proof uses a splitting argument on the domain, decomposition in spherical harmonics, new Carleman inequalities and refined Hardy inequalities.