Related papers: Carleman inequality for a linear degenerate parabo…
We consider a beam equation in presence of a leading degenerate operator which is not in divergence form. We impose clamped conditions where the degeneracy occurs and dissipative conditions at the other endpoint. We provide some conditions…
We establish new Carleman estimates for the wave equation, which we then apply to derive novel observability inequalities for a general class of linear wave equations. The main features of these inequalities are that (a) they apply to a…
We consider a degenerate hyperbolic equation of Kirchhoff type with a small parameter epsilon in front of the second-order time-derivative. In a recent paper, under a suitable assumption on initial data, we proved decay-error estimates for…
This review surveys previous and recent results on null controllability and inverse problems for parabolic systems with dynamic boundary conditions. We aim to demonstrate how classical methods such as Carleman estimates can be extended to…
This paper deals with the hierarchic control of a degenerate parabolic equation with missing initial condition. We present a Stackelberg strategy combining the concept of null controllability with low-regret control. We assume that we can…
We find an explicit form of weak solutions to a Riemann problem for a degenerate semilinear parabolic equation with piecewise constant diffusion coefficient. It is demonstrated that the phase transition lines (free boundaries) correspond to…
We consider the Cauchy problem in the Euclidean space for a doubly degenerate parabolic equation with a space-dependent exponential weight, where the exponent satisfies the doubling condition. In particular, both the so called logconvex and…
We consider the Dirichlet problem in a wedge for parabolic equation whose coefficients are measurable function of t. We obtain coercive estimates in weighted $L_{p,q}$-spaces. The concept of "critical exponent" introduced in the paper plays…
For a general class of divergence type quasi-linear degenerate parabolic equations with measurable coeffcients and lower order terms from non-linear Kato-type classes, we prove local boundedness and continuity of solutions, and the…
We consider a $2\times 2$ system of parabolic equations with first and zeroth coupling and establish a Carleman estimate by extra data of only one component without data of initial values. Then we apply the Carleman estimate to inverse…
A doubly degenerate parabolic equation in non-divergent form with variable growth is investigated in this paper. In suitable spaces, we prove the existence of weak solutions of the equation for cases $1\leq m < 2$ and $m\geq 2$ in different…
We explore how the analysis of the Carleman linearization can be extended to dynamical systems on infinite-dimensional Hilbert spaces with quadratic nonlinearities. We demonstrate the well-posedness and convergence of the truncated Carleman…
We consider elliptic transmission problems with complex coefficients across an interface. Under proper transmission conditions, that extend known conditions for well-posedness, and sub-ellipticity we derive microlocal and local Carleman…
The main objective of this paper is to establish the null controllability for the fourth order semilinear parabolic equations with the nonlinearities involving the state and its gradient up to second order. First of all, based on optimal…
In this article we establish the well-posedness, energy estimates, stability, and local null controllability for the thermistor system modeled by a parabolic-parabolic system using a control force acting on just one equation of the system.…
This is the first publication in which an ill-posed Cauchy problem for a quasi- linear PDE is solved numerically by a rigorous method. More precisely, we solve the side Cauchy problem for a 1-d quasilinear parabolc equation. The key idea is…
We establish a strong unique continuation property for stochastic parabolic equations. Our method is based on a suitable stochastic version of Carleman estimate. As far as we know, this is the first result for strong unique continuation…
In this paper we investigate the Hierarchical null controllability of a coupled degenerate semilinear parabolic equation in domains which are moving in time. We show the local null controllability of the semilinear system using Liusternik's…
The paper deals with the controllability of a degenerate beam equation. In particular, we assume that the left end of the beam is fixed, while a suitable control $f$ acts on the right end of it. As a first step we prove the existence of a…
In this article, we improve the classical Bukhgeim-Klibanov method presented in [1],which can be used to prove the conditional stability of inverse source problem for a hyperbolic equation from the measurement on the subboundary. A major…