Related papers: Global null-controllability and nonnegative-contro…

We consider a $4\times4$ nonlinear reaction-diffusion system posed on a smooth domain $\Omega$ of $\mathbb{R}^N$ ($N \geq 1$) with controls localized in some arbitrary nonempty open subset $\omega$ of the domain $\Omega$. This system is a…

In this paper, we consider the semilinear heat equations under Dirichlet boundary condition \[ u_{t}\left(x,t\right)=\Delta u\left(x,t\right)+f(u(x,t)), & \left(x,t\right)\in \Omega\times\left(0,+\infty\right), u\left(x,t\right)=0, &…

We give a boundary observability result for a $1$d wave equation with a potential. We then deduce with a Schauder fixed-point argument the existence of a Neumann boundary control for a semi-linear wave equation $\partial_{tt}y -…

Given a control region $\Omega$ on a compact Riemannian manifold $M$, we consider the heat equation with a source term $g$ localized in $Omega$. It is known that any initial data in $L^2(M)$ can be stirred to 0 in an arbitrarily small time…

We study the null-controllability of parabolic equations associated to non-autonomous Ornstein-Uhlenbeck operators. When a Kalman type condition holds for some positive time $T>0$, these parabolic equations are shown to enjoy a Gevrey…

This paper is addressed to a study of the null controllability for the semilinear parabolic equation with a complex principal part. For this purpose, we establish a key weighted identity for partial differential operators…

This paper establishes the local exact controllability of the quasilinear porous media equation with Dirichlet boundary condition.\\ Consider the equation $$\begin{aligned} &y_t - \Delta a(y) = mu+f \text{ on } Q\\ &y(0)=y_0,\ y…

In this work, we are interested in the small time global null controllability for the viscous Burgers' equation y_t - y_xx + y y_x = u(t) on the line segment [0,1]. The second-hand side is a scalar control playing a role similar to that of…

In this article we prove global propagation of analyticity in finite time for solutions of semilinear Schr\"odinger equations with analytic nonlinearity from a region $\omega$ where the Geometric Control Condition holds. Our approach…

We consider heat operators on a bounded domain $\Omega \subseteq \mathbb{R}^n$, with a critically singular potential diverging as the inverse square of the distance to $\partial \Omega$. While null boundary controllability for such…

This paper aims to answer an open problem posed by Morancey in 2015 concerning the null controllability of the heat equation on (-1, 1) with an internal inverse square potential located at x = 0. For the range of singularity under study,…

We consider in this paper a perturbation of the standard semilinear heat equation by a term involving the space derivative and a non-local term. We prove the existence of a blow-up solution, and give its blow-up profile. Our proof relies on…

We characterize the asymptotic behavior near blowup points for positive solutions of the semilinear heat equation \begin{equation*} \partial_t u-\Delta u =f(u), \end{equation*} for nonlinearities which are genuinely non scale invariant,…

We study the internal non null-controllability properties of the heat equation on 2-dimensional almost-Riemannian manifolds with an interior singularity, and under the assumption that the closure of the control zone does not contain the…

We consider the semilinear heat equation with Sobolev subcritical power nonlinearity in dimension $N=2$, and $u(x,t)$ a solution which blows up in finite time $T$. Given a non isolated blow-up point $a$, we assume that the Taylor expansion…

We consider the semilinear heat equation, to which we add a nonlinear gradient term, with a critical power. We construct a solution which blows up in finite time. We also give a sharp description of its blow-up profile. The proof relies on…

In this paper, we derive a bang-bang property of a kind of time optimal control problem for some semilinear heat equation on bounded $C^2$ domains (of the Euclidean space), with homogeneous Dirichlet boundary condition and controls…

We study the null controllability of three parabolic equations. The control is acting only on one of the three equations. The three equations are coupled by means of two cubic nonlinearities. The linearized control system around 0 is not…

In this article, we prove null-controllability results for the heat equation associated tofractional Baouendi-Grushin operators $$\partial_t u+\bigl(-\Delta_x-V(x)\Delta_y\bigr)^s u= \mathbb{1}_\Omega h$$ where $V$ is a potential that…

Let $\Omega$ be a bounded smooth domain in $\RR^N$. We consider the problem $u_t= \Delta u + V(x) u^p$ in $\Omega \times [0,T)$, with Dirichlet boundary conditions $u=0$ on $\partial \Omega \times [0,T)$ and initial datum $u(x,0)= M \phi…