On the optimality of the observability inequalities for parabolic and hyperbolic systems with potentials

Abstract

In this paper we prove the optimality of the observability inequality for parabolic systems with potentials in even space dimensions $n⩾2$. This inequality (derived by E. Fernández-Cara and the third author in the context of the scalar heat equation with potentials in any space dimension) asserts, roughly, that for small time, the total energy of solutions can be estimated from above in terms of the energy localized in a subdomain with an observability constant of the order of $\exp (C\Vert a{\Vert }_{\infty }^{2/3})$, a being the potential involved in the system. The problem of the optimality of the observability inequality remains open for scalar equations.

The optimality is a consequence of a construction due to V.Z. Meshkov of a complex-valued bounded potential $q=q(x)$ in ${\mathbb{R}}^2$ and a nontrivial solution u of $\mathrm{\Delta }u=q(x)u$ with the decay property $|u(x)|⩽\exp (-|x{|}^{4/3})$. Meshkov's construction may be generalized to any even dimension. We give an extension to odd dimensions, which gives a sharp decay rate up to some logarithmic factor and yields a weaker optimality result in odd space-dimensions.

We address the same problem for the wave equation. In this case it is well known that, in space-dimension $n=1$, observability holds with a sharp constant of the order of $\exp (C\Vert a{\Vert }_{\infty }^{1/2})$. For systems in even space dimensions $n⩾2$ we prove that the best constant one can expect is of the order of $\exp (C\Vert a{\Vert }_{\infty }^{2/3})$ for any $T>0$ and any observation domain. Based on Carleman inequalities, we show that the positive counterpart is also true when T is large enough and the observation is made in a neighborhood of the boundary. As in the context of the heat equation, the optimality of this estimate is open for scalar equations.

We address similar questions, for both equations, with potentials involving the first order term. We also discuss issues related with the impact of the growth rates of the nonlinearities on the controllability of semilinear equations. Some other open problems are raised.