Abstract
In this paper we investigate on a new strategy combining the logarithmic convexity (or frequency function) and the Carleman commutator to obtain an observation estimate at one time for the heat equation in a bounded domain. We also consider the heat equation with an inverse square potential. Moreover, a spectral inequality for the associated eigenvalue problem is derived.
Highlights
Introduction and main resultsWhen we mention the logarithmic convexity method for the heat equation in a bounded domain Ω ⊂ Rn :∂tu − ∆u = 0 in Ω × (0, T ),u = 0 on ∂Ω × (0, T ),u (·, 0) = u0 ∈ L2(Ω) \{0}, we have in mind that t →ln u (·, t) 2 L2 (Ω)is a convex function by evaluating the sign of the derivative of t →Ω |∇u (x, t)|2 dx Ω |u (x, t)|2 dx
We present an approach to get the observation estimate at one point in time for a model heat equation in a bounded domain Ω ⊂ Rn with Dirichlet boundary condition
We are interested in the following heat equation with an inverse square potential:
Summary
We present the strategy to get the observation at one point by studying the equation solved by f = ueΦ/2 for a larger set of weight functions Φ adapting the energy estimates style of computations in [4] (see [3, Section 4]). We present an approach to get the observation estimate at one point in time for a model heat equation in a bounded domain Ω ⊂ Rn with Dirichlet boundary condition.
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