Abstract
In this paper, we investigate the unique continuation properties for multi-dimensional heat equations with inverse square potential in a bounded convex domain Ω of mathbb{R}^{d}. We establish observation estimates for solutions of equations. Our result shows that the value of the solutions can be determined uniquely by their value on an open subset ω of Ω at any given positive time L.
Highlights
In this paper, we consider the quantitative unique continuation for multi-dimensional heat equations with a singular potential term
The heat equations studied in this article are described by
The study of unique continuation for the solutions of PDEs began at the beginning of the last century
Summary
We consider the quantitative unique continuation for multi-dimensional heat equations with a singular potential term. In [3], authors proved that if a non-negative initial value φ0 ∈ L2( ) is prescribed, there exists a unique global weak solution for equation (1.1) under assumption (1.2), but as μ > μ∗, the local solution may not exist. For any initial value φ0 ∈ L2( ), there exists a unique solution φ ∈ C([0, T]; L2( )) ∩ L2(0, T; H01( )) for equation (1.1) with (1.2). There exist two positive numbers α = α( , ω), C = C( , ω) such that, for each L > 0, φ(x, L)
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