Quantitative unique continuation for the semilinear heat equation in a convex domain

Abstract

In this paper, we study certain unique continuation properties for solutions of the semilinear heat equation ∂ t u − u = g(u), with the homogeneous Dirichlet boundary condition, over Ω × (0, T * ). Ω is a bounded, convex open subset of R d , with a smooth boundary for the subset. The function g : R → R satisfies certain conditions. We establish some observation estimates for (u − v), where u and v are two solutions to the above-mentioned equation. The observation is made over ω × {T }, where ω is any non-empty open subset of Ω, and T is a positive number such that both u and v exist on the interval [0, T ]. At least two results can be derived from these estimates: (i) if (u − v)(·, T ) L 2 (ω) = δ, then (u − v)(·, T ) L 2 (Ω) Cδ α where constants C > 0 and α ∈ (0, 1) can be independent of u and v in certain cases; (ii) if two solutions of the above equation hold the same value over ω × {T }, then they coincide over Ω × [0, T m ). T m indicates the maximum number such that these two solutions exist on [0, T m ).

Carleman estimate for stochastic parabolic equations and inverse stochastic parabolic problems

“…In deterministic setting, a result which is stronger than Theorem 1.2 was obtained in [26], where the authors study the following equation:…”

Quantitative unique continuation for the heat equations with inverse square potential

“…In [10], the authors established the unique continuation for parabolic equations with time independent coefficients by the properties of eigenfunctions of the corresponding elliptic operator, and this approach cannot be applied to parabolic equations with time dependent coefficients. From 1980s, there have been more results of unique continuation for parabolic equations, and we refer the readers to [5, 8, 9, 11, 14–16] and rich references cited therein. In our paper, we mainly study this property for the heat equations with the inverse square potential.…”

An Observability Inequality On A Kind of Linear Parabolic Equation

“…Observability can derive both exact controllability and approximate controllability directly or with some relevant conditions.Especially, in time optimal control problems, we often obtain null-controllability through observability estimate, to show the existence and bang-bang property of time optimal control problems [2][3][4][5].To the best of our knowledge, the observability estimates of parabolic equation system are main about the heat equation through Carleman inequality [6][7][8], or about semilinear parabolic equation with a discontinuous coefficient in one-dimension,or the case of coefficient with bounded variations of one-dimension,which proof relies on global Carleman estimate.But in the n-dimension case( 2 n ≥ ), the observability estimate of linear parabolic equation which coefficient has bounded variations is seldom talked about because the observability estimate dosen't completely rely on the corresponding Carleman inequality at all.…”