Abstract
In this note we prove the strong unique continuation property at the origin for the solutions of the parabolic differential inequality |Δu-ut|≤M|x|2|u|,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} |\\Delta u - u_t| \\le \\frac{M}{|x|^2} |u|, \\end{aligned}$$\\end{document}with the critical inverse square potential. Our main result sharpens a previous one of Vessella concerned with the subcritical case.
Highlights
The unique continuation property for second order elliptic and parabolic equations represents one of the most fundamental aspects of pde’s with a long history and several important ramifications
To the time-independent case in [7,8], our proof of Theorem 1.1 exploits the spectral gap on Sn−1. It relies in an essential way on a delicate a priori estimate which we prove in Lemma 2.2 below, and which we feel is of independent interest
We have not followed the tradition of skipping details, but we have carefully presented them in the proof of Theorem 1.2
Summary
The unique continuation property (ucp) for second order elliptic and parabolic equations represents one of the most fundamental aspects of pde’s with a long history and several important ramifications. We substitute the value of β given by (2.5) in the remaining integrals in the right-hand side of (2.3) obtaining the following conclusions. With such constant C2 in hands, we multiply (2.21) by C2α and add the resulting inequality to (2.20), obtaining α2 4 r −2α−4e2αrε u2 + (C2C0ε − C)α4ε r −2α−4+εe2αrε u2 (2.22)
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