Uniqueness of nonnegative solutions of the Cauchy problem for parabolic equations on manifolds or domains

• Kazuhiro Ishige ^[1] ; Minoru Murata ^[2]
  1. [1] Nagoya University

     Nagoya University

     Naka-ku, Japón

     
  2. [2] Tokyo Institute of Technology

     Tokyo Institute of Technology

     Japón

     
• Localización: Annali della Scuola Normale Superiore di Pisa. Classe di scienze, ISSN 0391-173X, Vol. 30, Nº 1, 2001, págs. 171-224
• Idioma: inglés
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• Resumen

  • We study uniqueness of nonnegative solutions of the Cauchy problem for parabolic equations on non-compact Riemannian manifolds or domains in We introduce two notions: (1) the parabolic Harnack principle with scale function p concerning inhomogeneity at infinity of manifolds and the second order terms of equations; and (2) the relative boundedness with scale function p concerning growth order at infinity of the lower terms of equations. In terms of this scale function, we give a general and sharp sufficient condition for the uniqueness of nonnegative solutions to hold. We also give a Tacklind type uniqueness theorem for solutions with growth conditions, which plays a crucial role in establishing our Widder type uniqueness theorem for nonnegative solutions. Our Tacklind type uniqueness theorem is of independent interest. It is new even for parabolic equations on in regard to growth rates at infinity of their lower order terms.