Premium
Estimates for Fundamental Solutions of Second‐Order Parabolic Equations
Author(s) -
Liskevich Vitali,
Semenov Yuli
Publication year - 2000
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/s0024610700001332
Subject(s) - order (exchange) , parabolic partial differential equation , mathematics , mathematical analysis , partial differential equation , economics , finance
In this paper we study the second‐order parabolic equation∂ t u ( t , x ) = ∇ ċ a ( t , x ) ċ ∇ u ( t , x ) ‐ b ( t , x ) ċ ∇ u ( t , x ) + ∇ b ∩ ( t , x ) u ( t , x ) u ( t , x ) + V ( t , x ) ( 1 )in a domain [0, T ]×R d ⊂ R d +1 , where a =( a i j)i , j = 1 dis matrix of bounded measurable coefficients, b =( b j )j = 1 d , andb ^ =(b ^ j )j = 1 dare measurable (in general, singular) vector fields, V is a measurable potential, T is a fixed positive number, and ∂ t u = ∂ u /∂ t , and we employ the notation ∇ ċ a ċ ∇ u = ∑ i , j = 1 d∂ x ia i j∂ x ju , b ċ ∇ u = ∑ j = 1 db j ∂ x ju , ∇ b ∩ u = ∑ j = 1 d∂ x j( b ∩ j u ) .We introduce a new class of coefficients in the lower‐order terms for which we prove the existence and the uniqueness of the weak fundamental solution, and for this we derive Gaussian upper and lower bounds.