Advances in Differential Equations

Abstract parabolic equations with applications to problems in cylindrical space domains

Michele Di Cristo, Davide Guidetti, and Alfredo Lorenzi

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In a Banach space $X$ we consider the partial differential equation $$ (*)\quad D_{t}u(t,x)+(-1)^ma(x)D_{x}^{2m}u(t,x)-A(x)u(t,x)=f(t,x) $$ where $m$ is a positive integer, related to the rectangle $(0,T)\times(0,L)$ and the family of closed linear operators $\{A(x)\}_{x\in[0,L]}$. Under suitable assumptions we uniquely solve certain initial and boundary-value problems associated with $(*)$. Some applications are given when,for each $x,$ $A(x)$ is an explicit linear uniformly elliptic differential operator.

Article information

Adv. Differential Equations, Volume 15, Number 1/2 (2010), 1-42.

First available in Project Euclid: 18 December 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 34G10: Linear equations [See also 47D06, 47D09] 47D06: One-parameter semigroups and linear evolution equations [See also 34G10, 34K30] 35K35: Initial-boundary value problems for higher-order parabolic equations 35K70: Ultraparabolic equations, pseudoparabolic equations, etc.


Di Cristo, Michele; Guidetti, Davide; Lorenzi, Alfredo. Abstract parabolic equations with applications to problems in cylindrical space domains. Adv. Differential Equations 15 (2010), no. 1/2, 1--42.

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