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 the initial and boundary value problem associated with $(*)$. Some applications are given when $A(x)$ are explicit linear uniformly elliptic differential operators.
Abstract parabolic equations with applications to problems in cylindrical space domains / M. Di Cristo, D. Guidetti, A. Lorenzi. - In: ADVANCES IN DIFFERENTIAL EQUATIONS. - ISSN 1079-9389. - 15:1-2(2010), pp. 1-42.
Abstract parabolic equations with applications to problems in cylindrical space domains
A. LorenziUltimo
2010
Abstract
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 the initial and boundary value problem associated with $(*)$. Some applications are given when $A(x)$ are explicit linear uniformly elliptic differential operators.
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