Asymptotical expansions of solutions of linear parabolic equations as \(t\to \infty\) (Q1118101)

The author deals with the following mixed problem: D\({}_ tu+Lu=0\) for \(t>0\), \(x\in \Omega\); \(u(x,0)=f(x)\) for \(x\in \Omega\); \(D^{\alpha}u=0\) on \(\partial \Omega\) for \(t>0\) and \(0\leq | \alpha | \leq m-1.\) Here \(\Omega \subset {\mathbb{R}}^ n\) is an external domain with smooth boundary \(\partial \Omega\), L is an uniformly elliptic differential operator of order 2m, defined on \({\mathbb{R}}^ n\), n is odd, \(\alpha\) is a multi-index. Let \(L_ 0\) be the operator L defined on the set of function \(u\in H_{2m}(\Omega)\), satisfying the above boundary conditions. Suppose \(L_ 0\) is a self-adjoint positive operator in \(L^ 2(\Omega)\) and denote by \(R_ z\) its resolvent. The asymptotic expansion of u as \(t\to \infty\) is determined by the asymptotics of the resolvent \(R_ z\) as \(z\to 0.\) In the present paper, under suitable hypotheses on the data and on the operator L, the author obtains sufficient conditions for the asymptotic expansion of \(R_ z\) in the form \[ R_ z=\sum^{M}_{j=0}F_ j\cdot k^ j+0(k^{M+1}) \] as \(k\to 0\), in the spaces \(B_{2m,s_ 1,s_ 2}\), \(s_ 1,s_ 2>0\), where \(B_{\ell,s_ 1,s_ 2}\) is the normed space of bounded operators A: \(L^ 2_{s_ 1}\to H_{-s_ 1,-s_ 2}\), with \[ L^ 2_ s\equiv \{\phi:\quad \int_{\Omega}(1+| x|)^ s | \phi (x)| dx<\infty \}. \]