Abstract

This paper deals with mixed boundary value problem of the form \[ \begin{gathered} u_t=(-1)^{m-1}L(x,t,D)u,\tag{1} \ u(0,x)=\varphi(x),\quad \partial^ju/\partial \nu^j\big|_{\partial \Omega}=0,\quad j=1,\ldots,m-1.\tag{2} \end{gathered} \] Here \(\Omega \) is an open domain in \(\mathbb R^n\), \(L(x,t,D)\) is a higher order elliptic operator with the main term \(\sum_{|p|,|q|=m}D^pa_{pq}(x,t)D^q\), each \(a_{pq}(x,t)\) being \((s\times s)\)-matrix-valued function. A generalized solution of the problem (1)-(2) is sought in the space \(H^{m,1}(\Omega \times[0,\infty) \mapsto \mathbb R^s)\) consisting of \(\mathbb R^s\)-valued functions which have all generalized derivatives in \(x\)-variables up to the \(m\)-th order inclusively and the first order generalized derivative in \(t\). The authors establish coefficient conditions under which such a solution exists, has the uniqueness property and, moreover, as the function of variable \(x\) satisfies the estimate \(\|u(t)\|_{H^m(\Omega)}\leq Ce^{\lambda t}\|\varphi \|_{H^m(\Omega)}\) for any \(t>0\) where \(C,\lambda \) are positive constants. Next, the authors investigate (1)-(2) as a Cauchy problem for the evolution equation \(\dot u=A(t)u\) in the Hilbert space \(H^m(\Omega)\) and use the obtained results to establish exponential stability of a semilinear evolution equation \(\dot u=A(t)u+f(t,u)\).