On boundary value problems for parabolic equations of higher order in time (Q1906994)

This paper is concerned with the initial-boundary value problem for the parabolic equations: \[ {\mathcal A}(t, x, \partial_t, \partial_x) u= f\quad\text{in } ]0, T]\times \Omega, \] \[ {\mathcal B}_\mu(t, x, \partial_t, \partial_x) u= g_\mu,\;\mu= 1,\dots, m,\text{ on } ]0, T]\times \partial \Omega, \] \[ u(0, x)= u_0(x),\dots, \partial^{\ell- 1}_t u(0, x)= u_{\ell- 1}(x)\text{ in }\Omega. \] Here \({\mathcal A}(t, x, \partial_t, \partial_x)= \sum^\ell_{k= 0} A_{\ell- k}(t, x, \partial_x) \partial^k_t\) is \(d(= 2m/\ell)\)-parabolic in the sense of Petrowskii, \(A_{\ell- k}\) being of order at most \(d(\ell- k)\) and \({\mathcal B}_\mu(t, x, \partial_t, \partial_x)= \sum^{\ell- 1}_{k= 0} {\mathcal B}_{\mu k}(t, x, \partial_x) \partial^k_t\), \({\mathcal B}_{\mu k}\) being of order at most \(\sigma_\mu- dk\). It is assumed that a complementing condition is satisfied in some suitable sense. The important novelty of this paper is that the above problem is solved by the method of analytic semigroups in case where the boundary conditions contain derivatives in the time variable. The problem is transformed to a system of first order in \(t\) by a usual method, and then solved in the space \(W^{(\ell-1)d, p}(\Omega)\times\dots\times W^{d, p}(\Omega)\times L^p(\Omega)\) under compatibility conditions on data using the author's previous result in [Lect. Notes Pure Appl. Math. 135, 227-242 (1991; Zbl 0781.35024)].