Regularity in time of the solution of parabolic initial-boundary value problem in \(L^ 1\) space
This paper is concerned with the regularity in t of the solution of the initial-boundary value problem of the linear parabolic partial differential equation \[ (1.1)\quad \partial u(x,t)/\partial t+A(x,t,D)u(x,t)=f(x,t),\quad \Omega \times (0,T], \] \[ (1.2)\quad B_ j(x,t,D)u(x,t)=0,\quad j=1,...,m/2,\quad \partial \Omega \times (0,T], \] \[ (1.3)\quad u(x,0)=u_ 0(x),\quad \Omega. \] Here \(\Omega\) is a not necessarily bounded domain in \(R^ n\) with boundary \(\partial \Omega\) satisfying a certain smoothness hypothesis. For each \(t\in [0,T]\) A(x,t,D) is a strongly elliptic linear differential operator of order m, and \(\{B_ j(x,t,D)\}^{m/2}_{j=1}\) is a normal set of linear differential operators of respective orders \(m_ j<m\). It is assumed that the realization \(-A_ p(t)\) of -A(x,t,D) in \(L^ p(\Omega)\) under the boundary conditions \(B_ j(x,t,D)u|_{\partial \Omega}=0\), \(j=1,...,m/2\), generates an analytic semigroup in \(L^ p(\Omega)\) for any \(p\in (1,\infty)\). A sufficient condition for that, which is also necessary when \(p=2\), is given in \textit{S. Agmon} [Commun. Pure Appl. Math. 15, 119-147 (1962; Zbl 0109.327)]. Assuming moreover that the coefficients of A(x,t,D), \(\{B_ j(x,t,D)\}^{m/2}_{j=1}\) and some of their derivatives in x belong to Gevrey's class \(\{M_ k\}\) [see for example \textit{J. L. Lions} and \textit{E. Magenes}, Ann. Mat. Pura Appl., IV. Ser. 68, 341-417 (1965; Zbl 0138.084) and ibid. 72, 343-394 (1966; Zbl 0173.432)] as functions of t and f also belong to the same class as a function with values in \(L^ 1(\Omega)\), we show that the same is true of the solution of (1.1)-(1.3) considered as an evolution equation in \(L^ 1(\Omega)\) for any initial value \(u_ 0\in L^ 1(\Omega)\). It should be noted here that if \(m_ j=m-1\), the boundary condition \(B_ j(x,t,D)u|_{\partial \Omega}=0\) is satisfied only in a variational sense.
scientific article; zbMATH DE number 4069420