Linear parabolic evolution equations in \(L^ p\)-spaces
The author studies regularity theory for the parabolic initial-boundary value problem \(u_ t(t,x)=A(x;D)u(t,x)+f(t,x),\) \(0<t\leq T\), \(x\in \Omega\), \(B_ i(x;D) u(t,x)=0\), \(1\leq i\leq m\), \(0<t\leq T\), \(x\in \partial \Omega\), \(u(0,x)=u_ 0(x)\). The inhomogeneous term f and the initial data are presumed to lie in \(L^ p\). The problem is studied abstractly as the Cauchy problem \(u'(t)=Au(t)+f(t),\) \(0<t\leq T\), \(u(0)=u_ 0\), where A is the infinitesimal generator of an analytic semigroup in a general Banach space E and \(f\in L^ p(0,T;E)\). A direct approach based upon the theory of interpolation spaces between D(A) and E is used to describe how the regularity of solutions depends on f and \(u_ 0\).
scientific article; zbMATH DE number 3906980
scientific article