Remarks on coerciveness in Besov spaces for abstract parabolic equations (Q1332785)

Several papers were devoted to the study of solutions in Besov spaces with values in a Banach space for the abstract linear evolution equation of parabolic type \[ D_ t u(t) + Au(t) = f(t), \quad 0 < t < T, \qquad u(0) = x. \tag{1} \] One of the main results there is that the mapping \(L:u \mapsto (D_ tu + Au, u(0))\) is a bijection from the intersection of two Besov spaces with values in the Banach space and the domain of the operator \(A\) and with exponents \(1 + \theta\) and \(\theta\) respectively to the space of data \((f,x)\) satisfying suitable compatibility relations, the coerciveness in Besov spaces for the equation (1). Let \(E\) and \(F\) be Banach spaces with \(F\) continuously embedded to \(E\) and a linear continuous operator \(A\) from \(F\) to \(E\) be given. Let us consider the evolution equation in \(E\) of the form (1). The aim in this note is to show that the parabolicity of the equation is characterized by the coerciveness in Besov spaces for the equation.