Abstract Cauchy problem. Semigroup, distribution, and regularization methods. (Q5948381)

Let \(u\) map the time interval, \([0,T)\) (where \(T\) is positive and possibly infinite), into a Banach space \(X\). Let \(D^k u\) denote the \(k\)th derivative of \(u\) with respect to time \(t\). Of concern is the differential equation \[ \sum^n_{k=0} A_k D^k u(t)= f(t),\tag{\(*\)} \] where \(A_k\) maps \(X\) into a Banach space \(Y\). If \(A_n\) is not injective, then \((*)\) reduces to a differential inclusion (or a multivalued differential equation). The theories of \(C_0\)-semigroups and cosine functions are special cases. In this survey paper, the author uses three approaches to \((*)\): semigroup methods, distribution theory methods, and regularization methods. Included are nonstrongly well-posed problems (including \(K\)-convolution semigroups). For much additional information, the interested reader should consult \textit{T. J. Xiao} and \textit{L. Jiang} [The Cauchy problem for higher-order abstract differential equations. Lecture Notes in Mathematics. 1701. Berlin: Springer (1998; Zbl 0915.34002)] (to which the author does not refer).