The phase space of a higher order Sobolev type equation (Q447720)

Let \(\mathcal U\) and \(\mathcal F\) be Banach spaces, the operators \(A, B_{n-1},\dotsc, B_0\) are from \(\mathcal L(\mathcal U,\mathcal F)\). Consider the Cauchy problem for a higher-order (Sobolev type) differential-operator equation NEWLINE\[NEWLINE\begin{aligned} &Av^{(n)}(t)=B_{n-1}v^{(n-1)}(t)+\dotsb+B_0v(t)+f(t),\tag{1}\\ &v(0)=v_0, v'(0)=v_1, \dotsc, v^{(n-1)}(0)=v_{n-1},\tag{2} \end{aligned}NEWLINE\]NEWLINE where the operator \(A\) is not invertible, \(\ker A\neq\{0\}\). It is also considered the corresponding homogeneous equation NEWLINE\[NEWLINEAv^{(n)}(t)=B_{n-1}v^{(n-1)}(t)+\dotsb+B_0v(t).\tag{3}NEWLINE\]NEWLINE The set \(\mathcal P\subset\mathcal U\) is called a phase space of the homogeneous equation (3) if, for any of its \(C^n(\mathbb R,\mathcal U)\) solutions \(v\), one has \(v(t)\in\mathcal P\) \(\forall t\in \mathbb R\), and, for any \(v_k\in\mathcal P\), \(k=0,\dotsb,n-1\), there exists a unique solution of problem (3), (2).NEWLINENEWLINEFirst, the author studies the phase space of (3), and then proves a unique solvability of the non-homogeneous problem (1), (2).