Complete second order differential equations in Banach spaces with dynamic boundary conditions
The authors consider the following second-order problem with dynamic boundary conditions in a Banach space \[ \begin{gathered} u''(t)+ Au(t)+ Bu'(t)= 0,\quad t\geq 0,\\ x''(t)+ A_1x(t)+ B_1x'(t)= G_0u(t)+ G_1u'(t),\quad t\geq 0,\\ x(t)= Pu(t),\quad t\geq 0,\\ u(0)= u_0,\;x(0)= x_0, u'(0)= u_1,\;x'(0)= x_1.\end{gathered} \] \(A\), \(B\), \(A_1\), \(B_1\), \(G_0\), \(G_1\) and \(P\) are linear operators. The second equation listed above acts as the dynamic boundary condition for \(u\). The authors overview a number of results for first-order problems with dynamic boundary conditions, but note that there has been less work on the second-order case. They also mention that while this problem can be converted into a first-order problem, this necessitates adding further assumptions. They deal with the second-order problem above by converting it into an associated abstract second-order Cauchy problem with operator matrices as coefficients. In Section 2, after recalling the definition of strong wellposedness for such problems, they introduce the notion of strong quasi-wellposedness. This definition is motivated by the fact that it is easier to verify it and is more general than strong wellposedness. They also specify conditions under which the two definitions are equivalent. In Section 3, they prove Theorem 3.2, which specifies necessary and sufficient conditions for the strong wellposedness of the associated Cauchy problem and Theorem 3.5, which gives sufficient conditions for strong quasi-wellposedness. Section 4 considers the inhomogeneous case. In Section 5, the authors give a number of initial boundary value problems involving partial differential equations to which these results can be applied. A lot of detail is given in these examples, which is of great convenience to the reader.
Second-order differential equations in Banach spaces
scientific article; zbMATH DE number 2104507
scientific article