Differential operators and \(C\)-wellposedness of complete second order abstract Cauchy problems
The authors consider the complete second-order Cauchy problem \[ u''(t)+ Bu'(t)+ Au(t)= 0,\quad t\geq 0,\quad u(0)= u_0,\quad u'(0)= u_1,\tag{\(*\)} \] where \(A\), \(B\) are closed linear operators in a Banach space \(E\). A solution to the Cauchy problem \((*)\) is (by definition) a map \(u(\cdot)\in C^2([0,\infty), E)\) such that \(u(t)\in D(A)\), \(u'(t)\in D(B)\) for each \(t\geq 0\) and \(Au(\cdot)\), \(Bu'(\cdot)\) are continuous, satisfying \((*)\). The Cauchy problem \((*)\) is called \(C\)-wellposed, if \((*)\) has a solution for any \(u_0\in C(D(A))\), \(u_1\in C(D(A)\cap D(B))\) and there exists a nondecreasing positive function \(M(t)\) defined on \([0,\infty)\) such that \(\|u(t)\|\leq M(t)(\|C^{-1}u_0\|+\|C^{-1}u_1\|)\), \(t\geq 0\) for any solution \(u(\cdot)\) to \((*)\) with \(u_0,u_1\in R(C)\), where \(C\) is a bounded injective operator on \(E\) such that \(A= C^{-1}AC\), \(B= C^{-1}BC\). The pair \((S_0(t),S_1(t))_{t\geq 0}\) of strongly-continuous families of bounded operators on \(E\) is called a strong \(C\)-propagation family for \((*)\), if some five conditions (too long for reproducing here) hold. The Cauchy problem \((*)\) is called strongly \(C\)-wellposed if there exists a strong \(C\)-propagation family for \((*)\). Definitions of analytically wellposed and solvable Cauchy problems \((*)\) in \(\Sigma_\theta\) \((0\leq\theta\leq{\pi\over 2})\) is also given, where \(\Sigma_\theta= \{z\in\mathbb{C}: z\neq 0, |\text{arg }z|< \theta\}\). The authors give necessary and sufficient conditions for the \(C\)-wellposedness and analytic wellposedness of the Cauchy problem \((*)\). An analytic solvability and a perturbation theorem are given as well. Next, the authors consider cases when \(E\) is one of the spaces \(L^p(\mathbb{R}^n)\) \((1\leq p\leq\infty)\), \(C_0(\mathbb{R}^n)\), \(C_6(\mathbb{R}^n)\), \(UC_6(\mathbb{R}^n)\) and \(A\), \(B\) are certain constant coefficient differential operators. A series of interesting theorems are then obtained. Finally, two examples are presented, one of them being the damped Klein-Gordon equation in one dimension.
scientific article; zbMATH DE number 1344266