Strongly damped semilinear equations
If \(\Omega\subset \mathbb{R}^n\) is a bounded domain with sufficiently smooth boundary \(\partial\Omega\) and if the differential operator \(Lu= \sum_{|\alpha|\leq 2m} a_\alpha(x) D^\alpha u\) is strongly elliptic of order \(2m\) in \(\Omega\), the strongly damped wave equation \[ u_{tt}+ (aL+ b) u_t+ (cL+ d)u= f(t, u,u_t)\quad \text{in}\quad \Omega\times (t_0, T), \] \[ u(x, t_0)= u_0\quad\text{and} \quad u_t(x, t_0)= u_1,\;x\in \Omega, \] \[ D^\alpha u= 0\quad\text{for} \quad (x, t)\in \partial \Omega\times [t_0, T),\;|\alpha|\leq m- 1 \] is considered as a special case of the abstract second-order semilinear differential equation in a Banach space \(X\), \[ u''(t)+ (aA+ bI) u'(t)+ (cA+ dI)u= f(t, u(t), u'(t)),\;t> t_0, \] \[ u(t_0)= u_0,\;u'(t_0)= u_1 \] with \(A= L\) and assuming that \(-A\) generates an analytic semigroup \(T(t)\) in \(X\). Sufficient conditions on \(T\), \(A\) and \(f\) are given to assure the existence and uniqueness of local and global classical solutions.
abstract second-order semilinear differential equation
scientific article; zbMATH DE number 847757