Nonlinear abstract wave equations with strong damping
The authors consider an initial-value problem for the equation \[ \ddot u- Au- B(u)= 0\tag{1} \] in a general Banach space \(X\). Here \(A\) is a closed linear operator in \(X\) satisfying the Hille-Yosida condition with the exception of the density of the domain \(D(A)\) of \(A\), and \(B\) is a nonlinear operator defined on \(D(A)\). The purpose of this paper is to study equation (1) under weaker conditions than those previously considered in literature. The authors prove the loal existence theorem and give a sufficient condition for the global existence. The authors' abstract results are applied to two examples, namely the quasilinear hyperbolic equation with linear viscosity, \[ \ddot u- \sigma(u_x)u_x- \mu\dot u_{xx}= 0,\quad \mu>0, \] with \(\sigma\in C^2(\mathbb{R})\), and a hyperbolic equation with strong damping: \[ \ddot u+|\sqrt A u|^{2\alpha}Au+ \mu A\dot u= 0,\quad \alpha\geq 1, \] where \(A\) is a nonnegative selfadjoint operator in a Hilbert space.
quasilinear hyperbolic equation with linear viscosity
scientific article; zbMATH DE number 1218636
scientific article