A class of second order quasilinear evolution equations
Given a nonnegative selfadjoint operator \(A\) on a Hilbert space \(H\), a nonlinear operator \(f\) from \(D(A^{{1\over 2}})\) to \(H\), and a \(C^ 1\)- function \(M(\cdot)\) satisfying \(M(s)\geq m_ 0> 0\) for all \(s\geq 0\). Using a local Lipschitz continuity assumption on \(f(u)\), the author proves the existence and the uniqueness of a local strong solution of the problem \[ u''(t)+ M(| A^{{1\over 2}} u(t)|^ 2) Au(t)= f(u(t)),\quad t\geq 0,\quad u(0)= u_ 0,\quad u'(0)= u_ 1, \tag{1} \] by applying the theory of nonlinear evolution equations developed by Kato. In the case, where \(A\) is the minus Laplacian with Dirichlet conditions on \(L^ 2(\Omega)\), \(f(u)=\mu u^ 3\) and \(M(s)= \alpha+ 2\beta s\) for \(s\geq 0\) and some constants \(\alpha> 0\), \(\beta\geq 0\) and \(\mu\in \mathbb{R}\), the blowing-up property of solutions to (1) is discussed.
scientific article; zbMATH DE number 696883
scientific article