On local existence of solutions of initial boundary value problems for the ``bad'' Boussinesq-type equation
The paper deals with the differential equation \[ u_{tt}-u_{xx}-bu_{xxxx}= \sigma(u)_{xx} \] together with certain appropriate initial and boundary conditions and where \(\sigma(s)\) is a given nonlinear function. In the case \(b < 0\) this equation is called the ``bad'' Boussinesq-equation since in this case even the linear part behaves bad. There are examples of not well-posed problems having this linear part. \textit{H. A. Levine} and \textit{B. D. Sleeman} [J. Math. Anal. Appl. 107, 206-210 (1985; Zbl 0591.35010)] showed the nonexistence of global positive solutions. In this interesting paper the existence of local generalized solutions is shown under rather weak conditions on \(\sigma(u)\) and the initial data, like \(\sigma\) twice continuously differentiable and \( \sigma'' (s) \) is locally Lipschitz continuous. If \(\sigma(s)\) is a concave function then, under certain conditions on the initial data, the solutions have a finite escape time (blow up).
scientific article; zbMATH DE number 1836644