Global nonexistence without blow-up for an evolution problem
The authors consider the equation \(u_{xx}-\lambda u+f(u)=0\) for \(|x|<1\), \(t>0\), where \(\lambda\geq 0\) and \(f\in C^\infty\). The equation is complemented by dynamic boundary conditions \(u_t\pm u_x=0\) for \(x=\pm 1\), \(t>0\) and initial conditions \(u(x,0)=u_0(x)\) for \(x=\pm 1\). The problem is reformulated as a constrained system of two ODEs and studied using the time-map. The main result is an example for a nonlinearity \(f\) such that the maximal existence interval of a solution is not right open. The authors construct also examples with local nonexistence and nonuniqueness. These examples contradict Theorem~1 by \textit{J.~Escher} [Math. Z. 210, No. 3, 413-439 (1992; Zbl 0759.35025)] and show that Theorem~2 in Escher's paper is close to optimal.
scientific article; zbMATH DE number 1384892