Local Solvability and Blow-Up of Classical Solution to Some Initial-Boundary Value Problem for a Nonlinear Equation Governing Ion-Acoustic Waves in a Plasma

An initial-boundary value problem for a Sobolev-type equation from the theory of ion-acoustic plasma waves is considered. The problem is reduced to an equivalent abstract integral equation. The local solvability of the equation is proved using the contraction mapping principle. Next, the bootstrap method is employed to improve the smoothness of the solution. Finally, the test function method with a certain sufficient condition is used to obtain a finite time blow-up result and an upper bound on the blow-up time is found.