Existence and nonexistence of global solutions for logarithmic hyperbolic equation

Abstract

This article is concerned with the initial-boundary value problem for a equation of quasi-hyperbolic type with logarithmic nonlinearity. By applying the Galerkin method and logarithmic Sobolev inequality, we prove the existence of global weak solutions for this problem. In addition, by means of the concavity analysis, we discuss the nonexistence of global solutions in the unstable set and give the lifespan estimation of solutions.

Keywords

• logarithmic hyperbolic equation
• stable and unstable sets
• existence and nonexistence
• global solutions
• initial-boundary value problem