Finite and Infinte Time Blow Up of Solutions to Wave Equations with Combined Logarithmic and Power-Type Nonlinearities

Abstract

In this paper, we investigate the global behavior of the weak solutions to the initial boundary value problem for the nonlinear wave equation in a bounded domain. The nonlinearity includes a logarithmic term and several power-type terms with nonnegative variable coefficients. Two new necessary and sufficient conditions for blow up of the weak solutions are established. The first one addresses the blow up of the global weak solutions at infinity. The second necessary and sufficient condition is obtained in the case of strong superlinearity and concerns blow up of the weak solutions for a finite time. Additionally, we derive new sufficient conditions on the initial data that guarantee blow up for either finite or infinite time. A comparison with previous results is also given.

Keywords

• nonlinear wave equation
• combined logarithmic nonlinearities
• blow up at infinity
• blow up for a finite time