Blow-up phenomena for a damped wave equation with logarithmic source term and variable-exponents

This paper investigates the wave equation with variable-exponent nonlinearity and logarithmic source term, given by the following: utt ? ?u + aut|ut|m(?)?2 = bu|u|p(?)?2 ln |u|, where a and b are positive constants, and the functions m(?) and p(?) satisfy certain required conditions. Using the energy method and several inequality techniques, we establish a finite-time global nonexistence result for specific solutions with positive initial energy, under appropriate conditions. This type of equation has significant applications in various fields, including fluid dy- namics, electrorheological fluids, quantum mechanics, nuclear physics, optics, and geophysics.