Global Conference on Mathematics and Applications (GCMA)

This paper studies a nonlinear wave equation with a variable-exponent logarithmic source term, distributed delay, and fractional damping in the internal feedback: $u_{tt}-\Delta u+a_1u_t+\int_{t_1}^{t_2}a_2(\kappa)\partial_t^{\alpha,\eta}u(x,t-\kappa)\,\mathrm{d}\kappa=|u|^{q(x)-2}u\ln|u|$, for $(x,t)\in\Omega\times(0,\infty)$, where $\Omega\subset\mathbb{R}^n$ is a bounded domain, $2\le q_1\le q(x)\le q_2<\infty$, and $q(\cdot)$ satisfies the log-Hölder continuity condition. Under a suitable positivity condition on the delay coefficient and negative initial energy $E(0)<0$, we prove that the weak solution blows up in finite time. The proof combines the perturbed energy method with the concavity method, adapted to variable-exponent Lebesgue and Sobolev spaces. More precisely, we derive the differential inequality $\mathcal{R}'(t)\ge \widehat{\gamma}\,\mathcal{R}^{1/(1-\alpha)}(t)$, which yields the upper bound for the blow-up time $T\le T^{*}=(1-\alpha)/[\widehat{\gamma}\alpha\,\mathcal{R}^{\alpha/(1-\alpha)}(0)]$. This result extends recent works on logarithmic wave equations by incorporating variable exponents in the source term while maintaining the distributed delay and fractional damping structure.