Global existence and finite-time blow-up for a strongly damped wave--MGT system with fully subcritical logarithmic nonlinearity

RESUMO

We study a coupled strongly damped wave--Moore--Gibson--Thompson (MGT) system with logarithmic source $f(u)=|u|^{\gamma-2}u\ln|u|$ in the full Sobolev-subcritical range $2<\gamma<2^*:=2n/(n-2)$. The main difficulty is that, in this full range, the logarithmic nonlinearity does not admit the standard compactness and difference estimates available in the lower subcritical regime. Using the augmented variable $w=v+\tau v_t$, we derive the exact energy-dissipation identity for the strongly damped system and exploit the associated coupled potential-well structure. For the local theory, a Faedo--Galerkin scheme combined with a closed nonlinear differential inequality and a spatial domain-splitting argument yields local existence. The strong damping provides the additional regularity needed to prove uniqueness throughout the full subcritical range and to establish a continuation principle. Below the well depth $d_\alpha$, we then obtain a sharp dynamical dichotomy: solutions with initial data in the stable set exist globally and decay exponentially, whereas solutions with initial data in the unstable set blow up in finite time. The blow-up argument reveals a structural cancellation specific to the augmented formulation: once the logarithmic contribution is reconstructed through the exact energy identity, the unfavorable MGT residual in Levine's concavity functional is absorbed algebraically. This allows the concavity method to close without imposing any additional sign condition on the initial velocities.