Global existence, blow-up, and lower bound Estimates for a variable exponent parabolic problem with high initial energy

Abstract

In this paper, we study an initial-boundary value problem involving the $p(x)$-Laplacian operator under Robin boundary conditions. By combining the potential well method with the Nehari manifold and the $\omega$-limit set, we prove the global existence and nonexistence of solutions when the initial energy ($J(u_0)$) is greater than the mountain pass level $d$. Additionally, we provide upper and lower bounds for the blow-up time. The present results of this paper complement and improve a recent paper investigated by EL MINSARI and OURRAOUI [São Paulo J. Math. Sci. 19, 32 (2025)].