Unboundedness of Solutions for a Class of Biharmonic Equations with Variable Exponents - Narayanan, Lakshmipriya; Soundararajan, Gnanavel (Discontinuity, Nonlinearity, and Complexity, 2026)
This paper investigates a nonlinear biharmonic equation involving variable exponent source terms, which arise naturally in models with non-standard growth conditions. The primary aim is to analyze the qualitative behavior of solutions, with particular focus on the existence and finite-time blow-up phenomena under certain energy constraints. We are especially interested in understanding how the spatial variability of the exponent influences the dynamics of the solution. We first establish the existence of weak solutions within an appropriate functional framework. The main objective, however, is to explore the conditions leading to the finite-time blow-up of solutions when the initial energy is negative. To this end, we demonstrate that such solutions cannot remain globally bounded and, in fact, blow up in finite time. Moreover, we derive sharp upper and lower bounds for the blow-up time, highlighting the dependence of these estimates on the initial data and the structure of the variable exponent. We also provide a precise characterization of the blow-up rate.