Well‐Posedness and Blow‐Up of Solutions to a p(x)‐Laplacian Plate Equation With Variable‐Exponent Nonlocal Damping and Logarithmic Source

This paper investigates a class of nonlinear plate equations with the ‐Laplacian operator and a nonlocal damping term in the presence of a variable‐exponent logarithmic source. Using the Faedo‐Galerkin approximation, we establish the existence of local weak solutions. Subsequently, after demonstrating that these local solutions extend globally in time, we prove their asymptotic stability. Finally, employing a contradiction method, we show that there exists a finite time at which the solutions blow up, provided the initial data satisfy appropriate conditions.