On Quenching Phenomena of Two-Dimensional Semilinear Wave Equation

Abstract In this paper, we investigate a two-dimensional abstract semilinear wave equation and its quenching behavior. With the help of Kaplan's first eigenvalue method, we prove that for a sufficiently large domain, the classic solution of this model approaches a finite value, while its associated second-order temporal derivative tends toward infinity. We employ a finite difference method with a uniform step size to analyze the quenching profile of the classical solution and estimate the quenching time. Unconditional stability of the proposed alternating direction implicit finite difference scheme is established. For the purpose of illustrating computational performance, we define two energy functionals and give their corresponding estimate of the classical solution so that the quality of approximation solutions can be evaluated by calculating the energy functional in its discrete sense. Finally, numerical simulations are provided to validate the theoretical analysis. Stability, oscillation, energy-preserving property, and convergence of the numerical scheme are also investigated through various numerical experiments.