Veröffentlichungen und Präsentationen

Introduction In this work, we investigate two nonlinear partial differential equation models with variable exponents and analyze their dynamical behavior using COMSOL Multiphysics. The first model is a coupled Klein–Gordon system exhibiting finite-time blow-up, while the second model is a p(x)-biharmonic equation char- acterized by energy decay. The Klein–Gordon system was implemented using the Wave Equation Interface, whereas the p(x)-biharmonic model was formu- lated through the Coefficient Form PDE Interface. Appropriate finite element discretizations and time-dependent solvers were employed to capture the evo- lution of the solutions. The obtained numerical simulations successfully repro- duced the theoretical predictions, demonstrating the effectiveness and flexibility of COMSOL Multiphysics for the investigation of nonlinear PDEs with variable exponents

Comsol Implementation

To implement Problem 1 in COMSOL Multiphysics, the governing equation was reformulated in accordance with the coefficient structure of the Wave Equation interface. The first- and second-order temporal derivative terms were repre- sented by backward and central difference approximations, respectively. Numer- ical simulations were subsequently carried out using the time-dependent solver module, where both the Generalized-α and BDF schemes were considered. A comparative analysis of the solutions obtained from these two integration meth- ods was performed to evaluate their numerical behavior and solution accuracy. The corresponding energy of the system was defined over the computational domain, and the total energy was obtained by performing the spatial integration of the energy density. In Problem 2, the nonlinear fourth-order equation is reformulated as a coupled system of second-order equations by introducing the auxiliary variable w = ∆u, making it compatible with the COMSOL Coefficient Form PDE interface. The resulting system is solved on a fine triangulated mesh over the unit disk domain with the prescribed initial conditions. Numerical simulations are carried out to analyze the solution dynamics and the corresponding energy decay behavior of the system.

Results

The numerical results obtained from COMSOL Multiphysics are presented for both problems. In Figure 1, the numerical simulation capture the rapid growth of the solution component u of Problem 1, indicating the finite-time blow-up behavior predicted by the theoretical analysis. For Problem 2, the numerical energy evolution is analyzed to examine the dissipative behavior of the system. The obtained results in Figures 2 and 3 demonstrate a decreasing energy profile over time, which is consistent with the theoretical energy decay property of the model.

Conclusion

In this study, the numerical behavior of two nonlinear partial differential equa- tion models with different dynamical properties was investigated using COM- SOL Multiphysics. The computational results obtained using COMSOL Mul- tiphysics demonstrate the capability of the software to successfully capture the characteristic behaviors of the considered nonlinear systems. Remark on the Considered Problems The first problem investigated in this work is derived from a previously pub- lished [1], and the corresponding reference is provided accordingly. The second problem is based on a recent research work that is currently under peer-review evaluation.