Analytical and computational results for the decay of solutions of a damped wave equation with variable-exponent nonlinearities - Messaoudi, Salim A.; Zahri, Mostafa (Topological Methods in Nonlinear Analysis, 2022)
With the advancement of science and technology, many physical andengineering models require more sophisticated mathematical functional spacesto be studied and well understood. For example, in fluid dynamics,electrorheological fluids (smart fluids) have the property that theviscosity changes (often drastically) when exposed to an electrical field.The Lebesgue and Sobolev spaces with variable exponents proved to beefficient tools to study such problems as well as other models like theimage processing. In this work, we consider the following nonlinear waveequation with variable exponents:\[ u_{tt}-\Delta u-\Delta u_{t}+|u_{t}|^{m(\cdot)-2}u_{t}=0,\quad \text{in }\Omega \times (0,T),\]where $\Omega $ is a bounded domain and $T> 0$, and show that weak solutions decay exponentially or polynomially depending on the range of the variable exponent $m$. We also give two numerical examples to illustrate our theoretical results.