Strongly damped wave equations of the p-Laplacian type
Damped wave equations have a rich history in PDEs and often occur in models of materials which exhibit both viscous and elastic properties. In this talk we shall investigate the challenging question of existence of solutions to strongly damped wave equations propagated by the p-Laplacian, a generalization of the Laplacian which is nonlinear when p ≠ 2.
We shall employ a constructive scheme known as the Galerkin method to create successive approximations to the equation
utt − ∆pu − ∆ut = 0
in a bounded domain Ω in ℝ3. This PDE is equipped with a nonlinear Robin- type boundary condition containing a source term which contributes to the energy of the system. Along the way we will encounter and discuss several useful tools from functional analysis and monotone operator theory which have parallels in the familiar real number system, and we shall then use these tools to rigorously pass to the limit in our approximations and demonstrate local existence of a solution.
Finally, we describe what conditions are sufficient to ensure that global solutions exist and survey some challenging open questions connected to this work.