I encountered the following partial differential equation (PDE) in a mathematical paper

$$\begin{array}{} u_{tt}+\Delta^2u-\nabla\cdot\left(|\nabla u|^{p-2}\nabla u\right)-\Delta u_{t}+\int_{0}^{t}g(t-s)\Delta u(x,s) ds=f(x,u,u_{t}) & \text{in} & \partial \Omega \times (0,T) \\ u=\frac{\partial u}{\partial n}=0 & \text{on} & \partial \Omega \times [0,T) \\ u(x,0)=u_0(x), \qquad u_{t}(x,0)=v_0(x) & \text{in} & \partial \Omega \end{array} $$

where $\Omega \subset \mathbb{R}^n$ is a bounded domain with Lipschitz-continuous boundary $\partial \Omega$. Also, $g \ge 0$ is called memory kernel that decays with a general rate and $f(x,u,u_t)$ is some nonlinear function. $p \ge 2$ is a real constant. The differential operators $\Delta$, $\Delta^2$ and $\nabla$ are the Laplacian, the Biharmonic and gradient operators, respectively.

We are interested in the case $n=3$ which has a physical meaning. Now, let us go into some physical insights.

Plates are initially flat structural members bounded by two parallel planes, called faces, and a cylindrical surface, called an edge or boundary. The generators of the cylindrical surface are perpendicular to the plane faces. The deck of a ship is an example of a plate. This PDE is describing the lateral displacement of a plate made of a homogeneous isotropic nonelinear viscoelastic material.

The function $u(x,t)$ is the lateral displacement of the plate at position $x$ and time $t$. I know that in classical linear elasticity, the following equation

$$u_{tt}+\Delta^2u=f(x) \tag{1}$$

describes the lateral displacement of a plate made of a homogeneous isotropic elastic material where $f(x)$ is an external force applied to the plate and $u_{tt}$ describes the inertia or acceleration term. It is also known as the equation for vibration of plates. I found from this paper that if we have structural damping then the term $\Delta u_t$ shows up in $(1)$. Also, an article in wikipedia revealed that the integral term $\int_{0}^{t}g(t-s)\Delta u(x,s) ds$ can show up in $(1)$ when viscoelasticity comes in. What remains unknown is

$$\Delta_p u \equiv \nabla\cdot\left(|\nabla u|^{p-2}\nabla u\right)$$

which is called the p-Laplacian operator. For $p=2$, it is the usual Laplacian operator $\Delta$. I really cannot find any background of this operator.


Can you please shed some light on the physical, mathematical or historical background of the p-Laplacian term? Where does it come from?


This question is answered on MathOverFlow. Interested readers can check it out.


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