\[u_t = \nabla\cdot\left( g\left(|\nabla u|\right)\nabla u\right),\qquad u(\mathbf{x},0)=f(\mathbf{x}).\]

For the moment the number of spatial dimensions is unspecified. In the equation above \(|\nabla u|\) is the vector 2-norm of the gradient, e.g. in one space dimension \(|u_x|\), or in two dimensions \(\sqrt{u_x^2+u_y^2}\). The function \(g:\mathbf{R}\rightarrow\mathbf{R}\) has not been given, but upon researching applications you will encounter appropriate choices. If you are lost to begin with, take \(g(s) = \frac{1}{1+s^2}\).

1D. In one space dimension the equation takes the form \[u_t = (g(|u_x|)u_x)_x \] As ever, the initial data and boundary conditions are an important part of the problem. As you understand the intended application of this equation, appropriate choices will become clearer. As a pointer, we suggest first solving in the domain \(0\leq x\leq 6\), with boundary conditions \(u(0,t)=u(6,t)=5\) and initial data \(u(x,0)=f(x)\), where \[f(x)=5-\tanh \left(\alpha(x-1)\right) -\tanh \left(\alpha(x-2)\right) +\tanh \left(\alpha(x-4)\right)+\tanh\left(\alpha(x-5)\right) + 0.1\, \sin^2 5\,x \sin 50\,x \] Take, for instance, \(\alpha = 30\), and observe the behaviour as you integrate forward a short distance in time.

Google: Perona-Malik equation, image restoration, diffusion equation, edge enhancement.

Challenges: Non-linearity, application.