An Example of Global Classical Solution for the Perona-Malik Equation

We consider the Perona-Malik equation u(t) = div(Delta u/(1+| Delta u |) in an open subset of R(n), with initial and Neumann boundary conditions. It is well known that in the one-dimensional case this problem does not admit any global C(1) solution if the initial condition u(0) is transcritical, namely when |Delta u(0)(x)| - 1 is a sign changing function In this paper we show that this result cannot be extended to higher dimension. We show indeed that for n >= 2 the problem admits radial solutions of class C(2,1) with a transcritical initial condition.