An example of global classical solution for the Perona-Malik equation

We consider the Cauchy problem for the Perona-Malik equation ut = div ( ∇u 1 + |∇u|2 ) in an open set Ω ⊆ R, with Neumann boundary conditions. It is well known that in the one-dimensional case this problem does not admit any global C solution if the initial condition u0 is transcritical, namely when |∇u0(x)| − 1 is a sign changing function in Ω. 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 with a transcritical initial condition. Mathematics Subject Classification 2000 (MSC2000): 35K55, 35K65, 35B05.