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

作者：Ghisi Marina; Gobbino Massimo^*

来源：Communications in Partial Differential Equations, 2011, 36(8): 1318-1352.

DOI：10.1080/03605302.2010.542672

• Anisotropic diffusion
• Degenerate parabolic equation
• Forward-backward parabolic equation
• Moving domains
• Overdetermined problem
• Perona-Malik equation
• Subsolutions and supersolutions

We consider the Perona-Malik equation
u(t) = div(Delta u/1+vertical bar Delta u vertical bar(2))
in an open set Omega 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 vertical bar Delta u(0)(x)vertical bar - 1 is a sign changing function in Omega. 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.

• 出版日期2011