A class of local classical solutions for the one-dimensional Perona-Malik equation

We consider the Cauchy problem for the one-dimensional Perona-Malik equation u t = 1 ― u 2 x/ (1 + u 2 x ) 2 u xx in the interval [―1,1], with homogeneous Neumann boundary conditions. We prove that the set of initial data for which this equation has a local-in-time classical solution u: [―1,1] x [0,T] → ℝ is dense in C 1 ([―1, 1]). Here "classical solution" means that u, u t , u x and u xx are continuous functions in [―1,1]×[0,T].

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