Some remarks about Perona–Malik equation

We study the Cauchy problem for the one-dimensional Perona–Malik equation (see (1.1) below) with periodic boundary conditions. It was proved (see [SIAM J. Appl. Math. 57(5) (1997) 1328–1342]) that, in general case, no global weak solution exists, in spite of good numerical results obtained by several authors (see [IEEE Trans. Pattern Anal. Machine Intell. 12 (1990) 629–639, SIAM J. Numer. Anal. 29(3) (1992) 845–866; 29(1) (1992) 182–193]). The present paper deals with the existence of solutions in a weaker sense. The outline of the paper, following some suggestions contained in [Su alcuni problemi instabili legati alla teoria della visione, Manuscript], is based on the study of a discretized form of the problem, involving ordinary differential equations in a suitable Banach space (jump discontinuities of the initial datum are allowed). We prove that the piecewise linear interpolations of the corresponding solutions obey the maximum principle, while their essential total variation is decreasing in time. We furthermore establish an a priori bound for their time derivative and obtain the convergence of a subsequence to an ultraweak solution (see definitions at the end of Section 1).