Abstract:

In this article, the authors consider equation u_t={\rm div}(\varphi (\Gamma u ) A(|D u|²)Du)-(u-I), where $\varphi$ is strictly positive and $\Gamma$ is a known vector-valued mapping, $A: {\Bbb R}_{+}\rightarrow {\Bbb R}^{+}$ is decreasing and $A(s)\sim 1/\sqrt{s}$ as $s\rightarrow +\infty$. This kind
of equation arises naturally from image denoising. For an initial datum $I \in {\rm BV}_{\rm loc}\cap L^{\infty},$ the existence of BV solutions to the initial value problem of the equation is obtained.

Key words: BV_loc function, BV_x function, strongly degenerate parabolic, denoising