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.