In this paper, we examine the nonlinear inhomogeneous parabolic equation, considering it as \(\begin{aligned} \left\{ \begin{array}{ll} v^{\prime }-\Delta _{1}v+\vert v\vert ^{s-1}v=f& \hbox {in }\Omega \times (0,T), \\ v=0 & \hbox {on }\partial \Omega \times (0,T), \\ v(x,0)=v_{0}(x) & \hbox {in }\Omega , \end{array} \right. \end{aligned}\) where \(\Delta _{1}v:= \operatorname {div}\left( \frac{D v}{|D v|}\right) \) is the 1-Laplacian operator, the set \(\Omega \subset \mathbb {R}^{N}\) is an open bounded domain with a boundary \( \partial \Omega \) satisfying the Lipschitz condition, \( T>0,\) \( s\ge 1,\) \( v_{0}\in L^{2}(\Omega )\) and f is a member of the Lebesgue space. We demonstrate the existence and uniqueness of a solution to the problem.
