Let \(\Omega \subset \mathbb{R}^d\) be a bounded domain, \(Q = (0,T)\times \Omega\). The following equation is considered: \[ j(v)_t - \text{ div } a(w,\nabla w)=f, \quad w=\varphi(v) \, \text{ in Q }, \quad j(v)|_{t=0} = j_0 \,\text{ on } \Omega \] with Dirichlet boundary conditions \(w|_{(0,T)\times \partial\Omega} = g\) or with the Neumann boundary condition \(a(w,\nabla w)\cdot n|_{(0,T)\times \partial\Omega} = s,\) where \(a(r,\xi) = S(r)\,a_0(\xi) + F(r)\) \( 1/C \leq S(r) \;\leq C \) or \(a(r,\xi) = a_0(\xi) + F(r)\), \( 0 \leq S(r) \;\leq C \), \(a_0: \mathbb{R}^d \to \mathbb{R}^d\), \(F: \mathbb{R} \to \mathbb{R}^d\), \(a_0(\xi) \cdot \xi \geq 1/C |\xi|^p\), \((a_0(\xi) - a_0(\eta)) \cdot (\xi - \eta) \geq 0\), \(|a_0(\xi)|^{p'} \leq C(1 + |\xi|^p)\), \(|F(r)|^{p'} \;\leq C(1 + |r|^p)\) \(p \in (1, +\infty)\), \(1/p + 1/p' = 1\), \(C = \text{const} > 0\), \(j, \, \varphi \) are continuous non-decreasing functions such that \(j(0) = \varphi(0) = 0, \) \(n\) is a unit normal to \(\partial\Omega\).
The authors prove the uniqueness of the weak solutions to the two problems with the help of the Kato inequality.