Let \(\Omega\) be a bounded domain in \(\mathbb{R}^N \), \(Q_T:=\Omega\times(0,T)\), \(\Sigma_T:=\partial\Omega\times(0,T)\). There is considered the following nonlinear problem with unknown functions \(\eta(x,t)\) and \(\theta(x,t)\): \[ \begin{aligned} \eta_t-\kappa \Delta \eta + g(\eta)+\alpha'(\eta)|\nabla \theta|&= 0 \text{ a.e. \;in}\;Q_T; \\ \alpha_0(\eta)\theta_t-\nu \Delta \theta - \text{div}\big( \alpha(\eta)\frac{\nabla\theta}{|\nabla \theta|}\big)+\partial I_{[-\theta^*,\,\theta^*]}(\theta) &\ni 0 \;\text{a.e. \;in} \;Q_T; \\ \partial \eta /\partial n=0, \;\theta&=0 \;\text{a.e. \;on} \;\Sigma_T; \\ \eta(x,0)=\eta_0(x),\;\theta(x,0)&=\theta_0(x) \;\text{\;a.e. \;in } \;\Omega,\end{aligned} \] where \(\kappa >0\) and \(\nu >0 \) are the small constants, \(\partial I_{[-\theta^*,\,\theta^*]}(\cdot)\) – subdifferential of the indicator function \(I_{[-\theta^*,\,\theta^*]}(\cdot)\) on \([-\theta^*,\,\theta^*]\).
The authors prove that this problem has at least one solution \(\eta, \theta\) such that
1. \(\eta \in C([0,T];\,H)\cap W^{1,2}_{loc}((0,T];\,H) \cap L^\infty_{loc} ((0,T];\,H^1) \cap L^2_{loc}((0,T];\,H^2)\), \(\theta \in C([0,T];\,H)\cap W^{1,2}_{loc}((0,T];\,H) \cap L^\infty_{loc} ((0,T];\,H^1_0)\) and \(0 \leq \eta \leq 1\), \(|\theta|\leq \theta^*\) a.e. in \(Q_T\), if \(\eta_0, \theta_0 \in H\) and \(0\leq \eta_0 \leq 1\), \(|\theta_0|\leq \theta^*\) a.e. in \(\Omega\);
2. \(\eta \in W^{1,2}(0,T;\,H) \cap L^\infty(0,T;\,H^1) \cap L^2(0,T;\,H^2)\), \(\theta \in W^{1,2}(0,T;\,H) \cap L^\infty(0,T;\,H^1_0)\) and \(0 \leq \eta \leq 1\), \(|\theta|\leq \theta^*\) a.e. in \(Q_T\), if \(\eta_0 \in H^1, \;\theta_0 \in H^1_0\) and \(0\leq \eta_0 \leq 1\), \(|\theta_0|\leq \theta^*\) a.e. in \(\Omega\),
where \(H:=L^2(\Omega)\); \(H^1:=H^1(\Omega)\), \(H^1_0:=H^1_0(\Omega)\), \(H^2:=H^2(\Omega)\) are the Sobolev spaces.
Corresponding estimates of the solution are also obtained.