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Global solvability of a model for grain boundary motion with constraint. (English) Zbl 1246.35100
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.

MSC:
 35K51 Initial-boundary value problems for second-order parabolic systems 35K55 Nonlinear parabolic equations 35R35 Free boundary problems for PDEs 35K86 Unilateral problems for nonlinear parabolic equations and variational inequalities with nonlinear parabolic operators
Keywords:
singular diffusion; subdifferential
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