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On the Cahn-Hilliard equation with irregular potentials and dynamic boundary conditions. (English) Zbl 1172.35417
Let $$\Omega$$ be a bounded and connected open subset of $${\mathbb R}^d$$, $$1\leq d \leq 3$$, with smooth boundary $$\Gamma$$ and consider the Cahn-Hilliard equation with (possibly) singular potential and dynamic boundary conditions $\begin{cases} \partial_t u = \Delta w , & (t,x)\in (0,\infty)\times\Omega, \\ w = \tau \partial_t u - \Delta u + \beta(u) + \pi(u) - f , & (t,x)\in (0,\infty)\times\Omega,\\ \partial_n w = 0 \;\;\text{ and }\;\; v = u_{|\Gamma}, & (t,x)\in (0,\infty)\times\Gamma, \\ \partial_t v = - \partial_n u + \nu \Delta_{\Gamma} v - \beta_\Gamma(v) - \pi_{\Gamma}(v) + f_\Gamma, & (t,x)\in (0,\infty)\times\Gamma, \end{cases}$ and initial condition $$u(0)=u_0$$ in $$\Omega$$. Here $$f$$, $$f_\Gamma$$ are given source terms, $$\tau$$, $$\nu$$ are non-negative real numbers, $$\pi$$, $$\pi_\Gamma$$ are Lipschitz continuous functions, $$\beta$$, $$\beta_\Gamma$$ are maximal monotone graphs, and $$\partial_n$$ and $$\Delta_\Gamma$$ denote the normal trace and the Laplace-Beltrami operator on the boundary $$\Gamma$$, respectively. Existence of a global solution is established for a general class of maximal monotone graphs $$\beta$$ and $$\beta_\Gamma$$ (including multivalued and singular ones) provided that $$\beta$$ dominates $$\beta_\Gamma$$ on $$D(\beta)$$ and $$\beta_\Gamma$$ is superlinear “at infinity” in a suitable sense. Uniqueness is also proved under weaker assumptions. Previous results were already available for smooth graphs $$\beta$$ and $$\beta_\Gamma$$ and are extended here to a class of non-smooth graphs. The existence proof relies on a compactness method: the existence of solutions is first shown for smooth graphs, including in particular the Yosida regularizations of $$\beta$$ and $$\beta_\Gamma$$. Passing to the limit as the regularization parameter converges to zero is then possible due to the available estimates.

MSC:
 35K50 Systems of parabolic equations, boundary value problems (MSC2000) 35K55 Nonlinear parabolic equations 35K35 Initial-boundary value problems for higher-order parabolic equations 80A22 Stefan problems, phase changes, etc.
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