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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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