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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.

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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