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Parabolic problems with mixed variable lateral conditions: an abstract approach. (English) Zbl 0890.35061

The author presents a new approach to initial-boundary value problems for linear parabolic differential equations of second order with mixed and time-dependent boundary conditions. In detail, the following boundary conditions are considered: If \(\Omega\) is a uniformly \(C^{1,1}\) open set of \(\mathbb{R}^N\) with boundary \(\Gamma\) and \(\Gamma^t_0\) is a uniform family of \(C^{1,1}\) submanifolds of \(\Gamma\) \((t\in [0,T])\), then \(u(x,t)= g_0(x, t)\) on \(\Sigma_0:= \bigcup_{t\in(0, T)}\Gamma^t_0\times \{t\}\) and \((\partial u/\partial\nu_A)(x,t)= g_1(x, t)\) on \((\Gamma\times (0,T))\backslash\overline\Sigma_0\). Here \(\nu_A\) is the conormal vector related to the elliptic operator \(A\) in \(\partial_tu+ Au=f\).
The author is interested in sufficiently wide conditions on the family \((\Gamma^t_0)_{t\in[0, T]}\) in order to obtain regularity of the type: If \(f\in L^2(\Omega\times (0,T))\) and \(u(x,0)\), \(g_0\), \(g_1\) are in suitable trace spaces, then \(\partial_tu\), \(Au\in L^2(\Omega\times (0,T))\), \(|\nabla u(.,t)|_{L^2(\Omega)}\in L^\infty(0, T)\). He proves a corresponding theorem if \(\sup_{x\in\Gamma^t_0} d(x,\Gamma^s_0)\leq \int^t_s\rho(\xi) d\xi\) for \(s<t\) (with \(\rho\in L^4(0,T)\)). This theorem follows from an abstract result on evolution equations on variable domains. The abstract theory is an essential part of the present paper. An application of the abstract theory to parabolic problems in non-cylindrical domains is also given.

MSC:

35K20 Initial-boundary value problems for second-order parabolic equations
47D06 One-parameter semigroups and linear evolution equations
35D10 Regularity of generalized solutions of PDE (MSC2000)
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