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On the operator of Dirichlet-Neumann type for some nonlinear parabolic equations. (Sur l’opérateur du type Dirichlet-Neumann pour certaines équations paraboliques non linéaires.) (French) Zbl 0914.35061
Équations aux dérivées partielles et applications. Articles dédiés à Jacques-Louis Lions. Gauthier-Villars: Paris. 655-665 (1998).
Let \(\Omega\subset\mathbb R^n\) be smoothly bounded domain with boundary \(\Gamma=\Gamma_1\cup\Gamma_0\cup\Gamma^*\), where \(\Gamma_1,\Gamma_0\) are disjoint and open in \(\Gamma\) and \(\Gamma^*\) is a smooth \((n-2)\)-dimensional manifold (\(\Gamma_0\) and \(\Gamma^*\) may be empty). Put \(Q=\Omega\times(0,T)\), \(\Sigma_i=\Gamma_i\times(0,T)\), \(i=1,2\), and let \(\nu\) denote the outward unit normal on \(\Gamma\). The author studies the operator \({\mathcal T}(u)={\partial\beta(u)/\partial\nu}| _{\Sigma_1}\), where \(u\) is the solution of the problem \(u_t-\Delta\beta(u)=0\) in \(Q\), \(\beta(u)=h\) on \(\Sigma_1\), \(\partial\beta(u)/\partial\nu=0\) on \(\Sigma_0\) and \(u(\cdot,0)=0\) in \(\Omega\). Denote by \(H^{1/2}_{00}(X)\) the interpolation space \([H^1_0(X),L^2(X)]_{1/2}\) and assume \(h\in H^1(0,T;L^2(\Gamma_1))\cap L^\infty(0,T;H^{1/2}_{00}(\Gamma_1))\). Under these assumption, it is shown that \({\mathcal T}(h)\) belongs to the dual of the space \(H^{1/2}_{00}(0,T;L^2(\Gamma_1))\cap L^2(0,T;H^{1/2}_{00}(\Gamma_1))\). In Section 3 of the paper, the author discusses the case \(\beta(r)=r\) and possible generalizations of his results.
For the entire collection see [Zbl 0899.00020].

35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
Stefan problem