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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].

MSC:
 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