Solution of a quasilinear parabolic-elliptic boundary-value problem (Q2708570)

Let \(\Omega\subset \mathbb{R}^N\), \(N\geq 2,\) be a simply connected, bounded domain with boundary \(\partial\Omega\in C^1\), and \(I=[0,T]\). For given nonnegative \(g(x,t,u):\Omega\times I\times \mathbb{R}\rightarrow \mathbb{R}\), \(\Omega_{\varepsilon}=\Omega\setminus \text{supp}_{x\in\Omega}g(x,t,u)\), \(\Omega_p=\Omega\setminus\overline{\Omega_{\varepsilon}}\), \(\Gamma=\partial{\Omega_{\varepsilon}}\cap\partial\Omega_p\). It is assumed that \(\Omega_{\varepsilon}, \Omega_p\) are independent of \(t\) and \(u\). The following quasilinear parabolic-elliptic boundary-value problem is studied NEWLINE\[NEWLINE g(x,t,u)u_t+A(t)u=f(x,t,u) \qquad \text{in}\quad \Omega\times I, NEWLINE\]NEWLINE NEWLINE\[NEWLINE u_{\varepsilon}(x,t)=u_p(x,t), \quad \partial_{\nu_{A}}u_{\varepsilon}(x,t)=\partial_{\nu_{A}}u_{p}(x,t) \quad \text{on} \quad\Gamma\times I, NEWLINE\]NEWLINE NEWLINE\[NEWLINE u(x,t)=0 \qquad \text{on} \quad\partial\Omega\times I, NEWLINE\]NEWLINE NEWLINE\[NEWLINE u(x,0)=U_0(x) \qquad x\in\Omega_p, NEWLINE\]NEWLINE where \(u_p=u|_{\Omega_p}, u_{\varepsilon}=|_{\Omega_{\varepsilon}}\), and \(A(t)u=-\sum_{i,k=1}^{N}\frac{\partial}{\partial x_k}\left (a_{ik}(x,t)\frac{\partial u}{\partial x_i}\right)+a_0(x,t)u\). Under the corresponding assumptions on the functions \(g, f, a_{ik}, a_0\), and \(U_0\), the existence of a local weak solution \(u\) for the problem under consideration with additional regularity is proved. To construct the solution the Rothe's method is used. The uniqueness of a weak solution for the problem under consideration is proved too.NEWLINENEWLINEFor the entire collection see [Zbl 0957.00037].