Parabolic problems with nonlinear dynamical boundary conditions and singular initial data
This paper is devoted to the following parabolic problem with dynamic boundary conditions: \[ \begin{cases} u_t+{\mathcal A}u=f(x,t,u,\nabla u),\quad & x \in \Omega,\;t>0,\\ (\gamma u)_t+ \beta u=g(x,t, \gamma u),\quad & x\in \Gamma,\;t>0.\\ u(x,0)= u_0(x),\quad & x\in\Omega,\\ (\gamma u)(x,0)=v_0(x),\quad & x\in \Gamma, \end{cases} \tag{1} \] where \(\Omega\) is a bounded domain in \(\mathbb{R}^n\) of class \(C^2\), \(\Gamma=\partial \Omega\), \(\gamma\) is the trace operator, and \({\mathcal A}u= -\Delta u+\omega u\), \(\beta u=u_\nu+\omega u\), \(\nu\) denotes the outer normal on \(T\). Here the authors are interested in obtaining the largest possible growth on the nonlinear terms such that problem (1) can be solved with initial data in Lebesgue spaces, \(L_{q_1}(\Omega)\times L_{q_2} (\Gamma)\), or even measures \({\mathcal M}(\Omega) \times{\mathcal M}(\Gamma)\), where \({\mathcal M}(\Omega)\) and \({\mathcal M}(\Gamma)\) denote the space of bounded Radon measures in \(\Omega\) and on \(\Gamma\), respectively.
scientific article; zbMATH DE number 1832879