Mixed nonlinear problems for parabolic equations with nonstationary boundary conditions and conditions of conjugacy
This paper is dedicated to mixed problems for nonlinear parabolical equations with the transmission and time-dependent boundary conditions. One of the problem is the following: Find the solution of the equation \[ \partial \mu /\partial t-\sum^{N}_{i=1}\partial a_ i(x,u,U_ x)/\partial x_ i+b_ 0(x,u)=f_ 0(x,u),x\in \Omega,\quad t\in (0,T) \] with initial \(u(x,0)=0\), \(x\in \Omega\), \(u(x,0)=u_{10}(x)\), \(x\in \Gamma_ 1\) and boundary conditions: \[ \partial u/\partial t=-\partial u/\partial \nu -b_ 1(x,u)+f_ 1(x,u),\quad x\in \Gamma_ 1, \] \[ 0=- \partial u/\partial \nu -b_ 2(x,0),\quad x\in \Gamma_ 2,\quad t\in (0,T), \] where \(u_ x\equiv (u_{x_ 1},...,u_{x_ N})\), \(\partial u/\partial \nu \equiv \sum^{N}_{i=1}a_ i(x,u,U_ x) \cos (\nu,x_ i)\), \(\Gamma_ 1\cap \Gamma_ 2=\phi\), \(\Gamma_ 1\cup \Gamma_ 2=\partial \Omega.\) The authors prove existence and uniqueness of the generalized solution of the problem by means of monotone semicontinuous operators.
scientific article; zbMATH DE number 4118809
scientific article