Mixed nonlinear problems for parabolic equations with nonstationary boundary conditions and conditions of conjugacy (Q1824765)

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.