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Global existence and nonexistence for semilinear parabolic systems with nonlinear boundary conditions. (English) Zbl 0652.35065
In a weak Hilbert space setting the semilinear parabolic Cauchy problem \[ \dot u+{\mathcal A}u=F(u),\quad u(0)=u_ 0 \] is considered, where -A generates a strongly continuous analytic semigroup. For certain classes of data \((F,u_ 0)\) regularity results and asymptotic behaviour of the solution are derived. These results are applied to prove global existence and “blow up” behaviour of semilinear parabolic systems with nonlinear boundary conditions of the form \[ \partial_ tu+{\mathcal A}u=f(\cdot,u)\quad in\quad \Omega \times (0,\infty);\quad {\mathcal B}u=g(\cdot,u)\quad on\quad \partial \Omega \times (0,\infty);\quad u(\cdot,0)=u_ 0\quad on\quad \Omega, \] where \(\Omega\) is a bounded domain in \({\mathbb{R}}^ n \)with smooth boundary \(\partial \Omega\), f and g are polynomially bounded smooth vector valued functions and (\({\mathcal A},{\mathcal B})\) defines a formally self-adjoint regular elliptic boundary value problem of second order.
Reviewer: J.Escher

MSC:
35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
47D03 Groups and semigroups of linear operators
35B40 Asymptotic behavior of solutions to PDEs
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
35K15 Initial value problems for second-order parabolic equations
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