an:03921100
Zbl 0576.35067
Amann, Herbert
Semilinear parabolic systems
EN
Commentat. Math. Univ. Carol. 26, 3-21 (1985).
00150471
1985
j
35K60 35B65 35A05 35A07
semilinear parabolic systems; local and global existence; time-dependent boundary conditions; regularity
The paper is concerned with semilinear parabolic systems of the form
\[
(P)\quad \partial u/\partial t+{\mathcal A}(t)u=f(t,x,u,Du,...,D^ ku)\quad in\quad \Omega \times]t_ 0,T];\quad {\mathcal B}(t)u=0\quad on\quad \partial \Omega \times]t_ 0,T];\quad u(.,t_ 0)=u_ 0\quad on\quad \Omega
\]
where \(\Omega\) is a smooth open subset of \({\mathbb{R}}^ n\), for every t in [0,T] \({\mathcal A}(t)\) is a linear differential operator 2m and \({\mathcal B}(t):=\{{\mathcal B}_{\Gamma}(t):\) \(\Gamma\) \(\subset \Gamma \}\) is a system of boundary operators associated with a decomposition \(\Gamma\) of \(\partial \Omega.\)
Then (\({\mathcal A}(t),{\mathcal B}(t),\Omega,\Gamma)\), \(0\leq t\leq T\), is said to be a regular parabolic initial value problem (IBVP) of order 2m, if some algebraic conditions, involving the coefficients of \({\mathcal A}\) and \({\mathcal B}\), and a functional condition, concerning the operaor \((\lambda +{\mathcal A}(t),{\mathcal B}(t))\), are satisfied. The class of IBVPs includes many known parabolic problems, such as strongly parabolic equations with boundary operators satisfying the complementing condition, etc.
In the main result the author shows that, if the nonlinearity f and the initial datum \(u_ 0\) are sufficiently smooth, there exists a unique local solution u of problem (P). An important feature is that no compatibility condition on f is required. Further regularity properties, under strong assumptions, are stated. Finally several conditions are provided, which ensure that the solution u is global, namely defined on all the interval \([t_ 0,T]\).
M.Degiovanni