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Semilinear parabolic systems. (English) Zbl 0576.35067
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 $$[t_ 0,T]$$.
Reviewer: M.Degiovanni

##### MSC:
 35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations 35B65 Smoothness and regularity of solutions to PDEs 35A05 General existence and uniqueness theorems (PDE) (MSC2000) 35A07 Local existence and uniqueness theorems (PDE) (MSC2000)
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