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Existence results for doubly nonlinear higher order parabolic equations on unbounded domains. (English) Zbl 0609.35048
We prove the existence of weak (or ”energy”) solutions of the homogeneous Dirichlet initial-boundary value problem for some equations of the form \(\partial (Bu)/\partial t+Au=f\), where A and B are nonlinear monotone operators deriving from convex functionals and the spatial domain is an arbitrary open set of \({\mathbb{R}}^ n\). In particular, our existence theorem applies (for any \(p,q>1\) and any m,n\(\geq 1)\) if A and B are defined by: \[ Au=(-1)^ m\sum _{| \alpha | =m}D^{\alpha}(| D^{\alpha}u| ^{p-1}sgn D^{\alpha}u),\quad (Bu)(x)=| u(x)| ^{q-1}sgn u(x). \] We start from an existence result of O. Grange and F. Mignot [J. Funct. Anal. 11, 77-92 (1972; Zbl 0251.35055)] and follow some methods of H. W. Alt and S. Luckhaus [Math. Z. 183, 311-341 (1983; Zbl 0497.35049)]. In addition, we use Nikol’skiĭ spaces (spaces involving Hölder conditions in the \(L^ p\) metric) to perform a key compactness argument of the proof. We also prove an abstract formula of integration by parts which allows to handle the very weak derivative \(\partial (Bu)/\partial t\) and implies an ”energy estimate for our weak solutions.

35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
47H05 Monotone operators and generalizations
35D05 Existence of generalized solutions of PDE (MSC2000)
35K65 Degenerate parabolic equations
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