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Critical and supercritical higher order parabolic problems in $$\mathbb R^N$$. (English) Zbl 1288.35281
Summary: Due to the lack of the maximum principle the analysis of higher order parabolic problems in $$\mathbb R^N$$ is still not as complete as the one of the second-order reaction-diffusion equations. While the critical exponents and then a dissipative mechanism in the subcritical case have already been satisfactorily described, for problems in the critical or supercritical regime the questions concerning well or ill-posedness, as well as possible dissipative properties of the solutions, have not yet been satisfactorily answered. This article is devoted to the analysis of the higher order parabolic problems in $$\mathbb R^N$$ in the latter case. Focusing on the critical and supercritical regimes we give sufficient “good”-sign conditions proving that the problem is then globally well posed in $$L^2(\mathbb R^N)$$ and even possesses a compact global attractor. On the other hand, for supercritically growing “bad”-signed nonlinearities we show that the problem is ill-posed.

##### MSC:
 35K30 Initial value problems for higher-order parabolic equations 35K58 Semilinear parabolic equations 35B33 Critical exponents in context of PDEs 35B40 Asymptotic behavior of solutions to PDEs 35B41 Attractors
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