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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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