Critical and supercritical higher order parabolic problems in RNRN

Due to the lack of the maximum principle the analysis of higher order parabolic problems in RNRN 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 (see Cholewa and Rodriguez-Bernal (2012)), 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 RNRN 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 L2(RN)L2(RN) 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.