Wide Oscillation Finite Time Blow Up for Solutions to Nonlinear Fourth Order Differential Equations

Blow-up profile for solutions of a fourth order nonlinear equation

“…Combining this last fact with [12,Lemma 9], one obtains the following proposition, the proof of which is straightforward and, therefore, omitted.…”

“…It could, however, be the case that blow-up occurs only in forward (or backward) time. If u is a non-trivial solution of (U q ) that is defined, say, for all r ≥ 0, then Gazzola and Pavani show (see [12]) that u must vanish for r → +∞. The next theorem not only shows that there is no lack of those solutions, but also provides a precise characterization of their asymptotic behavior.…”

“…Equation (1.1) is also connected to the dynamics of suspension bridges through a model proposed by Lazer and McKenna [16]. For further details and references on this subject, we also refer to the works [11,12,2,1] which contain a new point of view on the mathematical explanation of instability (and collapse) of suspension bridges. Much attention has received the case of the nonlinearity being superlinear, namely there exist q > 1 and c > 0 such that f (t)t ≥ c |t| q+1 for any t ∈ R , (1 2) and, in particular, the case f (t) = µt + |t| q−1 t , with µ ∈ R. In [7], the authors prove the following result.…”

“…Our results provide a precise blow-up rate for a "measure" of vertical displacement as well as vertical acceleration of the bridge, see Proposition 3.7. Again, we refer the interested reader to the papers [11,12] for a detailed description of this phenomenon. Furthermore, Theorem 1.5 gives a partial answer to some questions posed in [12] about the behavior of solutions of (1.6) for κ > 0, see Section 5.…”

“…Again, we refer the interested reader to the papers [11,12] for a detailed description of this phenomenon. Furthermore, Theorem 1.5 gives a partial answer to some questions posed in [12] about the behavior of solutions of (1.6) for κ > 0, see Section 5. In this direction see also the recent papers [10,19,8].…”