Blow up oscillating solutions to some nonlinear fourth order differential equations

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

“…Table 6. The second column of the table can be compared with [11,Numerical result 5]. We believe our results to be more accurate than the ones presented there because the auxiliary equation is easier to integrate numerically, and the transformation (2.11) preserves the relative error in mapping zeros of w (solution of the auxiliary equation) in zeros of u.…”

“…Motivated by the results we have obtained in Sections 3 and 5, and by the numerical experiments reported in [11,12], we expect many solutions of (1.1) to blow up in finite time through progressively wider oscillations, for several types of non-linearity f . This behavior poses several unavoidable difficulties in their numerical integration, which can be effectively mitigated by turning to an "auxiliary equation" obtained through (2.11).…”

“…one recognizes that, if u is non-trivial, then H is strictly increasing (see also [11]). Combining this last fact with [12,Lemma 9], one obtains the following proposition, the proof of which is straightforward and, therefore, omitted.…”

“…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.…”