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Blow-up profile for solutions of a fourth order nonlinear equation. (English) Zbl 1325.34045
The authors consider the equation $u''''(r)+\kappa u''(r)+f(u(r))=0,$ where $$\kappa\in\mathbb{R}$$ and $$f$$ is a locally Lipschitz nonlinear function. It is known that the nontrivial solutions of this equation blow up in finite time under suitable hypotheses on $$\kappa$$ and $$f$$. These solutions blow up with large oscillations. The main purpose of the paper is to draft a global picture of the behavior of the solutions by investigating their blow-up profile.
The typical result, in a simplified form, reads as follows: There exists a periodic function $$\gamma:\mathbb{R}\to\mathbb{R}$$, a set $$\Omega\subset\mathbb{R}^4$$ unbounded, arc-connected, symmetric with respect to the origin, with non-empty interior, and a constant $$a>0$$, such that, for any solution $$u$$ of $u''''(r)+u^3(r)=0$ with initial condition in $$\Omega$$, one has, up to a phase-shift of $$\gamma$$,
$\left| u(r)-\frac{1}{(T-r)^2}\gamma\left(\ln\left(\frac{T}{T-r}\right)\right)\right|< c(T-r)^a$
for all $$r\in[0,T)$$, for some $$T,c>0$$ that depend on the initial condition. Moreover, if $$\tau>0$$ is the least period of $$\gamma$$, then the function $$\gamma$$ satisfies $$\gamma(s+\tau/2)=-\gamma(s)$$, for all $$s\in\mathbb{R}$$, and $$\gamma$$ vanishes exactly twice on $$[0,\tau)$$. Further refinements enable to extend the result to a wider class of equations.
The main idea of the proofs lies in the following ansatz: if $$u$$ solves $$u''''(r)+| u(r)|^{q-1}u(r)=0$$ and blows up at $$T$$, then $$u(r)=cw(\varphi(r))/(T-r)^\eta$$, where $$\varphi$$ is a suitable change of variable and $$w$$ is a function defined on $$[0,\infty)$$ and satisfying an auxiliary equation; there is also a a computer assisted proof. Besides playing a key role in the theoretical analysis, this transformation turns out to be of great help also in the numerical investigation.
Reviewer: Pavel Rehak (Brno)

##### MSC:
 34C11 Growth and boundedness of solutions to ordinary differential equations
INTLAB
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##### References:
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