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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
Software:
INTLAB
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