On a global solution and asymptotic behaviour for the generalized damped extensible beam equation. (English) Zbl 0884.35105

The damped extensible beam referred to in the title motivates the study of a fourth order nonlinear PDE on a bounded domain \(\Omega\) with smooth boundary. The author considers the generalized Cauchy problem summarized here:
Let \(A\) be a linear operator in \(H=L^2 (\Omega)\) with dense domain \(V\). Let \(g\) be a non-decreasing continuous scalar function with \(g(0)=0\). Then, for all \(T>0\) and for appropriately regular initial conditions, there is a unique solution \(u\in W^{1,\infty} (0,T;V)\cap W^{2,\infty} (0,T;H)\) to the equation \[ u''+A^2 u+M \bigl(|A^{1/2} u|^2\bigr) Au+g(u')=0 \text{ in } L^{(q+1)/q} (0,T;V'). \] The existence of such a solution is established using the Faedo-Galerkin method, and much of that process is standard. The asymptotic behavior is derived via an energy estimate obtained from repeated (and precise) applications of Young’s Inequality.
The report is well written and easy to follow. The final chapter includes an example using the Laplace operator in \(\mathbb{R}^n\).


35L75 Higher-order nonlinear hyperbolic equations
74K10 Rods (beams, columns, shafts, arches, rings, etc.)
35B40 Asymptotic behavior of solutions to PDEs
Full Text: DOI


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