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Global solutions to some nonlinear dissipative mildly degenerate Kirchhoff equations. (English) Zbl 1028.35113
Let \(\Omega\subseteq \mathbb{R}^n\) \((n\leq 3)\) be an open domain, \(H:= L^2(\Omega)\), with norm \(\|\cdot\|\). The author considers the Cauchy problem \[ u''(t)+\delta u'(t)+ m(\|A^{1/2} u(t)\|^2) Au(t)+ f(u(t))= 0, \]
\[ u(0)= u_0,\quad u'(0)= u_1,\quad t\geq 0, \] where \(A=-\Delta\), with domain \(D(A)= H^1_0(\Omega)\cap H^2(\Omega)\). Under some natural conditions on the functions \(m\) and \(f\), the author proves existence and uniqueness of a global solution for \((u_0, u_1)\in D(A)\times D(A^{1/2})\). Moreover, she proves that \(u(t)\to 0\) as \(t\to\infty\).

35L80 Degenerate hyperbolic equations
35B40 Asymptotic behavior of solutions to PDEs
35L90 Abstract hyperbolic equations
35L15 Initial value problems for second-order hyperbolic equations
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