## 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 $$t\to\infty$$.

### MSC:

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

### Keywords:

longtime behavior; existence and uniqueness
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