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Convergence properties of the strongly damped nonlinear Klein-Gordon equation. (English) Zbl 0625.35058
The author studies the damped nonlinear Klein-Gordon equation $(1)\quad u_{tt}(x,t)+\alpha (-\Delta +\gamma)u_ t(x,t)+(-\Delta +m^ 2)u(x,t)+\lambda u(x,t)| u(x,t)|^{p-1}=0\quad x\in \Omega,\quad t\in {\mathbb{R}}$ with the initial and boundary conditions $(2)\quad u(x,0)=f(x)\quad for\quad x\in \Omega,\quad u_ t(x,0)=g(x)\quad and\quad u(x,t)=0\quad for\quad x\in \partial \Omega,\quad t\in {\mathbb{R}}$ where $$\Omega$$ is a bounded domain in $${\mathbb{R}}^ 3$$ with smooth boundary $$\partial \Omega$$; $$\alpha$$, $$\gamma$$, m, p, $$\lambda$$ are constants with $$\alpha,\lambda >0$$; $$\gamma$$,m$$\geq 0$$ and $$p\geq 1$$. Moreover f and g are smooth assigned functions.
It is known that for any $$\alpha >0$$ there exists a unique global strong solution $$u_{\alpha}$$ of (1), (2) [see P. Aviles and J. Sandefur, J. Differ. Equations 58, 404-427 (1985; Zbl 0572.34004)]. In this paper the author assumes $$p>3$$ and proves that there exists a sequence $$\{\alpha_ k\}$$ of positive, real numbers such that $$\alpha_ k\to 0$$ as $$k\to \infty$$ and a global weak solution v of (1), (2) with $$\alpha =0$$, such that $$u_{\alpha_ k}\to v$$ in $$(([0,T],L^ 2(\Omega))$$ for all $$T>0$$ a convenient summary of the undamped case (i.e. $$\alpha =0)$$ can be found by M. Reed [“Abstract nonlinear wave equations”, Lect. Notes Math. 507 (1976; Zbl 0317.35002)].
Reviewer: D.Fortunato

##### MSC:
 35L70 Second-order nonlinear hyperbolic equations 35L20 Initial-boundary value problems for second-order hyperbolic equations 35D05 Existence of generalized solutions of PDE (MSC2000)
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##### References:
 [1] Adams, R.A, Sobolev spaces, (1975), Academic Press New York · Zbl 0186.19101 [2] Aviles, P; Sandefur, J, Nonlinear second-order equations with applications to partial differential equations, J. differential equations, 58, 404-427, (1985) · Zbl 0572.34004 [3] Reed, M, Abstract non-linear wave equations, (1976), Springer-Verlag Berlin/Heidelberg/New York · Zbl 0317.35002 [4] Strauss, W, On weak solutions of semilinear hyperbolic equations, Anais acad. Brazil ciências, 42, 645-651, (1970) · Zbl 0217.13104
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