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On transitions to stationary states in one-dimensional nonlinear wave equations. (English) Zbl 0939.35030

The author considers the long-time asymptotics of solutions to one-dimensional nonlinear wave equations \[ \begin{aligned} \ddot u(x,t) & =u''(x,t)+ f \bigl(x,u(x,t) \bigr),\\ u|_{t=0} & =u_0(x),\;\dot u|_{t=0}= v_0(c), \end{aligned} \] where the solutions \(u\) take values in \(\mathbb{R}^d\). It is assumed that \[ f(x,u)= \chi(x)F(u), \quad F\in C^1(\mathbb{R}^d,\mathbb{R}^d), \quad\chi\in C(\mathbb{R}), \]
\[ F(u)= -\nabla V(u),\quad V(u)\to+ \infty\text{ as }|u|\to\infty, \]
\[ \chi(x)\geq 0,\;\chi(x)\equiv 0,\quad \chi(x)=0\text{ for }|x|\geq a, \] where \(a\) is some positive constant. The author shows that the orbit of the solution \(Y(t)=(u(\cdot,t)\), \(\dot u(\cdot,t))\) is precompact in some Fréchet topology, and the set of all stationary states \(S=(s(x),0)\) is its point attractor. Moreover, if \(d=1\) and the function \(F(u)\) is real analytic, then the limit \(\lim_{t\to\pm\infty}Y(t)\) exists.

MSC:

35B40 Asymptotic behavior of solutions to PDEs
35L70 Second-order nonlinear hyperbolic equations
35L15 Initial value problems for second-order hyperbolic equations
35B41 Attractors
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