## 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

### Keywords:

finite energy solution; Goursat problem; point attractor
Full Text: