Long time behavior of strongly damped nonlinear wave equations
It is studied the long time behaviour of the solution to the following nonlinear initial boundary problem \[ u_{tt}=\alpha u_{xxt}+\sigma(u_x)_x-f(u)+g(x),\quad x\in (0,1),\;t\in [0,\infty), \] \[ u(0)=u_0,\quad u_t(0)=u_1,\qquad u(0,t)=u(1,t)=0. \] Suppose that \(\sigma,f\) are smooth real valued functions on \(\mathbb{R}\) such that \(\sigma (0)=0\), \(\sigma '(s)\geq r_0\), \(s\in \mathbb{R}\), \(r_0>0\); \(\liminf_{| s| \to \infty} \frac {F(s)}{s^2}\geq 0\); and that there exists \(w>0\) such that \(\liminf_{| s| \to \infty} \frac {sf(s)-wF(s)}{s^2}\geq 0\), where \(F(s)=\int _0^s f(s)ds\). Let \(I=(0,1)\), \(Y=(H^2(I)\cap H^1_0(I)) \times L^2(I)\) be the Hilbert space with the norm \(\|(u,v)\|_Y=(\| \partial_{xx}\| ^2_{L^2(I)}+ \| v\|^2_{L^2(I)})^{\frac{1}{2}}\). If \(g\in L^2(I)\) and \((u_0,u_1)\in Y\) the authors prove the existence of a global solution \(u\) such that \((u,u_t)\in C([0,\infty),Y)\). This means that the problem generates a nonlinear semigroup \(S(t)\) on \(Y: S(t)(u_0,u_1)=(u(t),u_t(t))\), which is continuous in \(t\in [0,\infty)\) and \((u_0,u_1)\in Y\). If in addition to the above conditions \(g\in H^1(I)\), then the authors prove that the semigroup \(S(t)\) has a compact attractor with finite dimension in \(Y\).
scientific article; zbMATH DE number 1212344