Attractor of nonlinear evolutionary equation of viscoelasticity
Let the nonlinear evolution equation of viscoelasticity \[ \frac{\partial^ 2u(t,x)}{\partial t^ 2}-\frac{\partial}{\partial x}\sigma (\frac{\partial u}{\partial x})-\lambda \frac{\partial^ 3u}{\partial t\partial x^ 2}+\epsilon \frac{\partial u}{\partial t}+f(u)+g(x)=0,\quad t\geq 0,\quad x\in (0,1) \] and \[ u(t,x)|_{x=0}=u(t,x)|_{x=1}=0,\quad t\geq 0, \] \[ u(t,x)|_{t=0}=u_ 0(x),\quad \frac{\partial u(t,x)}{\partial t}|_{t=0}=p_ 0(x),\quad x\in (0,1); \] and let \[ E:=H^ 1_ 0(0,1)\cap H^ 2(0,1)\times H^ 1_ 0(0,1),\quad E_ 1:=H^ 1_ 0(0,1)\cap H^ 3(0,1)\times H^ 1_ 0(0,1)\cap H^ 2(0,1), \] where \(H^ 1_ 0(0,1)\), \(H^ k(0,1)\), \(k=2,3\), are Sobolev spaces. It is shown that if \(y_ 0=(u_ 0,p_ 0)\in E[E_ 1]\) and \(\sigma\),f,g satisfy some regularity condition then there exists a unique solution u such that \(y(t)=(u(t),\frac{du}{dt}(t))\in {\mathcal C}([0,\infty),E)\) and y(t) is continuous with respect to \(y_ 0\). The main result is the following: For the semigroup of operators \(S_ t:E\to E\) \([E_ 1\to E_ 1]\) defined by \(S_ ty_ 0=y(t)\) there exists an \((E_ 1,E)\)-attractor i.e. a bounded subset \(A\subset E\) such that for every bounded \(B\subset E_ 1:\) \[ \lim_{t\to \infty}dist(S_ tB,A)=0\text{ and } S_ tA=A,\quad t\geq 0. \]
nonlinear evolution equation of viscoelasticity
scientific article; zbMATH DE number 3986020
scientific article