On finite fractal dimension of the global attractor for the wave equation with nonlinear damping. (English) Zbl 1030.35017
The author considers the following nonlinear wave equation
\[
u_{tt}+g(u_t)-\Delta u + f(u) =0, \quad x\in \Omega,\;t>0,
\]
\[ u(x,t)=0, \quad x\in \partial \Omega,\;t>0, \]
\[ u(x,0)=u_0(x),\;u_t(x,0)=u_1(x), \quad x\in \Omega. \] If \(f\) and \(g\) are smooth and satisfy some polynomial growth conditions, then the global attractor associated with this equation has a finite fractal dimension.
\[ u(x,t)=0, \quad x\in \partial \Omega,\;t>0, \]
\[ u(x,0)=u_0(x),\;u_t(x,0)=u_1(x), \quad x\in \Omega. \] If \(f\) and \(g\) are smooth and satisfy some polynomial growth conditions, then the global attractor associated with this equation has a finite fractal dimension.
Reviewer: Gheorghe Moroşanu (Iaşi)
MSC:
35B41 | Attractors |
35L70 | Second-order nonlinear hyperbolic equations |
35L20 | Initial-boundary value problems for second-order hyperbolic equations |