Rate of stabilization of the solutions of an initial-boundary-value problem for a nonlocal evolution equation (Q1107727)

The author considers the nonlinear parabolic problem \[ (1)\quad u_ t=[\mu +\beta N(\| u_ t\|)]u_{xx}+f(x);\quad u(0,t)=u(1,t)=0,\quad u(x,0)=u_ 0(x), \] where \(u=u(x,t)\), N(\(\alpha)\) is a nonnegative function, \(\mu =Const>0\), \(\beta =Const\), \(0<\beta \leq \beta_ 0\), \(\| u_ t\|^ 2=\int^{1}_{0} u^ 2_ t dx\). Under some assumptions on \(\mu,f,u_ 0\) he obtains the existence, uniqueness and the asymptotic behavior of the solutions for the problem (1).