On semilinear evolution equations with many Lyapunov functionals (Q1198722)

The author considers differential equations of the form \(u_ t=u_{xx}+f(u)\), \(0\leq x\leq\pi\), \(t>0\), where \(u\) vanishes for \(x=0\) and \(x=\pi\) and initial values of \(u\) are given. Under fairly weak conditions on \(f(u)\), he provides bounds for the \(L^ p\)-norms \((1<p<\infty)\) of the solution. He applies the results obtained to parabolic equations with nonlinear transport terms, weakly coupled parabolic systems and the Cahn-Hilliard equation. The majority of the estimates obtained are in a time-asymptotic form.