Approximation of evolution equations with polynomial reproducing nonlinearities (Q1085339)

The author studies equations \(u'(t)=Au(t)+N(u(t))\) with polynomial nonlinearity N in a Banach space which lies in a Hilbert space. It is assumed that the nonlinear operator N is ''finitely reproducing'' relative to the orthonormal sequence generated by \(Au=\lambda u\). The major part of the paper deals with the problem \[ v_ t(x,t)=v_{xx}(x,t)+\sum^{3}_{j=0}a_ j(v(x,t))^ j,\quad t>0,\quad 0\leq x\leq \pi, \] \[ v_ x(0,t)=v_ x(\pi,t)=0,\quad t>0,\quad v(x,0)=\psi (x),\quad 0\leq x\leq \pi. \] The author proves the reproducing property of the nonlinear operator and gives numerical results on Faedo-Galerkin approximations.