On a product formula for a class of nonlinear evolution equations (Q762752)

This work is concerned with the initial value problem of the form \[ (IVP;u_ o)\quad (d/dt)u(t)+\partial \phi u(t)\ni Bu(t),\quad t>0,\quad u(0)=u_ 0\in D(\phi), \] where \(\phi\) is a proper lower semicontinuous convex functional on an abstract real Hilbert space H, \(\partial \phi\) is its subdifferential and B is a single-valued operator in H satisfying the following conditions. (A) For each \(r\geq 0\), the set \(L_{\phi}(r)=\{z\in H; | z|_ H\leq r, \phi (z)\leq r\}\) is compact in H. (B1) \(D(\phi)\subset D(B)\), where \(D(\phi)\) is the effective domain of \(\phi\). (B2) \(| Bz|^ 2_ H\leq \ell(\phi(z))\) for any \(z\in D(\phi)\), where \(\ell(\cdot)\) is a non-decreasing continuous function from R into \([0,\infty).\) (B3) B is demicontinuous on each level set of \(\phi\). Let \(S(\tau; 0\leq \tau <\infty\) be the nonlinear contraction semigroup on \(\overline{D(\phi)}\) generated by \(-\partial\phi.\) In this paper, the existence of the solution u of \((IVP;u_ 0)\) is proved by showing that, for suitably small t, \(u_ n(t)=[(1+\tau (n)B)S(\tau (n))P]^{[t/\tau (n)]}u_ 0\to u(t)\) in H as \(n\to \infty\), and \(\phi(S(\tau(n))Pu_ n(t))\to \phi(u(t))\) in R as \(n\to \infty\), for a suitable subsequence \(\{\tau(n)\}\) with \(\tau(n)\downarrow 0\) as \(n\to \infty\). Here, P is the projection from H onto \(\overline{D(\phi)}\) and [s] denotes the greatest integer in \(s\in R\). Further, if \((IVP;u_ 0)\) has a unique solution u on \([0,T^*)\), the above convergence occur without taking subsequence \(\{\tau(n)\}\) for every \(t\in [0,T^*)\).