Global solutions to semilinear parabolic systems for small data
We consider semilinear parabolic systems \(u_ t+Au+f(u)=g,\) \(u(0)=u_ 0\), -A being the generator of an analytic semigroup with spectrum \(\sigma\) (A) in the right half plane. For appropriate small data \((g,u_ 0)\) global solutions are obtained if 0 belongs to the resolvent set and local existence is proved if \(0\in \sigma (A)\). The inverse-function theorem is used in a class of functions which are Hölder continuous with respect to time t. The asymptotic behaviour of the solution is given too. As first application we prove the existence of global (respectively local) solutions to the Navier-Stokes equations in \(L_ p\)-spaces in bounded (respectively unbounded) domains. Secondly we consider the case where A is an elliptic operator of order 2m and f has bounded growth. Thirdly a quasilinear example where the nonlinearity involves derivatives of the same order as A is studied. Then an example of a nonautonomous nonlinearity is given and finally the convergence of solutions of the time-dependent system to solutions of the stationary (semilinear elliptic) one is derived for \(0\in \rho (A)\).
scientific article; zbMATH DE number 4096009
scientific article