Quasi-Linear Evolution Equations in Banach Spaces
Michael G. Murphy · Transactions of the American Mathematical Society · 1980
This dissertation is concerned with studying the quasilinear evolution equation u'(t) + A(t,u(t)) u(t) = 0 in [0,T], u(0) = xQ in a Banach space setting.The spirit of this inquiry follows that of T. Kato and his fundamental results concerning linear evolution equa tions .We feel that our results give a natural approach to dealing with the quasi-linear problem.Chapter I gives the preliminaries required for our work with abstract evolution equations.This consists of calculus in Banach spaces, the analytical theory of semigroups of bounded linear operators, and background material from the theory of linear evolution equations in Banach spaces.Chapter II gives the main result of this dissertation, an alternate proof of the main result, a related proposition, two corollaries to the main result including an application, and directions for related research.We assume that we have a family { A(t,w)} of operators in a Banach space X such that each -A(t,w) is the infini tesimal generator of a strongly continuous semigroup of bounded linear operators in X and that the family satisfies iv V continuity and stability conditions.We show that on a fixed subinterval of [0,T] that we have a family of approxi mate solutions to the quasi-linear problem that converge to a candidate function.The candidate function must be the solution to the quasi-linear problem if one exists.In fact, if u is the candidate function, it is enough that the linear problem v'(t) + A(t,u(t)) v(t) = 0, v(0) = x0 , have a solution in order that u will be the unique solution to the quasi-linear problem.We also show that the candidate function depends on the initial value in a strong way.The corollaries are concerned with the existence aspect.