Nonlinear Evolution Equations in Scales of Banach Spaces and Applications to PDEs
Nonlinear evolution problems are studied by means of an abstract integral equation in a general scale of Banach spaces. The range of spaces for which a suitable notion of solution exists is analyzed. Optimal uniqueness, blow up estimates, criteria for global existence and smoothing effects of the solution are obtained. Wide applicability of the theory is illustrated by a variety of examples, involving nested and not nested scales of spaces.