Estimates of Solutions to Nonlinear Evolution Equations

Abstract

Consider the equation

                 u’(t) = A (t, u (t)),   u(0)= U₀ ;   u' := du/dt     (1).  

Under some assumptions on the nonlinear operator A(t,u) it is proved that problem (1) has a unique global solution and this solution satisfies the following estimate                            

                  ||u (t)|| < µ (t) ^-1     for every t belongs to R₊ = [0,infinity).

Here µ(t) > 0,   µ belongs to  C¹ (R₊), is a suitable function and the norm ||u || is the norm in a Banach space X with the property ||u (t) ||’   <=  ||u’ (t) ||.