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Multivalued evolution equations with nonlocal initial conditions in Banach spaces. (English) Zbl 1145.35076

The authors prove the existence of integral solutions to the nonlocal Cauchy problem

u ' (t)-Au(t)+F(t,u(t)),0tT;u(0)=g(u)

in a Banach space X, where A:D(A)XX is m-accretive and such that A generates a compact semigroup, F:[0,T]×X2 X has nonempty, closed and convex values, and is strongly-weakly upper semicontinuous with respect to its second variable, and g:C([0,T];D(A) ¯)D(A) ¯. The case when A depends on time is also considered.

The proof of the basic result is constructed as follows: First, one considers the linear counterpart of the above equation, u ' (t)-Au(t)+f(t), on the same interval I=[0,T], with initial condition u(0)=u 0 . For the linear problem, the integral solution is defined by means of the integral inequality

u(t)-x 2 u(s)-x 2 +2 s t f(r)-y,u(r)-x + dr

for all xD(A), yAx and u(0)=u 0 , continuous. An integral solution for the nonlinear equation is defined as satisfying the linear equation for some integrable f(t)F(t,u(t)). Then, using the existence for the linear equation, one shows the existence in the nonlinear case. An application is provided for heat equation.


MSC:
35K90Abstract parabolic equations
34G25Evolution inclusions
47J35Nonlinear evolution equations