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Quasilinear evolution equations in Banach spaces

1980; American Mathematical Society; Volume: 259; Issue: 2 Linguagem: Inglês

10.1090/s0002-9947-1980-0567096-x

1088-6850

Michael G. Murphy,

Stability and Controllability of Differential Equations

This paper is concerned with the quasi-linear evolution equation u ′ ( t ) + A ( t , u ( t ) ) u ( t ) = 0 u’(t)\, + \,A(t,\,u(t))u(t)\, = \,0 in [ 0 , T ] , u ( 0 ) = x 0 [0,\,T],\,u(0)\, = \,{x_0} in a Banach space setting. The spirit of this inquiry follows that of T. Kato and his fundamental results concerning linear evolution equations. We assume that we have a family of semigroup generators that satisfies continuity and stability conditions. A family of approximate solutions to the quasi-linear problem is constructed that converges to a “limit solution.” The limit solution must be the strong solution if one exists. It is enough that a related linear problem has a solution in order that the limit solution be the unique solution of the quasi-linear problem. We show that the limit solution depends on the initial value in a strong way. An application and the existence aspect are also addressed.