Quasi-Linear Evolution Equations in Banach Spaces

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This paper is concerned with the quasi-linear evolution equation u'(t) + A(t, u(t))u(t) = 0 in [0, T], u(0) = xo 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 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. This paper is concerned with the quasi-linear evolution equation u'(t) + A(t, u(t))u(t) = 0 in[O, T], u(O) = xo in a Banach space setting. The spirit of this inquiry follows that of T. Kato. Kato wrote a fundamental paper on linear evolution equations in 1953 [9]; that is, investigation of u'(t) + A(t)u(t) = 0 on[O, T], u(0) = x0. He strengthened and extended his analysis of the linear problem in 1970 [11]. Kato also wrote on the quasi-linear problem in 1975 [13]. We feel that our results give a natural approach to dealing with the quasi-linear problem. After discussing the setting and method of attack, our theorem is stated and proved. We then give an application of the theorem using the Sobolevskii-Tanabe theory of linear evolution equations of parabolic type. A proposition relevant to our theorem is also given. Let X and Y be Banach spaces, with Y densely and continuously embedded in X. Let x0 E Y, T > 0, r > r, > 0, r2 > 0, W= Bx(xo; r), Z = Bx(xo; rl) n By(xo; r2), and for each t E [0, T] and w E W, let -A (t, w) be the infinitesimal generator of a strongly continuous semigroup of bounded linear operators in X, with Y c D(A(t, w)). We consider the quasi-linear evolution equation v'(t) + A(t, v(t))v(t) = 0. (QL) Received by the editors December 14, 1978 and, in revised form, July 6, 1979. AMS (MOS) subject classifications (1970). Primary 34G05, 47D05; Secondary 65J05, 41A65.