A Class of Abstract Quasi-Linear Evolution Equations of Second Order

Abstract

In this paper we study the abstract quasi-linear evolution equation of second order formula here in a general banach space z. it is well-known that the abstract quasi-linear theory due to kato [10, 11] is widely applicable to quasi-linear partial differential equations of second order and that his theory is based on the theory of semigroups of class (C0). (for example, see the work of hughes et al. [9] and heard [8].) however, even in the special case where a (t,w, v) = a is independent of (t, w, v), it is found in [2] and [14] that there exist linear partial differential equations of second order for which cauchy problems are not solvable by the theory of semigroups of class (C0) but fit into the mould of well-posed problems where the solution and its derivative depend continuously on the initial data if the initial condition is measured in the graph norm of a suitable power of a. (see also work by krein and khazan [13] and fattorini [6, chapter 8].) this kind of cauchy problem has recently been studied extensively, using the theory of integrated semigroups or regularized semigroups. the theory of integrated semigroups was studied intensively by arendt [1] and that of regularized semigroups was initiated by da prato [3] and renewed by davies and pang [4]. for the theory of regularized semigroups we refer the reader to [5] and [16]. (u(t),v(t))' = Au(t)(u(t),v(t)) for t∈[0,T] and (u(0),v(0)) = (φ,ψ) in a suitable Banach space X, where for each solution w of equation (1.1) the matrix operator Aw(t) in X is defined by Aw(t)(u,v)=(v,A(t,w(t),w'(t)) u). We are here interested in studying the case where each matrix operator Aw(t) is the (complete infinitesimal) generator of a regularized semigroup on X. In Section 3 we set up basic hypotheses on the operators appearing in equation (1.1), and prove a fundamental existence and uniqueness theorem (Theorem 3.6) for the Cauchy problem (1.1). The proof is based on the theory of regularized evolution operators developed by the author [15], and a method of successive approximations proposed by Kobayasi and Sanekata [12] is applied to construct a unique twice continuously differentiable function u satisfying equation (1.1). Our formulation includes the abstract quasi-linear wave equation of Kirchhoff type u(t)+­m(|A1/2u(t)|2)Au(t)=0 (1.2) in a real Hilbert space H, where A is a nonnegative selfadjoint operator in H. Section 4 presents a regularized semigroup theoretical approach to the local solvability of equation (1.2) in the `degenerate case' where the function m(r) has zeros (Theorems 4.1 and 4.2), by using the result obtained in Section 3. In Section 2 we summarize some results on the generation of a regularized evolution operator associated with the linearized equation of (1.1), under the `regularized stability ' condition, and show that the family of matrix operators used to solve the linearized equation (1.2) satisfies the regularized stability condition. This fact will be useful for our arguments in Section 4.