| Many problems in these fields can be described by the evolution equations in hyperbolic type in the Banach space, such as physics, chemistry, biology, economics, etc. And the associated evolution system of hyperbolic type is much different from the semigroup of bounded operators and the evolution system in parabolic type.Firstly, the conditions generating the evolution system in hyperbolic type and its' characters are discussed. The purpose of this paper is to introduce the strong solution of the first order linear evolution equations in hyperbolic type. The strong solution of equations is a more powerful kind of generalized solutions between the Y-valued solution and the mild solution. At the same time, the sufficient conditions of existence for strong solutions are given. So does the relation of the strong solution and the Y-valued solution, the strong solution and the mild solution.Secondly, consider the Cauchy problem of hyperbolic evolution equations:Also, some existence results of the Y-valued solution, the mild solution, the classic solution under the special condition are presented. The strong solution is introduced and some sufficient conditions of the strong solutions are proven. |
