Existence of strong solutions for a class of semilinear evolution equations with nonlocal initial conditions

Pengyu Chen,Yongxiang Li,Hongxia Fan

doi:10.1186/1687-1847-2012-79

Pengyu Chen, Yongxiang Li + Show 1 more

Open Access

https://doi.org/10.1186/1687-1847-2012-79

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Abstract

Abstract This paper discusses the existence of strong solutions for a class of semilinear evolution equations with nonlocal initial conditions in Hilbert spaces. The discussion is based on analytic semigroups theory and fixed point theorem. An application to a partial differential equation with nonlocal condition is also considered. Mathematics Subject Classification(2010): 34G20; 34K30; 35D35; 47D06.

Highlights

1 Introduction In this paper, we discuss the existence of strong solutions for a class of semilinear evolution equations with nonlocal initial conditions in a Hilbert space H
Since it is demonstrated that the nonlocal problems have better effects in applications than the classical Cauchy problems, differential equations with nonlocal conditions were studied by many authors and some basic results on nonlocal problems have been obtained, see [2,3,4,5,6,7,8,9,10] and the references therein
In [12], Byszewski pointed out that if gi ≠ 0, i = 1, 2, ..., m, the results can be applied to kinematics to determine the location evolution t ® u(t) of a physical object for which we do not know the positions u(0), u(t1), ..., u(tm), but we know that the nonlocal condition (2) holds

Summary

Introduction

We obtained the existence of strong solutions for the nonlocal problem (1)-(2) in a frame of abstract Hilbert spaces. By the maximal regularity of linear evolution equations with positive definite operator in Hilbert spaces (see [19], Chapter II, Theorem 3.3), when u(0) = u0 Î H1/2, the mild solution of the LIVP (7) has the regularity u ∈ W1,2(J, H) ∩ L2(J, H1) ∩ C(J, H1/2)

Results

Conclusion