ABSTRACT

In this paper, using the subvariant functional method due to Favard [Favard J. Sur les équations différentielles linéaires á coefficients presque-périodiques. ActaMathematica. 1928;51(1):31–81.], we prove the existence and uniqueness of compact almost automorphic solutions for a class of semilinear evolution equations in Banach spaces provided the existence of at least one bounded solution on the right half line. More specifically, we improve the assumptions in [Cieutat P, Ezzinbi K. Almost automorphic solutions for some evolution equations through the minimizing for some subvariant functional, applications to heat and wave equations with nonlinearities. J Funct Anal. 2011;260(9):2598–2634.], we show that the almost automorphy of the coefficients in a weaker sense (Stepanov almost automorphy of order $1\le p<\mathrm{\infty }$) is enough to obtain solutions that are almost automorphic in a strong sense (Bochner almost automorphy). For that purpose we distinguish two cases, $p=1$ and $p>1$. The main difficulty in this work, is to prove the existence of at least one solution with relatively compact range while the forcing term is not necessarily bounded. Moreover, we propose to study a large class of reaction-diffusion problems with unbounded forcing terms.

摘要

在本文中，由于 Favard [Favard J. Sur les équations différentielles linéairesá coefficients presque-périodiques，使用了子变函数方法。数学学报。1928;51(1):31-81.]，我们证明了 Banach 空间中一类半线性演化方程的紧致几乎自守解的存在性和唯一性，前提是在右半线上至少存在一个有界解。更具体地说，我们改进了 [Cieutat P, Ezzinbi K. 中的假设。通过最小化一些子变函数，应用到具有非线性的热和波动方程，一些演化方程的几乎自守解。J功能肛门。2011;260(9):2598–2634.]，我们展示了在较弱意义上的系数的几乎自同形（Stepanov 几乎自同形的阶$1\le p<\mathrm{\infty }$) 足以获得在强烈意义上几乎自同构的解（Bochner 几乎自同构）。为此，我们区分两种情况，$p=1$和$p>1$. 这项工作的主要困难是证明存在至少一个范围相对紧凑的解决方案，而强制项不一定是有界的。此外，我们建议研究一大类具有无界强迫项的反应扩散问题。