Existence of pseudo-almost automorphic solutions to some abstract differential equations with \(\mathbf S^p\)-pseudo-almost automorphic coefficients. (English) Zbl 1178.43004
A function \(f:\mathbb{R}\to\) Banach space \(X\) is called pseudo-almost automorphic if \(f= g+h\) with \(g\) (Bochner-) almost automorphic (aa) and \(h\) bounded \(eC(\mathbb{R},X)\) with
\[ {1\over 2T} \int^T_{-T}\| h(t)\|_X \,dt\to 0 \]
as \(T\to\infty\) (see [the author, Pseudo almost periodic functions in Banach spaces. New York, NY: Nova Science Publishers (2007; Zbl 1234.43002)]); \(f\in L^1_{\text{loc}}(\mathbb{R}, X)\) is called Stepanov \(S^p\) pseudo aa if \(f= g+ h\), with Bochner transform \(g^b: \mathbb{R}\to L^p:= L^p([0, 1],X)\) aa, Stepanov norm \(\| h\|^{L^p}< \infty\) and
\[ {1\over 2T} \int^T_{-T}\| h^b(t)\|_{L^p}\,dt\to 0, \]
where \((g^b(t))(s):= g(t+ s)\), \(0\leq s\leq 1\), \(r\in \mathbb{R}\).
For the equation \(u'(t)= Au(t)+ F(t,u(t))\), \(t\in\mathbb{R}\), it is outlined how the existence of a mild solution which is pseudo aa can be obtained, provided \(A\) is the infinitesimal generator of an asymptotically stable \(C^0\)-semigroup on \(X\) and the continuous \(F(t,u)\) is in \(t\) only Stepanov \(S^p\) pseudo aa, suitably locally uniformly in \(u\), \(p> 1\), further \(F\) satisfies in \(u\) a global Lipschitz condition with sufficiently small Lipschitz constant.
This is applied to the Dirichlet problem for a two-dimensional quasilinear heat equation. The non-autonomous case (Theorem 4.2) is somewhat mysterious to this reviewer.
\[ {1\over 2T} \int^T_{-T}\| h(t)\|_X \,dt\to 0 \]
as \(T\to\infty\) (see [the author, Pseudo almost periodic functions in Banach spaces. New York, NY: Nova Science Publishers (2007; Zbl 1234.43002)]); \(f\in L^1_{\text{loc}}(\mathbb{R}, X)\) is called Stepanov \(S^p\) pseudo aa if \(f= g+ h\), with Bochner transform \(g^b: \mathbb{R}\to L^p:= L^p([0, 1],X)\) aa, Stepanov norm \(\| h\|^{L^p}< \infty\) and
\[ {1\over 2T} \int^T_{-T}\| h^b(t)\|_{L^p}\,dt\to 0, \]
where \((g^b(t))(s):= g(t+ s)\), \(0\leq s\leq 1\), \(r\in \mathbb{R}\).
For the equation \(u'(t)= Au(t)+ F(t,u(t))\), \(t\in\mathbb{R}\), it is outlined how the existence of a mild solution which is pseudo aa can be obtained, provided \(A\) is the infinitesimal generator of an asymptotically stable \(C^0\)-semigroup on \(X\) and the continuous \(F(t,u)\) is in \(t\) only Stepanov \(S^p\) pseudo aa, suitably locally uniformly in \(u\), \(p> 1\), further \(F\) satisfies in \(u\) a global Lipschitz condition with sufficiently small Lipschitz constant.
This is applied to the Dirichlet problem for a two-dimensional quasilinear heat equation. The non-autonomous case (Theorem 4.2) is somewhat mysterious to this reviewer.
Reviewer: Hans F. Günzler (Kiel)
MSC:
| 43A60 | Almost periodic functions on groups and semigroups and their generalizations (recurrent functions, distal functions, etc.); almost automorphic functions |
| 34G20 | Nonlinear differential equations in abstract spaces |
| 47D06 | One-parameter semigroups and linear evolution equations |
| 34D10 | Perturbations of ordinary differential equations |
| 35B15 | Almost and pseudo-almost periodic solutions to PDEs |
Keywords:
almost automorphic; pseudo-almost automorphic; Stepanov almost automorphic; evolution equationCitations:
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