Well-posedness of non-autonomous linear evolution equations in uniformly convex spaces

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• 作者：J. Schmid and M. Griesemer
• 刊名：Mathematische Nachrichten
• 出版年：2017
• 出版时间：February 2017
• 年：2017
• 卷：290
• 期：2-3
• 页码：435-441
• 全文大小：159K
• ISSN：1522-2616

文摘

This paper addresses the problem of well-posedness of non-autonomous linear evolution equations math-equation-construct">mage="true" class="math-equation-image">mathml="true" class="math-equation-mathml" style="display:none">math xmlns:mml="http://www.w3.org/1998/Math/MathML">line-library/content/render" xmlns="http://www.w3.org/1998/Math/MathML">ẋ=A(t)xmath> in uniformly convex Banach spaces. We assume that math-equation-construct">mage="true" class="math-equation-image">mathml="true" class="math-equation-mathml" style="display:none">math xmlns:mml="http://www.w3.org/1998/Math/MathML">line-library/content/render" xmlns="http://www.w3.org/1998/Math/MathML">A(t):D⊂X→Xmath> for each t is the generator of a quasi-contractive, strongly continuous group, where the domain D and the growth exponent are independent of t. Well-posedness holds provided that math-equation-construct">mage="true" class="math-equation-image">mathml="true" class="math-equation-mathml" style="display:none">math xmlns:mml="http://www.w3.org/1998/Math/MathML">line-library/content/render" xmlns="http://www.w3.org/1998/Math/MathML">t↦A(t)ymath> is Lipschitz for all math-equation-construct">mage="true" class="math-equation-image">mathml="true" class="math-equation-mathml" style="display:none">math xmlns:mml="http://www.w3.org/1998/Math/MathML">line-library/content/render" xmlns="http://www.w3.org/1998/Math/MathML">y∈Dmath>. Hölder continuity of degree math-equation-construct">mage="true" class="math-equation-image">mathml="true" class="math-equation-mathml" style="display:none">math xmlns:mml="http://www.w3.org/1998/Math/MathML">line-library/content/render" xmlns="http://www.w3.org/1998/Math/MathML">α<1math> is not sufficient and the assumption of uniform convexity cannot be dropped.