Vector-valued Laplace Transforms and Cauchy Problems

1. Front Matter

   Pages i-xi

   PDF

2. Laplace Transforms and Well-Posedness of Cauchy Problems

   1. Front Matter

      Pages 1-4

      PDF

   2. The Laplace Integral

      • Wolfgang Arendt, Charles J. K. Batty, Matthias Hieber, Frank Neubrander

      Pages 5-63

   3. The Laplace Transform

      • Wolfgang Arendt, Charles J. K. Batty, Matthias Hieber, Frank Neubrander

      Pages 65-108

   4. Cauchy Problems

      • Wolfgang Arendt, Charles J. K. Batty, Matthias Hieber, Frank Neubrander

      Pages 109-240

3. Tauberian Theorems and Cauchy Problems

   1. Front Matter

      Pages 241-244

      PDF

   2. Asymptotics of Laplace Transforms

      • Wolfgang Arendt, Charles J. K. Batty, Matthias Hieber, Frank Neubrander

      Pages 245-332

   3. Asymptotics of Solutions of Cauchy Problems

      • Wolfgang Arendt, Charles J. K. Batty, Matthias Hieber, Frank Neubrander

      Pages 333-390

4. Applications and Examples

   1. Front Matter

      Pages 391-393

      PDF

   2. The Heat Equation

      • Wolfgang Arendt, Charles J. K. Batty, Matthias Hieber, Frank Neubrander

      Pages 395-410

   3. The Wave Equation

      • Wolfgang Arendt, Charles J. K. Batty, Matthias Hieber, Frank Neubrander

      Pages 411-422

   4. Translation Invariant Operators On L ^p(ℝⁿ)

      • Wolfgang Arendt, Charles J. K. Batty, Matthias Hieber, Frank Neubrander

      Pages 423-454

5. Back Matter

   Pages 455-523

   PDF

Linear evolution equations in Banach spaces have seen important developments in the last two decades. This is due to the many different applications in the theory of partial differential equations, probability theory, mathematical physics, and other areas, and also to the development of new techniques. One important technique is given by the Laplace transform. It played an important role in the early development of semigroup theory, as can be seen in the pioneering monograph by Rille and Phillips [HP57]. But many new results and concepts have come from Laplace transform techniques in the last 15 years. In contrast to the classical theory, one particular feature of this method is that functions with values in a Banach space have to be considered. The aim of this book is to present the theory of linear evolution equations in a systematic way by using the methods of vector-valued Laplace transforms. It is simple to describe the basic idea relating these two subjects. Let A be a closed linear operator on a Banach space X. The Cauchy problern defined by A is the initial value problern (t 2 0), (CP) {u'(t) = Au(t) u(O) = x, where x E X is a given initial value. If u is an exponentially bounded, continuous function, then we may consider the Laplace transform 00 u(>. ) = 1 e-). . tu(t) dt of u for large real>. .

• Bochner integral
• Evolution equations
• Fourier transform
• Laplace and Fourier transforms
• differential equation
• functional analysis
• partial differential equation
• partial differential equations

• Angewandte Analysis, Universität Ulm, Ulm, Germany

  Wolfgang Arendt

• St. John’s College, Oxford, UK

  Charles J. K. Batty

• Fachbereich Mathematik, TU Darmstadt, Darmstadt, Germany

  Matthias Hieber

• Department of Mathematics, Louisiana State University, Baton Rouge, USA

  Frank Neubrander

Policies and ethics