ABSTRACT
(a) In this section we consider Cauchy problems for first-order hyperbolic systems, and in a later section we will investigate initial-boundary-value problems for them. Let z = x + i y be a complex variable on the complex plane https://www.w3.org/1998/Math/MathML">ℂhttps://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003062233/9da0ee55-ab85-4900-bd1b-460e01b2ece9/content/ieq0409.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> and let t denote a real variable on the half-line t >0. We shall consider hyperbolic systems of first-order equations in the half-space https://www.w3.org/1998/Math/MathML">(t,z)∈ℝ+3https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003062233/9da0ee55-ab85-4900-bd1b-460e01b2ece9/content/ieq0410.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>. First we consider the system of two real equations, which may be put into the following single complex equation https://www.w3.org/1998/Math/MathML">u¯t−Lu=f(t,z),https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003062233/9da0ee55-ab85-4900-bd1b-460e01b2ece9/content/eqn4_1.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> where L is an elliptic operator https://www.w3.org/1998/Math/MathML">Lu≡a(z)uz¯+b(z)uz+c(z)u+d(z)u¯,2uz¯=ux+iuy,2uz=ux−iuyhttps://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003062233/9da0ee55-ab85-4900-bd1b-460e01b2ece9/content/eqn0596.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> and the bar over a function, as usual, denotes the complex conjugate. We assume that a(z), b(z) are functions vanishing at infinity with bounded first-order derivatives https://www.w3.org/1998/Math/MathML">a2¯≤k,bz≤lhttps://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003062233/9da0ee55-ab85-4900-bd1b-460e01b2ece9/content/ieq0411.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>, that c(z), d(z) are bounded in https://www.w3.org/1998/Math/MathML">ℂ:|c(z)|≤α,∣d(z)≤βhttps://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003062233/9da0ee55-ab85-4900-bd1b-460e01b2ece9/content/ieq0412.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> and that f(t, z) is the following function, having bounded https://www.w3.org/1998/Math/MathML">L2(ℂ)https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003062233/9da0ee55-ab85-4900-bd1b-460e01b2ece9/content/ieq0413.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>-norm: https://www.w3.org/1998/Math/MathML">‖f(t)‖L2(ℂ)=∫∫ℂ|f(t,z)|2 dx dy1/2<∞https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003062233/9da0ee55-ab85-4900-bd1b-460e01b2ece9/content/eqn0597.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> for any t ≥ 0.
