Lineare Evolutionsgleichungen

Cauchy problem in a product space (that may be particularly involved). For example, consider the heat equation  ∂u ∂t (t, x) = ∆u(t, x), t ≥ 0, x ∈ Ω, ∂w ∂t (t, z) = − ∂w ∂n (t, z), t ≥ 0, z ∈ ∂Ω, w(t, z) = u(t, z), t ≥ 0, z ∈ ∂Ω, where − ∂ ∂n denotes the outer normal derivative. This can be written as U ′(t) = ( ∆ 0 − ∂ ∂n 0 ) U(t), where U = ( u u|∂Ω ) . Observe that a relationship between the coordinates of U is intrinsic in the case of the wave equation (there, the first coordinate is the time derivative of the second one) but has to be imposed in the case of the heat equation with dynamic boundary conditions. Example 1.6. Consider the delayed partial differential equation ∂u ∂t (t, x) = ∆u(t, x) + u(t− 1, x) for t ≥ 0 and x ∈ R. The variation of u at time t also depends on what has happened to u at time t− 1 (think of a population model including pregnancy effects). Finding the right framework for such equations was an open problem for a long time. Finally, it was understood that the above equation cannot be written as an (ACP ) on a natural function space: indeed, a solution cannot be uniquely determined by a function in C0(R), but rather needs specification of an initial condition in the following form: u(s, x) = h(s, x), for s ∈ [−1, 0] and x ∈ R. Then, this initial value problem can be written as U ′(t) = AU(t) on the Banach space X := C([−1, 0];C0(R), where Af = f ′ and the domain of A consists of those continuously differentiable functions f : [−1, 0]→ C0(R) such that • f(0) ∈ C0(R) s.t. ∆f(0) ∈ C0(R) and • f ′(0) = ∆f(0) + f(−1). Exercise 1.7. Consider the Volterra integro-differential equation ∂u ∂t (t, x) = ∆u(t, x) + ∫ t