Semigroup - Semigroup Methods in Partial Differential Equations

Semigroup Methods in Partial Differential Equations

Further information: C0-semigroup

Semigroup theory can be used to study some problems in the field of partial differential equations. Roughly speaking, the semigroup approach is to regard a time-dependent partial differential equation as an ordinary differential equation on a function space. For example, consider the following initial/boundary value problem for the heat equation on the spatial interval (0, 1) ⊂ R and times t ≥ 0:

Let X be the Lp space L2((0, 1); R) and let A be the second-derivative operator with domain

Then the above initial/boundary value problem can be interpreted as an initial value problem for an ordinary differential equation on the space X:

On an heuristic level, the solution to this problem "ought" to be u(t) = exp(tA)u0. However, for a rigorous treatment, a meaning must be given to the exponential of tA. As a function of t, exp(tA) is a semigroup of operators from X to itself, taking the initial state u0 at time t = 0 to the state u(t) = exp(tA)u0 at time t. The operator A is said to be the infinitesimal generator of the semigroup.

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