Abstract Cauchy problem

An abstract Cauchy problem has the form $Z^{\prime }(t)=AZ(t)+F(t)$ in a Banach space, where $A$ may be unbounded and generates the homogeneous evolution.

A nonautonomous abstract Cauchy problem has the form $u^{\prime }(t)=A(t)u(t)$, with a time-dependent generally closed linear operator. Under stability, common-domain, and regularity hypotheses, it is propagated by an evolution family.

An evolution family consists of bounded operators $U(t,s)$ for $s\le t$ satisfying $U(s,s)=I$ and $U(t,r)U(r,s)=U(t,s)$. For sufficiently regular data it solves ${\partial }_{t}U(t,s)x=A(t)U(t,s)x$ and ${\partial }_{s}U(t,s)x=-U(t,s)A(s)x$.

For a partition $s=t_0<\cdots <t_N=t$, the frozen-generator approximation isUnder the hypotheses of the nonautonomous generation theorem, these products converge strongly and uniformly on compact time triangles to the evolution family.

A mild solution need only be continuous and satisfy the integrated semigroup formula. It need not lie in the generator domain or be differentiable.

If $A$ generates $U(t)$, the variation-of-constants formula isIt is also called Duhamel's formula.

A semilinear abstract Cauchy problem has $Z^{\prime }=AZ+N(Z)$, where the linear part generates a C0-semigroup and the nonlinear map is typically locally Lipschitz on the phase space.

A local mild solution is a fixed point of the nonlinear variation-of-constants map in $C([0,T];X)$. Local Lipschitz continuity of the nonlinearity gives existence and uniqueness for sufficiently small $T$.

For a locally Lipschitz semilinear evolution, a maximal mild solution either exists for all positive time or its phase-space norm becomes unbounded as the finite maximal time is approached.