Classical, strong and mild abstract Cauchy solutions

Assume Countable Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)) for the Lebesgue time integrals. Let $A:D(A)\subseteq X\to X$ be a linear operator on a real or complex Banach space $X$, with the Banach graph conventions of Infinitesimal generator of a C0-semigroup. Let $T_0>0$, $x\in X$ and let $f:(0,T_0)\to X$ be Bochner integrable (Bochner-integrable function). Consider the abstract Cauchy problem $u^{\mathrm{\prime }}(t)=Au(t)+f(t)$ for $0<t<T_0$, $u(0)=x$; endpoint equations are imposed only when $f$ is continuously extended to $[0,T_0]$. Derivatives use the underlying real structure, with one-sided derivatives at the endpoints. (1) A classical solution is a function $u\in C^1([0,T_0];X)$ with $u(t)\in D(A)$ for every $t\in [0,T_0]$, $Au\in C([0,T_0];X)$, $u\mathrm{\prime }(t)=Au(t)+f(t)$ for $0<t<T_0$ and $u(0)=x$; here membership in $D(A)$ is with the graph norm and the derivative is the Fréchet derivative of a curve (Fréchet derivative between Banach spaces). (2) A strong solution is a continuous $u\in C([0,T_0];X)$ with $u(t)\in D(A)$ for all $t$, $u(0)=x$, and $Au\in C([0,T_0];X)$, such that $u(t)=x+{\int }_0^t(Au(s)+f(s))ds$ for every $t$. (3) When $A$ generates a strongly continuous semigroup $T$ satisfying an exponential norm bound, a mild solution (variation-of-constants solution) is the continuous function given by the convergent Bochner integral $u(t)=T(t)x+{\int }_0^{t}T(t-s)f(s)ds,$ which is well defined by The variation-of-constants integral is continuous for integrable forcing. (4) An integral (integrated) solution is a continuous $u$ with ${\int }_0^{t}u(s)ds\in D(A)$ and $u(t)=x+A{\int }_0^{t}u(s)ds+{\int }_0^{t}f(s)ds$ for all $t$; for $f=0$ this is the integrated form of the homogeneous problem. These are four formulations, with overlaps and equivalences under additional hypotheses: neither differentiability nor membership of $u(t)$ in $D(A)$ is asserted by the mild definition, and the definitions do not assign a derivative to a mild solution.

Here “strong solution” means the graph-continuous integral notion explicitly stated above; some sources use that term for an almost-everywhere differential notion instead. Engel–Nagel II.6.3 calls the homogeneous integral formulation “mild”, whereas the variation-of-constants terminology here follows Schnaubelt Definition 2.11. The mild and integral formulations coincide for generators by the variation-of-constants theorem; they are not asserted to be different classes. Under DC the exponential bound required for the mild formulation is automatic, by Exponential bound for a C0-semigroup.

Classical solution. A classical solution is $C^1$ on the closed interval and takes values in the domain of $A$ at every time, including the endpoints; the equation $u\mathrm{\prime }(t)=Au(t)+f(t)$ is required pointwise on $(0,T_0)$, and the derivative is the Fréchet derivative of the curve (Fréchet derivative between Banach spaces). Since $A$ is generally unbounded, membership $u(t)\in D(A)$ and continuity of $Au$ are genuine restrictions. Together with continuity of $u$, they mean graph-norm continuity in $D(A)$ (Unbounded linear operators: domain, graph and extension).

Strong solution. A strong solution need not be differentiable; instead $Au$ is continuous, and the equation is imposed in integrated form $u(t)=x+{\int }_0^t(Au(s)+f(s))ds$, which makes sense because $Au$ is continuous and $f$ is Bochner integrable (Bochner-integrable function). If $Au+f$ on $(0,T_0)$ extends continuously to $[0,T_0]$, this identity gives $u\in C^1([0,T_0];X)$, with $u^{\mathrm{\prime }}=Au+f$ in the interior and one-sided endpoint derivatives equal to the extension values, by Fundamental theorem of calculus for Banach-valued continuous curves. Thus strong and classical solutions coincide when $f$ extends continuously to $[0,T_0]$; continuity only on $(0,T_0)$ does not ensure endpoint derivatives.

Mild solution. The mild solution is the explicit variation-of-constants function $u(t)=T(t)x+{\int }_0^{t}T(t-s)f(s)ds,$ which is a well-defined continuous $X$-valued function on $[0,T_0]$ by The variation-of-constants integral is continuous for integrable forcing; here $f$ need only be Bochner integrable, and no differentiability, no membership of $u(t)$ in $D(A)$ and no pointwise equation are asserted.

Integral (integrated) solution. An integral solution replaces differentiability by a weaker regularity: the primitive ${\int }_0^{t}u(s)ds$ lies in $D(A)$ for every $t$, and $u(t)=x+A{\int }_0^{t}u(s)ds+{\int }_0^{t}f(s)ds$ holds. For $f=0$ this is the integrated form of the homogeneous problem $u^{\mathrm{\prime }}=Au$, $u(0)=x$; its advantage is that it only evaluates $A$ on the primitive, which always lies in $D(A)$ when $u$ is a mild solution of the homogeneous problem.

The definitions do not by themselves assert implications between the four notions beyond the elementary ones visible above, and they do not assign a derivative to a mild solution. The precise equivalence results under additional hypotheses, and the separation of the notions when those hypotheses fail, are proved as theorems and exhibited by counterexamples on the companion pages.