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  "body": "\n## Definition\n\nLet $X$ be a [[Banach space|06GZ956657]], and let $A$ be the [[generator|60HSWZBTK9]] of a [[C0-semigroup|EVF180QJXY]] $(T(t))_{t \\ge 0}$ on $X$.\nLet $\\tau \u003e 0$, $u_0 \\in X$, and let $f \\colon [0, \\tau] \\to X$ be Bochner integrable.\nA *mild solution* of the initial value problem\n$$u'(t) = A u(t) + f(t) \\text{ for } 0 \u003c t \\le \\tau, \\qquad u(0) = u_0$$\nis the continuous function $u \\colon [0, \\tau] \\to X$ given by\n$$u(t) = T(t) u_0 + \\int_0^t T(t - s) f(s) \\, ds \\quad \\text{for } 0 \\le t \\le \\tau.$$\n\n## Remarks\n\nThe formula is the variation of constants formula, the analogue of [[Duhamel's principle|Q28DM6ZCMF]] for linear systems of ordinary differential equations.\nThe mild solution always exists and is unique.\nA classical solution, that is, a continuously differentiable $u$ with values in the domain of $A$ that satisfies the equation, is equal to the mild solution.\nConversely, if $u_0$ lies in the domain of $A$ and $f$ is continuously differentiable on $[0, \\tau]$, the mild solution is a classical solution.\n\nExample: let $X = \\R$, $A = a$ for a real number $a \\ne 0$, so $T(t) = e^{at}$, and let $f = 1$.\nThe mild solution is $u(t) = e^{at} u_0 + (e^{at} - 1)/a$, and it satisfies $u' = a e^{at} u_0 + e^{at} = a u + 1$ with $u(0) = u_0$.\n\nA mild solution need not be classical.\nLet $X = L^2(\\R)$, $A = d/dx$ with domain the Sobolev space $H^1(\\R)$, so $T(t)$ is the shift $(T(t) g)(x) = g(x + t)$, and let $f = 0$.\nFor $u_0$ the indicator function of $[0, 1]$, the mild solution $u(t) = u_0(\\cdot + t)$ is not differentiable as a map into $L^2(\\R)$, so the problem has no classical solution.\n\nNon-example: for $X = \\R$, $A = 0$ and $f = 0$, the function $u(t) = t$ is not a mild solution of $u' = 0$, $u(0) = 0$.\nHere $T(t) = 1$, so the formula forces $u(t) = 0$ for all $t$.\n",
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