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Let $\Omega \subset \mathbb{R}^n$ be open bounded. We find a function $u:\overline\Omega \rightarrow R$ satisfying $-\Delta u + u = f$ in $\Omega$, $u=0$ on $\partial\Omega$ ---(*)

Define strong solution (classical solution) of (*) in an appropriate Banach space.

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Strong and classical are not the same. Classical should be clear: a point-wise solution of the equation, and therefore a function $u$ of class (at least) $C^2$ that satisfies (*) at each point of $\Omega$.

A strong solution is something different, at least in principle: it is usually a twice weakly differentiable function $u$ that satisfies (*) almost everywhere. This is the definition in Gilbarg, Trudinger: Elliptic partial differential equations of second order, $2^{\mathrm{nd}}$ edition.

Edit: in particular, the concept of strong solutions is confined to variational equations. However, it seems that other Authors write strong to mean classical.

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What about in evolution equations? – Ylath Jan 15 at 3:46

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