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  "body": "\n## Definition\n\nLet $u$ be a solution of an evolution equation on $\\Omega \\times [0, T]$, with $\\Omega \\subseteq \\R^n$ open.\nThe *energy method* is the following kind of argument.\n\n1. Choose an energy $E(t) \\ge 0$, an integral over $\\Omega$ of a nonnegative quadratic expression in $u(\\cdot, t)$ and its derivatives.\n2. Differentiate $E$ in $t$, use the equation, and integrate by parts with the boundary conditions. The aim is a bound $E'(t) \\le C E(t) + F(t)$ with a constant $C \\ge 0$ and a known function $F \\ge 0$.\n3. Conclude by Grönwall's inequality that $E(t) \\le e^{Ct}\\big(E(0) + \\int_0^t F(s)\\, ds\\big)$ for $0 \\le t \\le T$.\n\nApply the argument to the difference of two solutions of a linear problem, with $F = 0$.\nIt gives uniqueness and continuous dependence on the data, as required for a [[well-posed problem|W09B9NZ9B5]].\n\n## Remarks\n\nExample: let $\\Omega$ be bounded with smooth boundary and let $u \\in C^2(\\overline{\\Omega} \\times [0, T])$ solve the [[wave equation|5EHD2XBQDG]] $u_{tt} = \\Delta u$ with $u = 0$ on $\\partial\\Omega$.\nTake $E(t) = \\frac{1}{2}\\int_\\Omega \\big(u_t^2 + \\abs{\\nabla u}^2\\big)\\, dx$, where $\\Delta$ is the [[Laplacian|0E0B6RA8K0]].\nThen $E'(t) = \\int_\\Omega (u_t u_{tt} + \\nabla u \\cdot \\nabla u_t)\\, dx = \\int_\\Omega u_t (u_{tt} - \\Delta u)\\, dx = 0$, since the boundary term contains $u_t = 0$ on $\\partial\\Omega$.\nSo $E$ is constant, and two solutions with the same data and boundary values coincide.\nFor the heat equation $u_t = \\Delta u$ with $u = 0$ on $\\partial\\Omega$, the energy $E(t) = \\frac{1}{2}\\int_\\Omega u^2\\, dx$ satisfies $E'(t) = -\\int_\\Omega \\abs{\\nabla u}^2\\, dx \\le 0$.\nNon-example: for the backward heat equation $u_t = -\\Delta u$ the same computation gives $E'(t) = \\int_\\Omega \\abs{\\nabla u}^2\\, dx$, which is not bounded by $C E(t)$, so step 2 fails; indeed that problem is ill-posed.\n",
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