hw4solns.pdf - APM346 HOMEWORK 4 Notes • Remember the...

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APM346 HOMEWORK 4NotesRemember thetriangle inequalityfor sums and integrals:|z1+· · ·+zk| ≤ |z1|+· · ·+|zk|,Zbaf(x)dxZba|f(x)|dx.Note that the fundamental solutionK(t, x) = (4πt)-12exp(-x24t)of the heat equationis strictlypositivefor allt >0,xR.Problem 4 will be a straightforward application of Thursday’s lecture (or read thefirst part of Strauss 3.4 if you’d like to get ahead).Problems1. Letgbe a bounded continuous function onRand letube the bounded solution to(t-2x)u= 0,u|t=s=g.Assume there is a constantM >0 such that|g(x)| ≤Mfor allx.Using therepresentation ofuin terms of the fundamental solution, show that|u(t, x)| ≤Mforallt > sandxR.Solution.Writeu(t, x) =g*Kt-s(x), whereKt(x) =K(t, x). Then using the trian-gle inequality for integrals, the positivity ofK, and the property thatR-∞K(t, x)dx=1 for anyt >0, we deduce|u(t, x)| ≤Z-∞|K(t-s, x-y)||g(y)|dyMZ-∞K(t-s, x-y)dy=M.

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