2 heat equation.pdf - NOTES ON THE HEAT EQUATION The function(1 \u03a6(t x:= 1 n(4\u03c0t 2 e\u2212 0 |x|2 4t t>0 t \u2264 0 x 6= 0 satisfies(\u2202t \u2212 \u2206)\u03a6 = 0 for

2 heat equation.pdf - NOTES ON THE HEAT EQUATION The...

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NOTES ON THE HEAT EQUATION The function Φ( t, x ) := ( 1 (4 πt ) n 2 e - | x | 2 4 t , t > 0 0 , t 0 , x 6 = 0 (1) satisfies ( t - Δ)Φ = 0 for all ( t, x ) 6 = (0 , 0). Need also that Z R n Φ( t, x ) dx = 1 for all t > 0; hence the family of functions { Φ( t, · ) } t> 0 is an approximate identity on R n . The initial value problem in Euclidean space The fundamental solution will allow us to fashion solutions to the initial value problem t u - Δ u = f, on (0 , ) × R n , u = g on { t = 0 } × R n , , (2) where f ( t, x ) and g ( x ) are given functions. We consider first the homogeneous problem, when f = 0: t u - Δ u = 0 , on (0 , ) × R n , u = g on { t = 0 } × R n , , (3) Theorem 0.1. Assume g C ( R n ) is bounded, and for t > 0 define u ( t, x ) := Z R n Φ( t, x - y ) g ( y ) dy. (4) Then (1) u is smooth on (0 , ) × R n . (2) ( t - Δ) u = 0 on (0 , ) × R n . (3) For any x 0 R n , lim ( t,x ) (0 ,x 0 ) = g ( x 0 ) . Proof. (1) ( t, x ) Φ( t, x ) is smooth all derivatves bounded on [ δ, ) × R n , and recall that by the properties of convolution, Φ( t, · ) * g is smooth as a function of x . Also, using the equation t Φ = ΔΦ to convert t derivatives to x derivatives, we see that Φ( t, · ) * g is smooth as a function of t as well. (2) t - Δ x Φ( t, x - y ) = 0 for each fixed y . (3) { Φ( t, · ) } t> 0 is an approximate identity on R n . Remarks. Smoothing property of the heat equation. Initial value problem with data given at time s . 1
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One often uses the notation u ( t, x ) = ( e t Δ g )( x ) to denote the solution u with initial condition u = g . The expression e t Δ
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