Partial Differentials Notes_Part_4

Partial Differentials Notes_Part_4 - Theorem 12.1. Suppose...

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Unformatted text preview: Theorem 12.1. Suppose u ( t, x, y ) is a solution to the forced heat equation u t = u + F ( t, x, y ) , for ( x, y ) , < t < c, where is a bounded domain, and > . Suppose F ( t, x, y ) for all ( x, y ) and t c . Then the global maximum of u on the set { ( t, x, y ) | ( x, y ) , t c } , occurs either when t = 0 , or at a boundary point ( x, y ) . Proof : First, let us prove the result under the assumption that F ( t, x, y ) < 0 ev- erywhere. At a local interior maximum, u t = 0, and, since its Hessian matrix must be negative semi-definite, u xx , u yy 0, which would imply that u t u 0. If the maxi- mum were to occur when t = c , then u t 0 there, and also u xx , u yy 0, leading again to a contradiction. To generalize to the case when F ( t, x, y ) 0, which includes the heat equation when F ( t, x, y ) 0, set v ( t, x, y ) = u ( t, x, y ) + ( x 2 + y 2 ) , where > . Then, v t = v 4 + F ( t, x, y ) = v + tildewide F ( t, x, y ) , where tildewide F ( t, x, y ) = F ( t, x, y ) 4 < . Thus, by the previous paragraph, the maximum of v occurs either when t = 0 or at a boundary point ( x, y ) . Now we let 0 and conclude the same for u . More precisely, let u ( t, x, y ) M on t = 0 or ( x, y ) . Then v ( t, x, y ) M + C where C = max braceleftbig x 2 + y 2 vextendsingle vextendsingle ( x, y ) bracerightbig is finite since is a bounded domain. Thus, u ( t, x, y ) v ( t, x, y ) M + C ....
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Partial Differentials Notes_Part_4 - Theorem 12.1. Suppose...

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