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# hw9 - Math 425 Spring 2011 Jerry L Kazdan Problem Set 9 D...

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Math 425, Spring 2011 Jerry L. Kazdan Problem Set 9 D UE : Thursday April 7 [ Late papers will be accepted until 1:00 PM Friday ]. 1. a) In a bounded region Ω R n , let u ( x , t ) satisfy the modified heat equation u t = Δ u + cu , where c is a constant , (1) as well as the initial and boundary conditions u ( x , 0 ) = f ( x ) , in Ω with u ( x , t ) = 0 for x ∂Ω , t 0 . (2) Let u ( x , t ) = v ( x , t ) e α t . Show that by picking the constant α cleverly, v satisfies equation (1) with c = 0 as well as (2). Moral: one can easily reduce understanding equations (1)-(2) to the special case c = 0. b) Generalize this to u t + a ( t ) u = Δ u where a ( t ) is any continuous function by seeking u ( x , t ) = ϕ ( t ) v ( x , t ) and picking the function ϕ ( t ) cleverly, 2. In a bounded region Ω R n , use the maximum principle to prove a uniqueness theorem for solutions u ( x , t ) of the inhomogeneous equation u t - Δ u = F ( x , t ) in Ω with u ( x , 0 ) = f ( x ) , in Ω and u ( x , t ) = ϕ ( x , t ) for x ∂Ω , t 0 . 3. Let Ω R n be a bounded region with smooth boundary ∂Ω and let u ( x , t ) satisfy the heat equation u t = Δ u for

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hw9 - Math 425 Spring 2011 Jerry L Kazdan Problem Set 9 D...

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