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Unformatted text preview: Homework Set 3: Math 716, Due: Wednesday, February 7th 1. Use energy method to prove uniqueness of classical solution to the initial value problem for the damped wave equation ( ǫ > 0): u tt + ǫu t = Δ u for x ∈ Ω ⊂ R n , t > , with u ( x , 0) = φ ( x ) , u t ( x , 0) = ψ ( x ) , u ( x , t ) = 0 on ∂ Ω Solution: Define the Energy the same way as for the undamped ( ǫ = 0 case) wave equation: E ( t ) = 1 2 integraldisplay Ω bracketleftbig u 2 t + ( ∇ u ) 2 bracketrightbig dx Then, using equation, dE dt = integraldisplay Ω [ u t u tt + [ ∇ u ] · [ ∇ u t ]] dx = integraldisplay Ω bracketleftbig u t Δ u − ǫu 2 t + [ ∇ u ] · [ ∇ u t ] bracketrightbig dx However, u t Δ u + [ ∇ u ] · [ ∇ u t ] = ∇· [ u t ∇ u ] Assuming ∂ Ω is regular enough to apply divergence theorem, we obtain from the boundary condition u ( x, t ) = 0 on ∂ Ω (and hence u t ( x, t ) = 0 on ∂ Ω), we obtain integraldisplay Ω [ u t Δ u + [ ∇ u ] · [ ∇ u t ]] dx = integraldisplay ∂ Ω ∂u ∂n u t dx = 0 Therefore, dE dt = − ǫ integraldisplay Ω u 2 t dx ≤ Therefore, E ( t ) ≤ E (0) To prove uniqueness, consider two solutions u 1 , u 2 of the wave equation satisfying homo- geneous BC on ∂ Ω and same initial condition. Then u = u 1 − u 1 will satisfy zero initial and boundary conditions. Hence from the above calculation E ( t ) ≤ E (0) = 1 2 integraldisplay Ω bracketleftbig u 2 t ( x, 0) + ( ∇ u ) 2 ( x, 0) bracketrightbig dx = 0 Therefore, E ( t ) = 0 implying from the expression for E that for any classical solution, u t ( x, t ) = 0 = ∇ u ( x, t ). This means u ( x, t ) = constant, independent of x and t . But since u ( x, 0) = 0, u ( x, t ) = 0, proving uniqueness of initial value problem....
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This note was uploaded on 07/26/2011 for the course MATH 716 taught by Professor Tanveer,s during the Spring '08 term at Ohio State.
- Spring '08