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220-HW1

# 220-HW1 - Problem 6 Consider the PDE u t uu x = 0 u x 0 =...

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MATH 220: PROBLEM SET 1 DUE THURSDAY, OCTOBER 1, 2009 Problem 1. Classify the following PDEs by degree of non-linearity (linear, semi- linear, quasilinear, fully nonlinear): (1) (cos x ) u x + u y = u 2 . (2) uu tt = u xx . (3) u x - e x u y = cos x. (4) u tt - u xx + e u u x = 0 . Problem 2. (1) Solve u x + (sin x ) u y = y, u (0 , y ) = 0 . (2) Sketch the projected characteristic curves for this PDE. Problem 3. (1) Solve yu x + xu y = 0 , u (0 , y ) = e y 2 . (2) In which region is u uniquely determined? Problem 4. (1) Solve u x + u t = u 2 , u ( x, 0) = e x 2 . (2) Show that there is T > 0 such that u blows up at time T , i.e. u is continuously differentiable for t [0 , T ), x arbitrary, but for some x 0 , | u ( x 0 , t ) | → ∞ as t T - . What is T ? Problem 5. Solve u t + uu x = 0 , u ( x, 0) = - x 2 for | t | small.
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Unformatted text preview: Problem 6. Consider the PDE u t + uu x = 0 , u ( x, 0) = φ ( x ) . Suppose that φ ′ ≥ -C , where C > 0. Show that the PDE has a C 1 solution on R x × [0 , 1 C ) t . Show also that for t ∈ [0 , 1 C ), u x satis±es the estimate u x ( x, t ) ≥ 1 t-C − 1 . (Note that the right hand side is negative!) (Hint: Consider the diFerence quotients u ( ξ 2 ( t ) ,t ) − u ( ξ 1 ( t ) ,t ) ξ 2 ( t ) − ξ 1 ( t ) , where x = ξ j ( t ) are the projected characteristic curves emanating from the point x j on the x axis.) 1...
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