Subject: Partial differential equations Kindly solve it as soon as possible , kindly give it a try at least Thank you!!
0. "/>
Extracted text: 3) Consider the non-linear initial value problem Ofu – 0zu = (du)² – (d¿u)², x, t e R, (2) u(x, 0) = 0, du(x, 0) = g(x), g :R → R given. (a) Prove that if w = w(x, t) satisfies { Ofw – aw = 0, w(x,0) = 1, dw(x, 0) = g(x), x,t e R, then w(x, t) = e"(z,t). (b) Show that pI+t u(7, t) = log (1 + 5 9(e)ds). (c) Prove that if (3) | I9(s) ds < 2,="" then="" the="" solution="" u(x,="" t)="" of="" (2)="" is="" defined="" for="" all="" t="" e="" r.="" notice="" that="" if="" (3)="" fails,="" then="" the="" solution="" u(x,="" t)="" of="" (2)="" may="" only="" exist="" in="" a="" finite="" time="" interval="" [0,t),="" t=""> 0.