ADVANCED Partial Differential Equations QUESTION: Let f be a smooth function with continuous second order derivatives with f''=>1

Problem 2 (20 pts) Let f be a smooth function with continuous second order derivatives with f" > 1. Consider the following initial value probem S l Ut + f(u)x = -U, X ER, t > 0, u(x,0) = uo(a) e Cl(R). (a) Determine the sufficient and necessary conditions on the initial data Uo(2) for this problem to have a unique global smooth solution. (b) If yo(x) is continuously differentiable and is uniformly bounded in 2, and satisfies the conditions found in part a), show that the global solution u(x, t) satisfies that ||u(x, t) || [20(R) Converges to zero as t goes to infinity. Problem 2 (20 pts) Let f be a smooth function with continuous second order derivatives with f" > 1. Consider the following initial value probem S l Ut + f(u)x = -U, X ER, t > 0, u(x,0) = uo(a) e Cl(R). (a) Determine the sufficient and necessary conditions on the initial data Uo(2) for this problem to have a unique global smooth solution. (b) If yo(x) is continuously differentiable and is uniformly bounded in 2, and satisfies the conditions found in part a), show that the global solution u(x, t) satisfies that ||u(x, t) || [20(R) Converges to zero as t goes to infinity