(a,b). Prove that for Suppose u solves utt - UTT 0 for x < x 0...

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(a,b). Prove that for Suppose u solves utt - ²UTT 0 for x < x <∞ and t> 0 with initial data u(x,0) = f(x) and ut(x,0) = g(x). Suppose that both f and g have compact support, meaning that there is a finite interval (a, b) such that f(x) = 0 and g(x) = 0 whenever x any t > 0, the function x→u(x, t) also has compact support, in the sense that u(x, t) = 0 whenever lies outside some finite interval. Can you describe the smallest such interval in terms of a, b,c and t? Assume as in question 3 that f and g have compact support. Establish that the energy of a solution u(x, t) defined by ² E(t) := [ ½ (0)² + = 2 (0₂)²³ d² -(ur)² dx is conserved, i.e. constant for all positive time. (a,b). Prove that for Suppose u solves utt - ²UTT 0 for x < x <∞ and t> 0 with initial data u(x,0) = f(x) and ut(x,0) = g(x). Suppose that both f and g have compact support, meaning that there is a finite interval (a, b) such that f(x) = 0 and g(x) = 0 whenever x any t > 0, the function x→u(x, t) also has compact support, in the sense that u(x, t) = 0 whenever lies outside some finite interval. Can you describe the smallest such interval in terms of a, b,c and t? Assume as in question 3 that f and g have compact support. Establish that the energy of a solution u(x, t) defined by ² E(t) := [ ½ (0)² + = 2 (0₂)²³ d² -(ur)² dx is conserved, i.e. constant for all positive time.