Teaching CSIR NET Mock Test Series Mathematical Science Partial Differential Equations Laplace Equation, Heat and Wave Equations
Let u be the solution of
\(\left.\begin{array}{rl} \frac{\partial^2 u}{\partial t^2}=\frac{\partial^2 u}{\partial x^2}, & (x, t) \in \mathbb{R} \times(0, \infty), \\ u(x, 0)=f(x), & x \in \mathbb{R}, \\ u_t(x, 0)=g(x), & x \in \mathbb{R}, \end{array}\right\}\)
where f, g are in C2 (ℝ) and satisfy the following conditions.
(i) f(x) = g(x) = 0 for x ≤ 0,
(ii) 0 < f(x) ≤ 1 for x > 0,
(iii) g(x) > 0 for x > 0
(iv) \(\int_0^{∞}\)g(x)dx < ∞.
Then, which of the following statements are true?
1
u(x, t) = 0 for all x ≤ 0 and t > 0
2
u is bounded on ℝ × (0, ∞)
3
u(x, t) = 0 whenever x + t < 0
4
u(x, t) = 0 for some (x, t) satisfying x + t > 0