differentiel eqution

[mona123]

Problem. Consider an initial value problem for transport equation with nonlinear source
u_t − u_x = u²
x ∈ R, t > 0,
u(x, t = 0) = g(x).
Where g(x) is non-negative continuous function with continuous derivative that attains it’s maximum at a single point x = 0 and g(0) = 1.
a) solve initial value problem.
b) Show that solution becomes unbounded in finite time and thus solution for the problem exists only on
a finite time interval. Find maximal interval of existence.
c) Find coordinate of a point (x∗, t∗) at which solution of this problem becomes unbounded

 

[Ishuda]

You might consider the method of characteristics which leads to the 1D
u' = u²

 

[mona123]

hi
i tried with the first question and i find u(x,t)= g(x+t)/(1-tg(x,t))
is it the right answer ?.if it is please help me to solve the flowing questions.thanks

 

[Ishuda]

hi
i tried with the first question and i find u(x,t)= g(x+t)/(1-tg(x,t))
is it the right answer ?.if it is please help me to solve the flowing questions.thanks

Other than a minor mistake, I think it is. Correcting that mistake we have
u(x, t) = g(x+t) / [ 1 - t g(x+t) ]

Since g has only the single maximum, and is differentiable (and thus continuous) it can never increase. Since it is positive, it has some lower bound which is non-negative and less than 1 (the single max at zero is 1). Call that lower bound g₁ and we have
0 \(\displaystyle \le\) g₁ \(\displaystyle \le\) 1.
Since g is continuous, there is an s₂ such that
g(s₂) = (1+g₁)/2
(actually any value between 1 and g₁). If we let
t = 2/(1 + g₁) = 1/g(s₂) > 0
and
x = s₂ - t + \(\displaystyle \delta\),
then
x+t = s₂ + \(\displaystyle \delta\),
and
u(x,t) = g(s₂ + \(\displaystyle \delta\)) / [1 - g(s₂ + \(\displaystyle \delta\))/g(s₂)]
= g(s₂) g(s₂ + \(\displaystyle \delta\)) / [g(s₂) - g(s₂ + \(\displaystyle \delta\)) ]

What happens as \(\displaystyle \delta \to 0\)

EDIT: Fix subscript and note that since g(s) is monotonic for s from 0 to a (possibly unbounded) s₁ (g from 1 to g₁) we can take use any 1-1 function mapping s to f, i.e. t = 1/g(f(s)), s \(\displaystyle \epsilon\) [0,s₁) and f is 1-1 and go thought the above argument. Thus u is undefined everywhere along the curves t = 1/g(f(s)) and x = f(s) - t

 

[mona123]

differential equation

hi Ishuda
thank you for your help.can you please put the number correspanding of my problem to your answers.since i didnt understand the limit of each answer