Help Implicit funtions theorem question

[Beren1936]

Let

\begin{cases}
u_t(x,t) + u(x,t)u_x(x,t) = 0, & \text{for } x\in\mathbb{R},\; t > 0 \\
u(x,0) = h(x), & \text{for } x\in\mathbb{R}.
\end{cases}

Suppose that h ∈ C²(R) and that h′(x)>0 for all x ∈ R. Check, with the help of implicit function theorem, for (x , t)sufficiently close to (x₀ , 0) (where x₀ ∈ R is a fixed number), that the equation

u=h(x−tu)

defines a solution to the Cauchy problem (for x sufficiently close to x₀ and t close to zero).

 

[Deleted member 4993]

Let

\begin{cases}
u_t(x,t) + u(x,t)u_x(x,t) = 0, & \text{for } x\in\mathbb{R},\; t > 0 \\
u(x,0) = h(x), & \text{for } x\in\mathbb{R}.
\end{cases}

Suppose that h ∈ C²(R) and that h′(x)>0 for all x ∈ R. Check, with the help of implicit function theorem, for (x , t)sufficiently close to (x₀ , 0) (where x₀ ∈ R is a fixed number), that the equation

u=h(x−tu)

defines a solution to the Cauchy problem (for x sufficiently close to x₀ and t close to zero).

Since you did not show any work, we need to make sure that we all agree on definitions. Please tell us:

• What is implicit function theorem?
  
  
• What is the statement of Cauchy problem?

Please show us what you have tried and exactly where you are stuck.

Please follow the rules of posting in this forum, as enunciated at:

READ BEFORE POSTING​

Please share your work/thoughts about these problems.

 

[Beren1936]

Since you did not show any work, we need to make sure that we all agree on definitions. Please tell us:

• What is implicit function theorem?
  
  
• What is the statement of Cauchy problem?

Please show us what you have tried and exactly where you are stuck.

Please follow the rules of posting in this forum, as enunciated at:

READ BEFORE POSTING​

Please share your work/thoughts about these problems.

i try that:
du(x₀ , 0) + du/dx(x₀ , 0). h(x) = 0


du/dx(x₀ , 0). h(x) = du(x₀ , 0)

from here, I don't know how to proceed

I thought something like that too to start ...

F(x, t , u(x,t)) = 0

F(x, t , u(x,t)) = u_t(x,t)+u(x,t)u_x(x,t)

Some advice?

 

[Deleted member 4993]

i try that:
du(x₀ , 0) + du/dx(x₀ , 0). h(x) = 0


du/dx(x₀ , 0). h(x) = du(x₀ , 0)

from here, I don't know how to proceed

I thought something like that too to start ...

F(x, t , u(x,t)) = 0

F(x, t , u(x,t)) = u_t(x,t)+u(x,t)u_x(x,t)

Some advice?

But you did not answer:

• What is implicit function theorem?
  
  
• What is the statement of Cauchy problem?