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Jeff Dodd
Jeff Dodd
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I need to know about this non-linear logarithmic fast diffusion equation for a function $u(x,t)$ of one space variable $x$ and time $t$: $$ u_t = (\ln u)_{xx}$$ which is to run on an interval $ a \leq x \leq b $ with periodic boundary conditions $$ u(a) = u(b) $$$$ u(a,t) = u(b,t) $$ $$ u_x(a) = u_x(b) $$$$ u_x(a,t) = u_x(b,t) $$ andfor $t \geq 0$ and an initial condition $$ u(x,0) = f(x) $$
where $f$ is a smooth, strictly positive function defined for $ a \leq x \leq b $ (which itself satisfies the above boundary conditions).

In particular, I would like to be able to say that there is a smooth solution of this initial-boundary-value problem which approaches the constant equilibrium solution as $t \rightarrow \infty$. An extensive literature search has turned up similar results for similar problems, but nothing I can quote for this particular problem.

QUESTION: Does anyone know of any literature that addresses existence, uniqueness, regularity, and\or behavior as $t \rightarrow \infty$ of solutions of this initial-boundary-value problem? (Failing that, any advice or insight about this problem would be greatly appreciated!)

I need to know about this non-linear logarithmic fast diffusion equation for a function $u(x,t)$ of one space variable $x$ and time $t$: $$ u_t = (\ln u)_{xx}$$ which is to run on an interval $ a \leq x \leq b $ with periodic boundary conditions $$ u(a) = u(b) $$ $$ u_x(a) = u_x(b) $$ and an initial condition $$ u(x,0) = f(x) $$
where $f$ is a smooth, strictly positive function defined for $ a \leq x \leq b $ (which itself satisfies the above boundary conditions).

In particular, I would like to be able to say that there is a smooth solution of this initial-boundary-value problem which approaches the constant equilibrium solution as $t \rightarrow \infty$. An extensive literature search has turned up similar results for similar problems, but nothing I can quote for this particular problem.

QUESTION: Does anyone know of any literature that addresses existence, uniqueness, regularity, and\or behavior as $t \rightarrow \infty$ of solutions of this initial-boundary-value problem? (Failing that, any advice or insight about this problem would be greatly appreciated!)

I need to know about this non-linear logarithmic fast diffusion equation for a function $u(x,t)$ of one space variable $x$ and time $t$: $$ u_t = (\ln u)_{xx}$$ which is to run on an interval $ a \leq x \leq b $ with periodic boundary conditions $$ u(a,t) = u(b,t) $$ $$ u_x(a,t) = u_x(b,t) $$ for $t \geq 0$ and an initial condition $$ u(x,0) = f(x) $$
where $f$ is a smooth, strictly positive function defined for $ a \leq x \leq b $ (which itself satisfies the above boundary conditions).

In particular, I would like to be able to say that there is a smooth solution of this initial-boundary-value problem which approaches the constant equilibrium solution as $t \rightarrow \infty$. An extensive literature search has turned up similar results for similar problems, but nothing I can quote for this particular problem.

QUESTION: Does anyone know of any literature that addresses existence, uniqueness, regularity, and\or behavior as $t \rightarrow \infty$ of solutions of this initial-boundary-value problem? (Failing that, any advice or insight about this problem would be greatly appreciated!)

Source Link
Jeff Dodd
Jeff Dodd
  • 23
  • 4

The logarithmic fast diffusion equation in one space variable with periodic boundary conditions.

I need to know about this non-linear logarithmic fast diffusion equation for a function $u(x,t)$ of one space variable $x$ and time $t$: $$ u_t = (\ln u)_{xx}$$ which is to run on an interval $ a \leq x \leq b $ with periodic boundary conditions $$ u(a) = u(b) $$ $$ u_x(a) = u_x(b) $$ and an initial condition $$ u(x,0) = f(x) $$
where $f$ is a smooth, strictly positive function defined for $ a \leq x \leq b $ (which itself satisfies the above boundary conditions).

In particular, I would like to be able to say that there is a smooth solution of this initial-boundary-value problem which approaches the constant equilibrium solution as $t \rightarrow \infty$. An extensive literature search has turned up similar results for similar problems, but nothing I can quote for this particular problem.

QUESTION: Does anyone know of any literature that addresses existence, uniqueness, regularity, and\or behavior as $t \rightarrow \infty$ of solutions of this initial-boundary-value problem? (Failing that, any advice or insight about this problem would be greatly appreciated!)