Mixing
Blow-up conditions for an ODE with perspective nonlinearity
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I would like to obtain necessary and sufficient conditions on $f$ (within a certain class of $f$, defined below) for finite-time blow-up of ALL positive solutions of the non-autonomous ODE
$$x^{prime}=tf(x/t),qquad x(t_0)=x_0 qquadqquad (*)$$
i.e. that the solution blows-up in finite (forwards) time ("BUFT") for every choice of $t_0>0$ and $x_0>0$. The RHS of (*) is actually the perspective of the function $f$ and is a convex function in $(x,t)$ whenever $f$ is convex.
Here $f:[0,infty)to [0,infty) $ is locally Lipschitz continuous (but $C^1$ would be ok), non-decreasing, $f(0)=0$ and convex. We also assume that the ODE $x^{prime}=f(x)$ BUFT, i.e.,
$$int_{1}^infty frac{dx}{f(x)}<infty .$$
With these assumptions one can easily show that not all positive solutions BUFT if
$$int_{0^{+}} frac{f(x)}{x^3}<infty . qquadqquad (**)$$
My question is: does failure of (**) imply BUFT of all positive solutions of (*)?
It is easy to show that (**) implies that $x(t)toinfty$ as $ttoinfty$, if $x$ exists globally. With additional assumptions upon $f$ one can also show BUFT, but this is not the sharp result I seek. (It may be instructive to think of $f(x)=x^p$ ($p>1$) as a special case first.)
ordinary-differential-equations analysis convex-analysis blowup
asked Feb 2 at 12:36
RL18RL18
112
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add a comment |
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I would like to obtain necessary and sufficient conditions on $f$ (within a certain class of $f$, defined below) for finite-time blow-up of ALL positive solutions of the non-autonomous ODE
$$x^{prime}=tf(x/t),qquad x(t_0)=x_0 qquadqquad (*)$$
i.e. that the solution blows-up in finite (forwards) time ("BUFT") for every choice of $t_0>0$ and $x_0>0$. The RHS of (*) is actually the perspective of the function $f$ and is a convex function in $(x,t)$ whenever $f$ is convex.
Here $f:[0,infty)to [0,infty) $ is locally Lipschitz continuous (but $C^1$ would be ok), non-decreasing, $f(0)=0$ and convex. We also assume that the ODE $x^{prime}=f(x)$ BUFT, i.e.,
$$int_{1}^infty frac{dx}{f(x)}<infty .$$
With these assumptions one can easily show that not all positive solutions BUFT if
$$int_{0^{+}} frac{f(x)}{x^3}<infty . qquadqquad (**)$$
My question is: does failure of (**) imply BUFT of all positive solutions of (*)?
It is easy to show that (**) implies that $x(t)toinfty$ as $ttoinfty$, if $x$ exists globally. With additional assumptions upon $f$ one can also show BUFT, but this is not the sharp result I seek. (It may be instructive to think of $f(x)=x^p$ ($p>1$) as a special case first.)
ordinary-differential-equations analysis convex-analysis blowup
asked Feb 2 at 12:36
RL18RL18
112
$endgroup$
add a comment |
$begingroup$
I would like to obtain necessary and sufficient conditions on $f$ (within a certain class of $f$, defined below) for finite-time blow-up of ALL positive solutions of the non-autonomous ODE
$$x^{prime}=tf(x/t),qquad x(t_0)=x_0 qquadqquad (*)$$
i.e. that the solution blows-up in finite (forwards) time ("BUFT") for every choice of $t_0>0$ and $x_0>0$. The RHS of (*) is actually the perspective of the function $f$ and is a convex function in $(x,t)$ whenever $f$ is convex.
Here $f:[0,infty)to [0,infty) $ is locally Lipschitz continuous (but $C^1$ would be ok), non-decreasing, $f(0)=0$ and convex. We also assume that the ODE $x^{prime}=f(x)$ BUFT, i.e.,
$$int_{1}^infty frac{dx}{f(x)}<infty .$$
With these assumptions one can easily show that not all positive solutions BUFT if
$$int_{0^{+}} frac{f(x)}{x^3}<infty . qquadqquad (**)$$
My question is: does failure of (**) imply BUFT of all positive solutions of (*)?
It is easy to show that (**) implies that $x(t)toinfty$ as $ttoinfty$, if $x$ exists globally. With additional assumptions upon $f$ one can also show BUFT, but this is not the sharp result I seek. (It may be instructive to think of $f(x)=x^p$ ($p>1$) as a special case first.)
ordinary-differential-equations analysis convex-analysis blowup
asked Feb 2 at 12:36
RL18RL18
112
$endgroup$
I would like to obtain necessary and sufficient conditions on $f$ (within a certain class of $f$, defined below) for finite-time blow-up of ALL positive solutions of the non-autonomous ODE
$$x^{prime}=tf(x/t),qquad x(t_0)=x_0 qquadqquad (*)$$
i.e. that the solution blows-up in finite (forwards) time ("BUFT") for every choice of $t_0>0$ and $x_0>0$. The RHS of (*) is actually the perspective of the function $f$ and is a convex function in $(x,t)$ whenever $f$ is convex.
Here $f:[0,infty)to [0,infty) $ is locally Lipschitz continuous (but $C^1$ would be ok), non-decreasing, $f(0)=0$ and convex. We also assume that the ODE $x^{prime}=f(x)$ BUFT, i.e.,
$$int_{1}^infty frac{dx}{f(x)}<infty .$$
With these assumptions one can easily show that not all positive solutions BUFT if
$$int_{0^{+}} frac{f(x)}{x^3}<infty . qquadqquad (**)$$
My question is: does failure of (**) imply BUFT of all positive solutions of (*)?
It is easy to show that (**) implies that $x(t)toinfty$ as $ttoinfty$, if $x$ exists globally. With additional assumptions upon $f$ one can also show BUFT, but this is not the sharp result I seek. (It may be instructive to think of $f(x)=x^p$ ($p>1$) as a special case first.)
ordinary-differential-equations analysis convex-analysis blowup
ordinary-differential-equations analysis convex-analysis blowup
asked Feb 2 at 12:36
RL18RL18
112
asked Feb 2 at 12:36
RL18RL18
112
edited Feb 7 at 10:05
RL18
asked Feb 2 at 12:36
RL18RL18
112
asked Feb 2 at 12:36
RL18RL18
112
asked Feb 2 at 12:36
RL18RL18
112
112
add a comment |
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