A nonlinear differential inequality
I was trying to prove a generalisation of maximum principle and for that purpose I added a correction term. After some manipulations the condition I was looking for was reduced the following nonlinear differential inequality for some $C^2$ function $x : mathbb{R} to mathbb{R}$
$$x'' + (x')^2 < 0 qquad text{in} quad mathbb{R} $$
I haven't been able to construct any such $x$. Is the corresponding ODE some standard form? My knowledge of ODEs is very limited so any ideas/hints are welcome.
real-analysis differential-equations
asked Dec 4 '18 at 10:47
OhDaeSuOhDaeSu
1173
add a comment |
I was trying to prove a generalisation of maximum principle and for that purpose I added a correction term. After some manipulations the condition I was looking for was reduced the following nonlinear differential inequality for some $C^2$ function $x : mathbb{R} to mathbb{R}$
$$x'' + (x')^2 < 0 qquad text{in} quad mathbb{R} $$
I haven't been able to construct any such $x$. Is the corresponding ODE some standard form? My knowledge of ODEs is very limited so any ideas/hints are welcome.
real-analysis differential-equations
asked Dec 4 '18 at 10:47
OhDaeSuOhDaeSu
1173
add a comment |
I was trying to prove a generalisation of maximum principle and for that purpose I added a correction term. After some manipulations the condition I was looking for was reduced the following nonlinear differential inequality for some $C^2$ function $x : mathbb{R} to mathbb{R}$
$$x'' + (x')^2 < 0 qquad text{in} quad mathbb{R} $$
I haven't been able to construct any such $x$. Is the corresponding ODE some standard form? My knowledge of ODEs is very limited so any ideas/hints are welcome.
real-analysis differential-equations
asked Dec 4 '18 at 10:47
OhDaeSuOhDaeSu
1173
I was trying to prove a generalisation of maximum principle and for that purpose I added a correction term. After some manipulations the condition I was looking for was reduced the following nonlinear differential inequality for some $C^2$ function $x : mathbb{R} to mathbb{R}$
$$x'' + (x')^2 < 0 qquad text{in} quad mathbb{R} $$
I haven't been able to construct any such $x$. Is the corresponding ODE some standard form? My knowledge of ODEs is very limited so any ideas/hints are welcome.
real-analysis differential-equations
real-analysis differential-equations
asked Dec 4 '18 at 10:47
OhDaeSuOhDaeSu
1173
asked Dec 4 '18 at 10:47
OhDaeSuOhDaeSu
1173
asked Dec 4 '18 at 10:47
OhDaeSuOhDaeSu
1173
asked Dec 4 '18 at 10:47
OhDaeSuOhDaeSu
1173
asked Dec 4 '18 at 10:47
OhDaeSuOhDaeSu
1173
1173
add a comment |
add a comment |
1 Answer
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Note that you are demanding the function $gcolon tmapstoexp(x(t))$ to have strictly negative second derivative on the whole of $mathbb{R}$, in particular, it must be concave and bounded below. This cannot happen:- the derivative must be nonzero somewhere, WLOG at $t=0$, then the graph lies below the support line at $0$ so must $to-infty$ at one of $pminfty$.
answered Dec 4 '18 at 11:39
user10354138user10354138
7,4122925
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1 Answer
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1 Answer
1
active
oldest
votes
active
oldest
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active
oldest
votes
Note that you are demanding the function $gcolon tmapstoexp(x(t))$ to have strictly negative second derivative on the whole of $mathbb{R}$, in particular, it must be concave and bounded below. This cannot happen:- the derivative must be nonzero somewhere, WLOG at $t=0$, then the graph lies below the support line at $0$ so must $to-infty$ at one of $pminfty$.
answered Dec 4 '18 at 11:39
user10354138user10354138
7,4122925
add a comment |
Note that you are demanding the function $gcolon tmapstoexp(x(t))$ to have strictly negative second derivative on the whole of $mathbb{R}$, in particular, it must be concave and bounded below. This cannot happen:- the derivative must be nonzero somewhere, WLOG at $t=0$, then the graph lies below the support line at $0$ so must $to-infty$ at one of $pminfty$.
answered Dec 4 '18 at 11:39
user10354138user10354138
7,4122925
add a comment |
Note that you are demanding the function $gcolon tmapstoexp(x(t))$ to have strictly negative second derivative on the whole of $mathbb{R}$, in particular, it must be concave and bounded below. This cannot happen:- the derivative must be nonzero somewhere, WLOG at $t=0$, then the graph lies below the support line at $0$ so must $to-infty$ at one of $pminfty$.
answered Dec 4 '18 at 11:39
user10354138user10354138
7,4122925
Note that you are demanding the function $gcolon tmapstoexp(x(t))$ to have strictly negative second derivative on the whole of $mathbb{R}$, in particular, it must be concave and bounded below. This cannot happen:- the derivative must be nonzero somewhere, WLOG at $t=0$, then the graph lies below the support line at $0$ so must $to-infty$ at one of $pminfty$.
answered Dec 4 '18 at 11:39
user10354138user10354138
7,4122925
answered Dec 4 '18 at 11:39
user10354138user10354138
7,4122925
answered Dec 4 '18 at 11:39
user10354138user10354138
7,4122925
answered Dec 4 '18 at 11:39
user10354138user10354138
7,4122925
7,4122925
add a comment |
add a comment |
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