Increasing property of solution of heat equation
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Let $$u_t - Delta u = f$$
with $u(t,0) = u(t,1) = 0$ and $u(0,x) = u_0(x)$ given, and $f(t,x)$ is also given. This is the heat equation on the interval $(0,1)$.
How can I choose $f$ and $u_0$ to ensure that $u$ is increasing with respect to time? I.e. if $t geq s$ then $u(t,x) geq u(s,x)$ for a.e. $x$?
If $f equiv 0$, then $u$ is decreasing with respect to $t$. But I don't know how to ensure that it becomes increasing. Does anyone know?
functional-analysis pde sobolev-spaces heat-equation
asked Nov 14 at 11:21
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up vote
0
down vote
favorite
Let $$u_t - Delta u = f$$
with $u(t,0) = u(t,1) = 0$ and $u(0,x) = u_0(x)$ given, and $f(t,x)$ is also given. This is the heat equation on the interval $(0,1)$.
How can I choose $f$ and $u_0$ to ensure that $u$ is increasing with respect to time? I.e. if $t geq s$ then $u(t,x) geq u(s,x)$ for a.e. $x$?
If $f equiv 0$, then $u$ is decreasing with respect to $t$. But I don't know how to ensure that it becomes increasing. Does anyone know?
functional-analysis pde sobolev-spaces heat-equation
asked Nov 14 at 11:21
StopUsingFacebook
1678
1
For example if $f=f(t)$ then $u$ is independent on $x$ and satisfies $u_t=f$, so it is increasing if $fge 0$.
– Giuseppe Negro
Nov 14 at 11:40
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Let $$u_t - Delta u = f$$
with $u(t,0) = u(t,1) = 0$ and $u(0,x) = u_0(x)$ given, and $f(t,x)$ is also given. This is the heat equation on the interval $(0,1)$.
How can I choose $f$ and $u_0$ to ensure that $u$ is increasing with respect to time? I.e. if $t geq s$ then $u(t,x) geq u(s,x)$ for a.e. $x$?
If $f equiv 0$, then $u$ is decreasing with respect to $t$. But I don't know how to ensure that it becomes increasing. Does anyone know?
functional-analysis pde sobolev-spaces heat-equation
asked Nov 14 at 11:21
StopUsingFacebook
1678
Let $$u_t - Delta u = f$$
with $u(t,0) = u(t,1) = 0$ and $u(0,x) = u_0(x)$ given, and $f(t,x)$ is also given. This is the heat equation on the interval $(0,1)$.
How can I choose $f$ and $u_0$ to ensure that $u$ is increasing with respect to time? I.e. if $t geq s$ then $u(t,x) geq u(s,x)$ for a.e. $x$?
If $f equiv 0$, then $u$ is decreasing with respect to $t$. But I don't know how to ensure that it becomes increasing. Does anyone know?
functional-analysis pde sobolev-spaces heat-equation
functional-analysis pde sobolev-spaces heat-equation
asked Nov 14 at 11:21
StopUsingFacebook
1678
asked Nov 14 at 11:21
StopUsingFacebook
1678
asked Nov 14 at 11:21
StopUsingFacebook
1678
asked Nov 14 at 11:21
StopUsingFacebook
1678
asked Nov 14 at 11:21
StopUsingFacebook
1678
1678
1
For example if $f=f(t)$ then $u$ is independent on $x$ and satisfies $u_t=f$, so it is increasing if $fge 0$.
– Giuseppe Negro
Nov 14 at 11:40
add a comment |
1
For example if $f=f(t)$ then $u$ is independent on $x$ and satisfies $u_t=f$, so it is increasing if $fge 0$.
– Giuseppe Negro
Nov 14 at 11:40
1
1
For example if $f=f(t)$ then $u$ is independent on $x$ and satisfies $u_t=f$, so it is increasing if $fge 0$.
– Giuseppe Negro
Nov 14 at 11:40
For example if $f=f(t)$ then $u$ is independent on $x$ and satisfies $u_t=f$, so it is increasing if $fge 0$.
– Giuseppe Negro
Nov 14 at 11:40
add a comment |
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For example if $f=f(t)$ then $u$ is independent on $x$ and satisfies $u_t=f$, so it is increasing if $fge 0$.
– Giuseppe Negro
Nov 14 at 11:40