The solution of heat equation satisfies $u(x,t)rightarrow 0$ as $|x|+trightarrow infty$
up vote
0
down vote
favorite
Suppose $u$ solution of the heat equation $u_{t}=u_{xx}$, where $fin S(mathbb{R})$ belongs to Schwartz space, $u(x,t)=left(f*mathcal{H}_{t}right)(x)$ setting $u(x,0)=f(x)$.
I need to prove that $u$ is continuous on the closure of the upper half-plane, and vanishes at infinity, that is,
$$u(x,t)rightarrow 0quadtextrm{as}quad |x|+trightarrow infty. $$
The hint is to the second part is to prove that:
$displaystyle (i)quad |u(x,t)|leqfrac{C}{sqrt{t}}$
$displaystyle (ii)quad |u(x,t)|leqfrac{C}{left(1+|x|^{2}right)}+Ct^{-1/2}e^{-cx^{2}/t}$
and using $(i)$ when $|x|leq t$, and using $(ii)$ otherwise.
$u$ is continuous because it is a convolution of two continuous function. But the second part is confuse to me.
All I have is
$$displaystyle|u(x,t)|leq int_{-infty}^{infty}|hat{f}(xi)|cdot |e^{-4pi^{2}txi^{2}}|dxi$$
pde fourier-analysis convolution heat-equation
asked Nov 17 at 22:51
Mateus Rocha
781116
add a comment |
up vote
0
down vote
favorite
Suppose $u$ solution of the heat equation $u_{t}=u_{xx}$, where $fin S(mathbb{R})$ belongs to Schwartz space, $u(x,t)=left(f*mathcal{H}_{t}right)(x)$ setting $u(x,0)=f(x)$.
I need to prove that $u$ is continuous on the closure of the upper half-plane, and vanishes at infinity, that is,
$$u(x,t)rightarrow 0quadtextrm{as}quad |x|+trightarrow infty. $$
The hint is to the second part is to prove that:
$displaystyle (i)quad |u(x,t)|leqfrac{C}{sqrt{t}}$
$displaystyle (ii)quad |u(x,t)|leqfrac{C}{left(1+|x|^{2}right)}+Ct^{-1/2}e^{-cx^{2}/t}$
and using $(i)$ when $|x|leq t$, and using $(ii)$ otherwise.
$u$ is continuous because it is a convolution of two continuous function. But the second part is confuse to me.
All I have is
$$displaystyle|u(x,t)|leq int_{-infty}^{infty}|hat{f}(xi)|cdot |e^{-4pi^{2}txi^{2}}|dxi$$
pde fourier-analysis convolution heat-equation
asked Nov 17 at 22:51
Mateus Rocha
781116
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Suppose $u$ solution of the heat equation $u_{t}=u_{xx}$, where $fin S(mathbb{R})$ belongs to Schwartz space, $u(x,t)=left(f*mathcal{H}_{t}right)(x)$ setting $u(x,0)=f(x)$.
I need to prove that $u$ is continuous on the closure of the upper half-plane, and vanishes at infinity, that is,
$$u(x,t)rightarrow 0quadtextrm{as}quad |x|+trightarrow infty. $$
The hint is to the second part is to prove that:
$displaystyle (i)quad |u(x,t)|leqfrac{C}{sqrt{t}}$
$displaystyle (ii)quad |u(x,t)|leqfrac{C}{left(1+|x|^{2}right)}+Ct^{-1/2}e^{-cx^{2}/t}$
and using $(i)$ when $|x|leq t$, and using $(ii)$ otherwise.
$u$ is continuous because it is a convolution of two continuous function. But the second part is confuse to me.
All I have is
$$displaystyle|u(x,t)|leq int_{-infty}^{infty}|hat{f}(xi)|cdot |e^{-4pi^{2}txi^{2}}|dxi$$
pde fourier-analysis convolution heat-equation
asked Nov 17 at 22:51
Mateus Rocha
781116
Suppose $u$ solution of the heat equation $u_{t}=u_{xx}$, where $fin S(mathbb{R})$ belongs to Schwartz space, $u(x,t)=left(f*mathcal{H}_{t}right)(x)$ setting $u(x,0)=f(x)$.
I need to prove that $u$ is continuous on the closure of the upper half-plane, and vanishes at infinity, that is,
$$u(x,t)rightarrow 0quadtextrm{as}quad |x|+trightarrow infty. $$
The hint is to the second part is to prove that:
$displaystyle (i)quad |u(x,t)|leqfrac{C}{sqrt{t}}$
$displaystyle (ii)quad |u(x,t)|leqfrac{C}{left(1+|x|^{2}right)}+Ct^{-1/2}e^{-cx^{2}/t}$
and using $(i)$ when $|x|leq t$, and using $(ii)$ otherwise.
$u$ is continuous because it is a convolution of two continuous function. But the second part is confuse to me.
All I have is
$$displaystyle|u(x,t)|leq int_{-infty}^{infty}|hat{f}(xi)|cdot |e^{-4pi^{2}txi^{2}}|dxi$$
pde fourier-analysis convolution heat-equation
pde fourier-analysis convolution heat-equation
asked Nov 17 at 22:51
Mateus Rocha
781116
asked Nov 17 at 22:51
Mateus Rocha
781116
asked Nov 17 at 22:51
Mateus Rocha
781116
asked Nov 17 at 22:51
Mateus Rocha
781116
asked Nov 17 at 22:51
Mateus Rocha
781116
781116
add a comment |
add a comment |
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3002914%2fthe-solution-of-heat-equation-satisfies-ux-t-rightarrow-0-as-xt-rightarr%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
