A problem about fundamental solution of Laplace equation
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I would like to derive a solution to the following equation in $R^{3}$:
$$-Delta u(x) + e^{u(x)} - e^{-u(x)} = delta(x)$$where $Delta$ is the Laplace Operator and $delta(x)$ is the Dirac's delta function.
I know how to solve $-Delta u(x) = delta(x)$ in the sense of weak convergence but I'm stuck on the exponential term of the equation above.
pde heat-equation fundamental-solution
asked Nov 18 at 12:14
YC_Xu
213
add a comment |
up vote
2
down vote
favorite
I would like to derive a solution to the following equation in $R^{3}$:
$$-Delta u(x) + e^{u(x)} - e^{-u(x)} = delta(x)$$where $Delta$ is the Laplace Operator and $delta(x)$ is the Dirac's delta function.
I know how to solve $-Delta u(x) = delta(x)$ in the sense of weak convergence but I'm stuck on the exponential term of the equation above.
pde heat-equation fundamental-solution
asked Nov 18 at 12:14
YC_Xu
213
3
This problem is highly nonlinear. The problem is thus much harder than the simple Laplace equation which you compare it to. Could you explain a bit more about the context? (Otherwise, you might not obtain any answer to this rather hard problem)
– Fabian
Nov 18 at 12:22
One of the consequences of the nonlinearity of the problem, pointed out by @Fabian, is that its solvability, when the initial/boundary/source data is a measure is not guaranteed. See for example H. Brezis and A. Friedman; Nonlinear parabolic equations involving measures as initial conditions, J. Math. Pures Appl. 62 (1983), p. 73-97 for parabolic equations.
– Daniele Tampieri
Nov 18 at 13:02
I guess it is reasonable to derive a solution that is radially symmetric, which reduces the equation to a nonlinear ODE. My attempt is to first solve the homogeneous ODE with right hand side being zero, i.e $-frac{d^{2}u}{dr^{2}}-frac{2}{r}u(r)+e^{u(r)}-e^{-u(r)}=0$. Then hopefully I can pick some terms in the general solution which has a singularity at zero leading to a candidate solution. However, I don't know how to solve the ODE.
– YC_Xu
Nov 18 at 14:12
add a comment |
up vote
2
down vote
favorite
up vote
2
down vote
favorite
I would like to derive a solution to the following equation in $R^{3}$:
$$-Delta u(x) + e^{u(x)} - e^{-u(x)} = delta(x)$$where $Delta$ is the Laplace Operator and $delta(x)$ is the Dirac's delta function.
I know how to solve $-Delta u(x) = delta(x)$ in the sense of weak convergence but I'm stuck on the exponential term of the equation above.
pde heat-equation fundamental-solution
asked Nov 18 at 12:14
YC_Xu
213
I would like to derive a solution to the following equation in $R^{3}$:
$$-Delta u(x) + e^{u(x)} - e^{-u(x)} = delta(x)$$where $Delta$ is the Laplace Operator and $delta(x)$ is the Dirac's delta function.
I know how to solve $-Delta u(x) = delta(x)$ in the sense of weak convergence but I'm stuck on the exponential term of the equation above.
pde heat-equation fundamental-solution
pde heat-equation fundamental-solution
asked Nov 18 at 12:14
YC_Xu
213
asked Nov 18 at 12:14
YC_Xu
213
edited Nov 18 at 13:47
asked Nov 18 at 12:14
YC_Xu
213
asked Nov 18 at 12:14
YC_Xu
213
asked Nov 18 at 12:14
YC_Xu
213
213
3
This problem is highly nonlinear. The problem is thus much harder than the simple Laplace equation which you compare it to. Could you explain a bit more about the context? (Otherwise, you might not obtain any answer to this rather hard problem)
– Fabian
Nov 18 at 12:22
One of the consequences of the nonlinearity of the problem, pointed out by @Fabian, is that its solvability, when the initial/boundary/source data is a measure is not guaranteed. See for example H. Brezis and A. Friedman; Nonlinear parabolic equations involving measures as initial conditions, J. Math. Pures Appl. 62 (1983), p. 73-97 for parabolic equations.
– Daniele Tampieri
Nov 18 at 13:02
I guess it is reasonable to derive a solution that is radially symmetric, which reduces the equation to a nonlinear ODE. My attempt is to first solve the homogeneous ODE with right hand side being zero, i.e $-frac{d^{2}u}{dr^{2}}-frac{2}{r}u(r)+e^{u(r)}-e^{-u(r)}=0$. Then hopefully I can pick some terms in the general solution which has a singularity at zero leading to a candidate solution. However, I don't know how to solve the ODE.
– YC_Xu
Nov 18 at 14:12
add a comment |
3
This problem is highly nonlinear. The problem is thus much harder than the simple Laplace equation which you compare it to. Could you explain a bit more about the context? (Otherwise, you might not obtain any answer to this rather hard problem)
– Fabian
Nov 18 at 12:22
One of the consequences of the nonlinearity of the problem, pointed out by @Fabian, is that its solvability, when the initial/boundary/source data is a measure is not guaranteed. See for example H. Brezis and A. Friedman; Nonlinear parabolic equations involving measures as initial conditions, J. Math. Pures Appl. 62 (1983), p. 73-97 for parabolic equations.
– Daniele Tampieri
Nov 18 at 13:02
I guess it is reasonable to derive a solution that is radially symmetric, which reduces the equation to a nonlinear ODE. My attempt is to first solve the homogeneous ODE with right hand side being zero, i.e $-frac{d^{2}u}{dr^{2}}-frac{2}{r}u(r)+e^{u(r)}-e^{-u(r)}=0$. Then hopefully I can pick some terms in the general solution which has a singularity at zero leading to a candidate solution. However, I don't know how to solve the ODE.
– YC_Xu
Nov 18 at 14:12
3
3
This problem is highly nonlinear. The problem is thus much harder than the simple Laplace equation which you compare it to. Could you explain a bit more about the context? (Otherwise, you might not obtain any answer to this rather hard problem)
– Fabian
Nov 18 at 12:22
This problem is highly nonlinear. The problem is thus much harder than the simple Laplace equation which you compare it to. Could you explain a bit more about the context? (Otherwise, you might not obtain any answer to this rather hard problem)
– Fabian
Nov 18 at 12:22
One of the consequences of the nonlinearity of the problem, pointed out by @Fabian, is that its solvability, when the initial/boundary/source data is a measure is not guaranteed. See for example H. Brezis and A. Friedman; Nonlinear parabolic equations involving measures as initial conditions, J. Math. Pures Appl. 62 (1983), p. 73-97 for parabolic equations.
– Daniele Tampieri
Nov 18 at 13:02
One of the consequences of the nonlinearity of the problem, pointed out by @Fabian, is that its solvability, when the initial/boundary/source data is a measure is not guaranteed. See for example H. Brezis and A. Friedman; Nonlinear parabolic equations involving measures as initial conditions, J. Math. Pures Appl. 62 (1983), p. 73-97 for parabolic equations.
– Daniele Tampieri
Nov 18 at 13:02
I guess it is reasonable to derive a solution that is radially symmetric, which reduces the equation to a nonlinear ODE. My attempt is to first solve the homogeneous ODE with right hand side being zero, i.e $-frac{d^{2}u}{dr^{2}}-frac{2}{r}u(r)+e^{u(r)}-e^{-u(r)}=0$. Then hopefully I can pick some terms in the general solution which has a singularity at zero leading to a candidate solution. However, I don't know how to solve the ODE.
– YC_Xu
Nov 18 at 14:12
I guess it is reasonable to derive a solution that is radially symmetric, which reduces the equation to a nonlinear ODE. My attempt is to first solve the homogeneous ODE with right hand side being zero, i.e $-frac{d^{2}u}{dr^{2}}-frac{2}{r}u(r)+e^{u(r)}-e^{-u(r)}=0$. Then hopefully I can pick some terms in the general solution which has a singularity at zero leading to a candidate solution. However, I don't know how to solve the ODE.
– YC_Xu
Nov 18 at 14:12
add a comment |
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This problem is highly nonlinear. The problem is thus much harder than the simple Laplace equation which you compare it to. Could you explain a bit more about the context? (Otherwise, you might not obtain any answer to this rather hard problem)
– Fabian
Nov 18 at 12:22
One of the consequences of the nonlinearity of the problem, pointed out by @Fabian, is that its solvability, when the initial/boundary/source data is a measure is not guaranteed. See for example H. Brezis and A. Friedman; Nonlinear parabolic equations involving measures as initial conditions, J. Math. Pures Appl. 62 (1983), p. 73-97 for parabolic equations.
– Daniele Tampieri
Nov 18 at 13:02
I guess it is reasonable to derive a solution that is radially symmetric, which reduces the equation to a nonlinear ODE. My attempt is to first solve the homogeneous ODE with right hand side being zero, i.e $-frac{d^{2}u}{dr^{2}}-frac{2}{r}u(r)+e^{u(r)}-e^{-u(r)}=0$. Then hopefully I can pick some terms in the general solution which has a singularity at zero leading to a candidate solution. However, I don't know how to solve the ODE.
– YC_Xu
Nov 18 at 14:12