Questions tagged [parabolic-pde]
This tag is for questions relating to "Parabolic partial differential equation", are usually time dependent and represent diffusion-like processes. Solutions are smooth in space but may possess singularities. However, information travels at infinite speed in a parabolic system.
433
questions
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10
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Existence of solution to a parabolic bvp
I should study some parabolic PDEs, but I'm not an expert, so I would like to ask your advice.
First, could you give me some useful references concerning PARABOLIC PDES?
I started reading DiBenedetto'...
- 63
0
votes
1
answer
59
views
The exact, formal definition of a prabolic problem/equation
While searching for information about properties of parabolic problems, I stumbled upon a publication titled "Study of Nonlinear Parabolic Problems".
This made me wonder why the writers ...
- 55
0
votes
0
answers
18
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numerically compute eigenfunctions of $a(u,v)=\langle f(\nabla u)\nabla u,\nabla v\rangle_{L^2}$
Let $D:=(0,1)^2$ and consider the nonnegative form $$a(u,v):=\langle f(\nabla u)\nabla u,\nabla v\rangle_{L^2(D;\:\mathbb R^2)}$$ for $u,v\in L^2(D)$ where $f:\mathbb R^2\to(0,\infty)$ is a smooth ...
- 13.2k
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0
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75
views
Strong Maximum Principle for the heat Equation - Evans - Theorem 4 - Why do we need U to be bounded.
I had a doubt in following theorem of Evans
THEOREM 4 (Strong maximum principle). Suppose $u \in C^2_1(U) \cap C(\bar{U})$ is satisfies $u_t + \Delta u = 0$ within $U_T$. (i) Then
$$
\max _{\bar{U_T}} ...
- 790
2
votes
1
answer
43
views
Reference to a Theorem (or book) about parabolic PDEs
I am studying free-boundary problems, and I see that many authors face the following standard problem from the theory of parabolic PDEs:
Given a domain $[x_1, x_2] \times [t_0, t_1)$, find a function $...
- 173
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votes
0
answers
31
views
Exponential decay for a smooth solution to a parabolic PDE
Consider the following problems.
Problem 1. Let $\Omega \subset \mathbb{R}^n$ be open and bounded with smooth boundary, L an elliptic operator defined by $$Lu=-D_j(a^{ij}D_iu)$$ where $a^{ij}=a^{ji}$ ...
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0
answers
29
views
Tricky confusion with the definition of a weak solution for the linear parabolic equations
Let us consider the linear parabolic PDE:
\begin{equation}
\partial_t u - \Delta u=f \text{ and }u(0)=u_0
\end{equation}
where $u_0 \in L^2(\mathbb{R})$ and $f \in L^2_t([0,T], L^2_x(\mathbb{R}))$.
...
- 7,088
0
votes
0
answers
20
views
Linear fourth-order parabolic equations
I have this parabolic equation
$$u_t=u_{xx}-\epsilon u_{xxxx}$$
Where can I find references on the existence and uniqueness of the solution?
- 33
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votes
0
answers
34
views
What is $W_2^{1,1/2}(Q_T)$?
In the book "linear and quasi-linear equations of parabolic type", it defines the space $W_q^{2l,l}(Q_T)$. But what is $W_2^{1,1/2}(Q_T)$. Is it the special case when $l=1/2$?
- 689
1
vote
0
answers
47
views
De Giorgi-Nash-Moser estimattes for non-symmetric a_ij
Is there any references on De Giorgi-Nash-Moser estimates for the parabolic operator
$$ \partial_t - \partial_j[a_{ij}(t, x)\partial_i] $$
where $a_{ij}$ is not symmetric?
- 29
3
votes
0
answers
92
views
Heat equation with sharp inhomogenous Dirichlet conditions
Consider the standard heat equation in with Dirichlet boundary data.
$$\begin{cases}
\partial_tu - \Delta u = 0 & \text{in } \Omega_T := (0,T) \times \Omega, \\
\hfill u = g & \text{in } ...
- 7,052
1
vote
0
answers
29
views
Well-posedness of linear parabolic PDE
Is there any reference that explicitely comments on the well-posedness (existence of unique solution) for a PDE of the form?
$$\begin{cases}
\frac{\partial}{\partial s}v(s,x)+\varepsilon u\frac{\...
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votes
0
answers
43
views
How to prove the comparison principle of this parabolic partial differential equation?
Here $H$ is a positive uniformly convex function. I came across this comparison principle while studying the paper. Maybe the author thought it was easy to prove so he omitted the proof, but I failed ...
- 25
1
vote
0
answers
39
views
Positiveness of weak solutions of a parabolic PDE
Given $\Omega \subset \mathbb{R}^n$ a bounded smooth open set, $T>0$ and $a,b \in L^\infty(\Omega)$ and positive. I'm trying to prove the positiveness of weak solutions to the following PDE
$$
\...
- 318
2
votes
0
answers
38
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A non-separable heat-convection differential equation with 1 space-variable
There is a problem I have much difficulties to solve. It is about a temperature difference induced by pressure variation. It is this equation solved by kieransquared, but this time with specific ...
- 149
0
votes
0
answers
41
views
Time Discretisation of a System of PDEs
Suppose we have an arbitrary system of PDEs
$$ \partial_t u - D_1 \Delta u + a(v) u = f(t)$$
$$\partial_t v - D_2 \Delta v + b(u) v = g(t) $$
We want to discretise the system in time. We use the ...
- 75
0
votes
0
answers
75
views
A reference for solution of non homogeneous heat equation on bounded domain
In PDE's book from Evans, is said that
$$
u(x,t) = \int_0^t \int_{\mathbb{R}^N} \frac{1}{(4\pi (t-s))^{N/2}} e^{-\frac{|x-y|^2}{4(t-s)}} f(y,s) dy ds
$$
is a solution of
$$
\begin{cases}
u_t - \Delta ...
- 891
0
votes
1
answer
63
views
Physical meaning of a heat equation with a term $\alpha u$ [closed]
I would like to know if there is any physical meaning for the equation
$$
\begin{cases}
u_t - u_{xx} + \alpha u = f(x), (a,b)\times(0, +\infty) \\
u(x,0)=u_0(x), x \in (a,b)\\
u(a,t)=u(b,t)=0, t \in (...
- 891
1
vote
0
answers
55
views
Translating parabolic PDEs (e.g. heat equation) to geometric flows (e.g. mean curvature flow)
Suppose $X:\mathcal S\to\mathbb R^3$ is an embedding of a surface $\mathcal S\subset\mathbb R^3$. Then, the mean curvature normal of $\mathcal S$ is the Laplacian applied to the coordinates $\Delta ...
- 880
1
vote
0
answers
27
views
Reference request for the analysis of a nonlinear Fokker-Planck type PDE
It is well-known that the linear Fokker-Planck equation (written in one space dimension for simplicity) $$\partial_t \rho = \partial_x \left(\rho_\infty \partial_x\left(\frac{\rho}{\rho_\infty}\right)\...
- 2,724
0
votes
0
answers
40
views
Why is the wick rotated heat equation equivalent to Mellin transform followed by Fourier transformed heat equation?
I was studying this equation:
$$\frac{\partial^2}{\partial x^2}\Psi(t,x)=c(t,x)\frac{\partial}{\partial t}\Psi(t,x) \tag{1}$$
where $c(t,x)=-t/x.$
I found a particular solution to this equation, $\Psi(...
- 892
2
votes
0
answers
22
views
Continuity bounds of heat kernel
I am currently trying to prove the following bound on the difference in first exit time distributions:
$$P_x(\tau_D\le t+\delta)-P_x(\tau_D\le t)\le C_D\delta$$
where $x\in D$, $t,\delta>0$, and $D$...
- 55
0
votes
0
answers
43
views
Partial differential equations: Mellin transform in space and time?
This is related to: Partial Differential Equations: Fourier Transform in Space and Time?, and The equation $r^2 \frac{\partial ^3\Psi(r,s)}{\partial r^3}=s^2 \frac{\partial \Psi(r,s)}{\partial s}.$ ...
- 892
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votes
0
answers
25
views
PDEs: Is there a model to combine simulateous diffusive and congestive(concentrating) forces?
I am still pretty new to the field of PDEs, but I wanted to develop a novel economics model to study the spatial congestion of economic activity. For example, we see that cities tend to accumulate ...
- 2,159
2
votes
1
answer
69
views
References for regularity of solution to fourth order elliptic/parabolic PDEs
After searching, all I could find was regularity of solutions to second-order PDEs like in Partial Differential Equations by Evans and Functional Analysis, Sobolev Spaces and Partial Differential ...
- 75
0
votes
1
answer
46
views
Diffusion equation derivation from Strauss
In an intermediary step in the proof of the condition that $u(x,0) = \phi(x)$, Strauss Page 49, writes:
$u(x,t) = \int_{-\infty}^{\infty} \frac{\partial Q}{\partial x}(x-y,t)\phi(y) dy$.
On the next ...
- 1,214
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votes
0
answers
22
views
Space-variant diffusion with infinite speeds: eigendecomposition and matrix exponential
The heat diffusion equation on some domain $\Omega$ with Neumann boundary conditions on $\partial\Omega$ and normal $n$ is given as:
\begin{alignat}{3}
\partial_t u(t,x) &= \Delta u(t,x), &\...
- 2,027
0
votes
0
answers
52
views
Heat equation: proving that smaller diffusion leads to bigger solution via energy methods
Let $\Omega$ be a bounded Lipschitz domain and denote by $u_\alpha$ the solution of the heat equation
$$u_t -\alpha \Delta u = f$$
with $f \in L^2(0,T;L^2(\Omega))$, $u(0) = u_0$ given and $u|_{\...
- 411
0
votes
0
answers
26
views
Cauchy problem for parabolic equation
Let $\mathcal{L}=a \partial_{xx}+b\partial_{x}+c$, where $a>0,c\in\mathbb{R},b\in \mathbb{R}$, I want to know if the following parabolic equation gets a smooth and unique solution
\begin{align}
&...
- 73
0
votes
0
answers
28
views
Parabolic PDEs: is it possible to simulate blow-up phenomena in MATLAB?
I want to learn to use matlab to study the graphical behavior of parabolic partial differential equation solutions, in the sense of trying to simulate when the solution will be global or when it will ...
- 2,847
0
votes
1
answer
52
views
Is this a coupled system of PDEs?
I came across this question
Set of coupled partial differential equations
And the answer is clear. However, what if the solution of the second equation does not appear in the first equation for ...
- 337
5
votes
1
answer
175
views
Particular solution of Schrodinger equation, satisfying time dependent boundary conditions
I am considering the Schrodinger equation on $[0,1]$:
$$iu_t=-u_{xx}$$
I am looking for a particular solution $u(x,t)$ which satisfies the boundary condition $$\cos(t)\cdot u_x(0,t)=\sin(t)\cdot u(0,t)...
- 4,158
1
vote
0
answers
22
views
Convergence of Weak Time Derivative in Bochner Space
I'm currently working with two sequences of functions, $u$ and $u_n$, where $u, u_n \in L^{2}(0,T,H_{0}^{1}(\Omega)) \cap C([0,T],L^{2}(\Omega))$ and their derivatives $u', u_n' \in L^{2}(0,T,H^{-1}(\...
- 330
0
votes
1
answer
55
views
Inequality for a parabolic pde
I am trying to solve the following exercise.
Consider the linear parabolic equation
\begin{equation}
\partial _t u -\nabla (A(x,t)\nabla u)=f(x,t), x \in \Omega,t>0,
\end{equation}
subject to the ...
- 139
0
votes
1
answer
59
views
Inequality for parabolic pde
I am trying to solve the following exercise.
Consider the linear parabolic equation
\begin{equation}
\partial _t u -\nabla (A(x,t)\nabla u)=f(x,t), x \in \Omega,t>0,
\end{equation}
subject to the ...
- 139
1
vote
0
answers
103
views
Is there any functional f(s) such that this system of differential equations has an analytical solution?
I am dealing with the following system of partial differential equations that describes the effects of random motility on bacteria that consume a diffusible substrate:
\begin{cases}
b_{t} = \mu b_{xx} ...
- 355
1
vote
1
answer
56
views
Parabolic PDE whose solution is a product of lower dimensional solutions?
Consider the partial differential equation:
$$ \bigg(\sum_{i=1}^n s_i\bigg) \Delta \Phi=nu \Phi_u $$
for $\Phi(u,s_1,s_2,\cdot\cdot\cdot,s_n).$
This is a linear parabolic partial differential equation ...
- 892
0
votes
1
answer
80
views
(Evans) Incomplete proof of the existence and uniqueness of weak solutions to a reaction-diffusion system
In Example 1 in section 9.2 (Fixed Point Methods), Evans employs Banach's Fixed Point Theorem to prove the existence and uniqueness of a weak solution to a system of reaction-diffusion equations. A ...
0
votes
0
answers
28
views
Reversal of Variable Transformation in the Solution of Inhomogeneous Dirichlet PDE
Initially, we are dealing with a non-homogeneous Dirichlet problem:
\begin{equation}
\begin{cases}
q_t(z,t)-D(z,t)q_{zz}(z,t) = 0 & \text{for $0<z<L,t>0$} \\
q(0,t)=a &...
- 1
0
votes
0
answers
15
views
Solution for a non-linear parabolic pde
I want to know how to deal with the following non-linear parabolic pde
$$\begin{cases}
W_t(t,x)+W+W_x-W_{xx}-\mathrm{e}^xW_x^{-1}W_{xx}-\mathrm{e}^x=0, \quad (t,x)\in (0,T]\times(0,\infty)\\
W(0,x)=\...
- 73
2
votes
1
answer
51
views
Uniformly bounded sequence in $W^{2,1}_{q}(\Omega^{\prime})$ has a subsequence in $C^{1+\alpha,\frac{1+\alpha}{2}}_{\text{loc}}(\Omega)$
Let $D$ be a bounded domain in $\mathbb{R}^n$ and $\Omega=D\times (0,T)$. For any $q>1$ and any $\Omega^{\prime}\Subset\Omega$, $\left\|v_k\right\|_{W_{q}^{2,1}(\Omega^\prime)<+\infty}$, where $...
- 21
1
vote
1
answer
150
views
General solution of $x^2 u_{xx} - 2xy u_{xy} + y^2 u_{yy} = e^x$
I am trying to find the general solution of $x^{2} u_{xx} -2xy u_{xy} + y^{2} u_{yy} = e^{x}$. The equation is parabolic as the only root of the characteristic equation is $\lambda = \frac{y}{x}$. ...
- 37
2
votes
0
answers
40
views
Finite time blow-up for solutions of a quasi-parabolic system
I am reading a book about the Harmonic Map flow and in the proof of the existence of solution the author states something I am not totally familiar with. Let me recall the statement of the theorem.
...
- 3,913
4
votes
2
answers
161
views
A doubt on Theorem 2.6 from Pazy's book
I have been very confused about an argument on Theorem 2.6 from Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations. Here is the theorem and part of the proof:
I ...
- 891
1
vote
0
answers
92
views
Maximum principle or comparison principle
the equation
$\varphi_t + D_p H(x, Du ^\varepsilon)\cdot D \varphi = \varepsilon\Delta \varphi$ is a linear parabolic equation. Thus, by the comparison principle for parabolic equations, we have for ...
- 33
0
votes
0
answers
84
views
Comparison principle of parabolic equation
Let $\varepsilon > 0$, consider the following viscous Hamilton-Jacobi equation
$$
\begin{cases}
u ^\varepsilon_t + H(x, Du ^\varepsilon) = \varepsilon\Delta u^\varepsilon& \text{in }\mathbb{R}^...
- 33
0
votes
0
answers
100
views
Find the energy of a PDE, and show it is conserved
In the domain $(0,1) \times [0,T], T>0$, we consider the boundary value problem
$$V_{tt} + \eta V = (\xi V_x - \beta V_{xxx})_x,\,\,\,\,\,V(0,t)=0, V(1,t)=0, V_x(0,t)=0, V_x(1,t)=0,$$
where $\eta, \...
- 1,384
0
votes
0
answers
38
views
Cauchy Problem as Continuous Operator between Normed Spaces
Let $\lambda \colon [0, T] \times (0, \infty) \to [0, \infty)$ as well as some reasonable $f \colon (0, \infty) \to [0, \infty)$ and consider the following Cauchy boundary value problem:
$$\partial_t ...
- 1,200
0
votes
0
answers
15
views
Using the truncation method to prove weak solutions of Lotka-Volterra with diffusion system are nonnegative
I'm reading Perthame's Parabolic equations in biology book and I'm confused at what he is doing in this proof.
For setup we are considering weak solution $$u\in C(\mathbb{R}^+;L^2(\mathbb{R}^d))$$ to ...
- 1
5
votes
0
answers
81
views
PDE with a non-classical boundary condition
Assume that one has a classical PDE, say: $u_t(t,x)-u_{xx}(t,x)=0$, $t\in (0,1)$, $x\in (0,2)$, and $u(0,x)=0$. Then we can prove existence (and uniqueness) of solution when boundary conditions: $u(t,...
- 271