Questions tagged [parabolic-pde]
Questions about partial differential equations of parabolic type. Often used in combination with the top-level tag ap.analysis-of-pdes.
444
questions
1
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0
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30
views
Reference: Result of interior parabolic regularity theory for Hamilton–Jacobi equations
Does anyone know the parabolic regularity result that Ben-Artzi used in the article The local theory for viscous Hamilton–Jacobi equations in Lebesgue spaces used to prove that the solution to the ...
- 337
0
votes
0
answers
44
views
Traveling wave solutions for epidemic mathematical models
We know that traveling wave solutions are studied in order to obtain the speed of dengue dissemination in partial differential equation systems related to epidemic diseases such as Dengue. On the ...
- 31
0
votes
0
answers
27
views
Solution for a non-linear parabolic pde
I want to know how to deal with the following non-linear parabolic pde
$$\begin{cases}
W_t(t,x)+W+W_x-W_{xx}-\mathrm{e}^xW_x^{-1}W_{xx}-\mathrm{e}^x=0, \quad (t,x)\in (0,T]\times(0,\infty)\\
W(0,x)=\...
- 103
1
vote
0
answers
39
views
Comparison principle for porous medium equation in Fourier variables
Let $V:[0,\infty) \to[0,\infty)$ be convex, $C^2$ with $V(0)=0$. Define $F(u): = uV'(u)-V(u) $. Let $v\in L^1 (\Bbb R^d)$, $v\geq0$ so that $F\circ v\in L^1 (\Bbb R^d)$. For fixed $\varepsilon>0$, ...
- 973
1
vote
0
answers
46
views
A parametrix construction for heat boundary value problem using Fourier transformation
Let $\Omega$ be a smooth bounded open subset in $\mathcal{R}^{d}$, with $d \geqslant 3
$ and $T>0$. Consider the linear parabolic initial Dirichlet boundary value problem with $f\in H^{-1}(\Omega)$...
- 11
0
votes
0
answers
19
views
Is there a principle of comparison for mild-solutions?
We consider the equations of the form
\begin{equation}\label{Eq.un}
\left\{
\begin{array}{rll}
u_t - \Delta u &= |\cdot|^{\gamma}u^{p}& \mbox{ in } \mathbb{R}^n \times (0,T),...
- 337
6
votes
2
answers
217
views
Existence of solutions to the heat equation on nonsmooth domains
Let $\Omega \subset \mathbb{R}^n$ be a compact domain and for given functions $g: \partial \Omega \times [0,T] \to \mathbb{R}$ and $h: \Omega \to \mathbb{R}$ consider the heat equation
$$
\begin{cases}...
2
votes
0
answers
47
views
Well-posedness or existence for a Poisson problem in Orlicz spaces
I know that the problem
\begin{equation}
\Delta_p u = f
\end{equation}
make sense if $f \in L^q$ with $n/p<q<n$ and that is there a existence theory for
$$
u_t -\Delta_p u = f
$$
For a given ...
- 231
1
vote
0
answers
17
views
Free programs suggestions to simulate parabolic EDPs
I'm interested in learning how to computationally simulate the behavior of parabolic partial differential equations, but I don't know where to start, what are the best free programs to use and where ...
- 337
0
votes
0
answers
72
views
Convergence of Solutions of Integral Equations with Weakly Converging Forcing Terms
Let $\Omega$ be a bounded interval of $\mathbb{R}$ and let $y\in L^\infty(\Omega \times (0,T))$ be a mild solution of the integral equation
$$
y(\cdot,t)=S(t) y_0+\int_0^t S(t-s) \left[u(\cdot,s)y(...
- 21
1
vote
0
answers
83
views
For the solvability of the poisson equation $\Delta u = f$ on manifold with boundary
For poisson equation $\Delta u = f$ in bounded domain in $\mathbb{R}^n$, we can directly get the solution by Green function. For poisson equation $\Delta u = f$ on closed Riemannian manifold, the ...
- 391
0
votes
0
answers
54
views
Symmetric formulation for the heat equation
Originally posted on MSE
Consider the heat equation on a domain $U$: $$\partial_t u - \operatorname{div}(A\,\nabla u)=f$$
with $u(0)=0, u=0$ on the boundary of the domain of definition. Consider a ...
- 1
0
votes
0
answers
26
views
Boundary regularity for heat equation
Consider the heat equation $u_t - \Delta u=0$ with $u = u_0$ on $\partial B \times (0,T) \cup B \times \{t=0\}$. We consider weak solutions $u \in C^0(0,T;L^2(B)) \cap L^2(0,T;u_0 + W_0^{1,2}(B))$ ...
- 473
5
votes
1
answer
161
views
Convergence of heat flow on non-compact manifolds?
Consider the heat equation $\partial_t u= \Delta u+\lambda_1 u$ on a non-compact complete manifold $M$ (with nonpositive curvature) where $\lambda_1$ is the first eigenvalue and we start with some ...
- 591
1
vote
1
answer
226
views
Physical relevancy of two curious PDE's
My research has brought me to the following linear parabolic second order PDE:
$$ \frac{\partial^2}{\partial x^2}\Psi(t,x)=c(t,x)\frac{\partial}{\partial t}\Psi(t,x) $$
for $c(t,x)=-\frac{t}{x}$ and $...
1
vote
0
answers
38
views
Wellposdeness of some HJB equation
Consider the non-linear PDE for $u:[0,1]\times [-1,1]\to\mathbb R$ as follows:
$$u_t= \inf_{b\ge 1/e} \big(-b u_{xx} - \log b - 1\big), \quad \forall (t,x) \in (0,1) \times (-1,1),$$
together with the ...
- 437
1
vote
0
answers
54
views
Solution to $u_t = A(t)u + f(t)$ on bounded domain
I am dealing with the problem
\begin{align}u_t &= \nabla \cdot (a(x,t) \nabla u) + f(x,t) &\text{ on } \Omega \times (0,T)\\
\partial_{\nu} u &= 0 &\text{ on } \partial \Omega \...
- 11
2
votes
0
answers
126
views
Function is in $L^2$ . how to show that gradient is also in $L^2$?
I am dealing with diffusion-reaction equation with three species. I have $L^2$ bound of concentrations. Now I want $L^2$ bound of gradient of concentrations. Somehow if I get $L^4$ or $L^\infty$ bound ...
- 21
5
votes
1
answer
293
views
The decay rate of a degenerate heat equation in torus $\mathbb{T}^2$
Consider the degenerate heat equation on torus $\mathbb{T}^2=\mathbb{R}^2/\mathbb{Z}^2$:
$$ \frac{\partial}{\partial t} u(x,t)= \left( \sin^2(\pi x_1) \frac{\partial^2}{\partial^2 x_2} + \sin^2(\pi ...
- 51
4
votes
0
answers
89
views
Continuity of solutions of Elliptic PDE with respect to parameters
Let $\alpha \in \mathbb{R}$ and $u_\alpha$ satisfy
$$ \Delta u_\alpha+e^{u_\alpha}=\alpha f(x), \ \ \ \ x\in \mathbb{R}^2$$
where $f$ is a fast decaying smooth function.
I would like to know how the ...
- 765
4
votes
0
answers
58
views
Systems of parabolic equations -- Petrovskii's condition
Consider the flat torus $\mathbf{T}^d:=\mathbf{R}^d/\mathbf{Z}^d$ and define the corresponding periodic-parabolic cylinder $Q_T:=(0,T)\times\mathbf{T}^d$.
Given a matrix field $A:Q_T\rightarrow\text{M}...
- 2,410
3
votes
0
answers
86
views
Comparison principle for Elliptic PDE with exponential nonlinearity
Suppose $\varphi$ is a radial (and radially decreasing) solution of
$$\Delta \varphi+e^{\varphi}=0, \ \ \text{on} \ \ r \in (0,R), $$
with $ R>0$, and $\psi$ is a decreasing radial function ...
- 765
4
votes
1
answer
355
views
Periodicity and Burger's equation
Consider the 1-dimensional Burger's equation on a finite interval $I=(0,1)$,
$$u_t+uu_x=u_{xx}$$
with initial condition
$$u(x,0)=f(x)$$
and boundary conditions
$$u(0,t)=A(t) \qquad u(1,t)=B(t).$$
...
- 40.2k
1
vote
0
answers
24
views
Singular asymptotic limits of mean-convex MCF
Let $(M_t \mid t \geq 0)$ be a mean-convex mean curvature flow of hypersurfaces in ambient Riemannian manifold $(N^{n+1},g)$. Brian White proved that this flow (defined 'weakly' as a level set flow ...
- 4,536
1
vote
1
answer
82
views
Fractional reaction-diffusion with Caputo derivative
I'm interested in the following Cauchy problem for a linear diffusion equation
$$
\begin{cases}
{^C}\!D^{a}_tu(t,x) = \sigma\Delta u(t,x),\\
u(0)=u_0\in X.
\end{cases}
$$
where ${^C}\!D^{\sigma}_t$
...
2
votes
1
answer
284
views
Property so that $f(t)\equiv 0$ for all $t\geq T$ for some finite $T>0$?
Let $f:[0, \infty)\to [0, \infty)$ be non-decreasing and satisfy for all $t>t_{0}$, $$f(t)+C\int_{t_{0}}^{t}f^{\gamma}(s)ds\leq \frac{1}{t-t_{0}}\int_{t_{0}}^{t}f(s)ds,$$ where $0<\gamma<1$ ...
- 439
2
votes
0
answers
69
views
Convergence of Green's function of fractional heat equation
For the fractional heat equation
\begin{equation} \partial_t u + (-\Delta^s)u=0 \text{ in } \mathbb{R}^d \times (0,\infty),
\end{equation} where $s \in (0,1)$ where the fractional laplacian is the ...
- 21
1
vote
0
answers
80
views
Hölder regularity in a quantitative manner
Why the investigation pertaining to sharp estimates in diffusion pde is important? In what scenarios the quantitative way reveals important nuances of a problem and play a decisive role in a finer ...
- 361
0
votes
1
answer
159
views
Looking for English version of a paper of Jean Ginibre
I am in serious need of an English translation for the following paper:
Séminaire Bourbaki by Jean Ginibre, Le problème de Cauchy pour des EDP semi-linéaires
périodiques en variables d’espace, d'après ...
- 79
1
vote
0
answers
157
views
Maximal regularity heat equation
Considering the heat equation on the flat torus $\mathbf{T}^d$, we have the maximal regularity estimate
\begin{align*}
\forall \varphi\in\mathscr{D}(\mathbf{R}\times\mathbf{T}^d),\quad \|\Delta \...
- 2,410
0
votes
1
answer
53
views
Reference and hint for L^p estimates of the gradient of solutions to parabolic equation in divergence form
Considering a weak solution $u\in L^2(0,1;H^1(B_1))$ with $\partial_t u \in L^2(0,1;H^{-1}(B_1))$ to
$$\partial_t u-\operatorname{div}(A(x,t)\nabla u)=f+\operatorname{div}(F) \hspace0.5cm \text{in} \...
- 61
2
votes
0
answers
41
views
Are there results that relate the restriction of the heat semigroup in $\mathbb{R}^n$ and the semigroup in a domain?
I'm thinking about the following situation:0 suppose that
$$
S_{\Omega}(t)f = \int_{\Omega} K_{\Omega}(x,y,t)f(y)dy
$$ where $K_\Omega(x,y ,t)$ is the Dirichlet heat kernel in the domain $\Omega$. It ...
- 337
3
votes
0
answers
184
views
A non-linear PDE $v^2v_t=v_{xx}v-v_{x}^2$
PS : Indeed, there is a typo in my equation. Thanks to Zachary's observation.
Consider a PDE for $v: [0,1]^2\to (-\infty,0]$ satisfying
$$v_t(t,x) = \frac{v_{xx}(t,x)v(t,x)-v_{x}(t,x)^2}{v(t,x)^2},\...
- 437
1
vote
0
answers
58
views
Parabolic PDE: Zero now means zero anytime before
Studying some mathematical models I came across a simple-looking question that I do not know how to handle.
If we have the following problem:
$$\begin{cases} \dfrac{\partial Z}{\partial t}-\Delta Z=aZ-...
- 1,154
3
votes
0
answers
94
views
Uniqueness continuation property for parabolic equation
Consider the following parabolic equation:
$$\DeclareMathOperator{\Div}{div}
\begin{cases}
\dfrac{\partial \rho }{\partial t}-\Div\left( a\left( x\right) \nabla
\rho \right) +p(x)\rho = 0 & \...
0
votes
0
answers
39
views
Evolution operator pointwise continuity
For any $f\in L^{\infty}(\Omega)$ denote by $U(\tau, t)f$ the evolution operator, i.e. the solution of the following problem (I choose this logistic equation one because it has bounded solution on $(0,...
- 1,154
0
votes
1
answer
108
views
Distance function to mean curvature flow
In Ilmanen's monograph on elliptic regularization and mean curvature flow, the following claim is made. (Just for reference, this is on page 60.) For the proof—apparently standard calculations—the ...
- 4,536
4
votes
1
answer
84
views
Interior Sobolev regularity of parabolic solutions
In Evans book (and many others) there are a classic result about interior regularity in Sobolev spaces for solutions to uniformly elliptic problem (Theorem 1, p. 309). That is, let $\Omega\subset\...
- 61
1
vote
1
answer
146
views
Reference for representation of heat equation with Neumann boundary condition on smooth domain using reflected Brownian motion
We know that the solution of the heat equation $\partial_tu=\frac 12\Delta u$ with Dirichlet boundary condition $u\rvert_{\partial\Omega}=g$ is $u(t,x)=\mathbb{E}[g(B_\tau)\mid B_t=x]$, with $\tau$ ...
- 587
6
votes
0
answers
100
views
Heat Flows and spatial singularities
While working on an abstract problem, I came up with the following question:
Let $\Omega_1 := \mathrm B(-1, 1)$ and $\Omega_2 := \mathrm B(1, 1)$, where $\mathrm B(x, r) \subseteq \mathbb R^2$ denotes ...
5
votes
1
answer
174
views
Two dimensional oscillatory integral
I am having a little confusion in verifying the two dimensional oscillatory integral in Lemma 2.1 in This paper, namely
$$I_t (x,y) = \int_{\mathbb{R}^2} |\xi|^{\epsilon + i \beta} e^{i t(\xi^3 + \xi \...
- 79
2
votes
0
answers
78
views
On Fredholm alternative for Neumann conditions
Let $\Omega$ be a bounded Lipschitz domain in $\mathbb{R}^n$ and $f \in L^2(\Omega)$. It is well known that if $\lambda$ is a Dirichlet Laplacian eigenvalue, then the equation $$\begin{cases}
-\Delta ...
- 1,320
2
votes
1
answer
116
views
Periodic solution for linear parabolic equation - existence, regularity
I am interested in proving the existence and regularity of solution to the following problem:
$$\begin{cases} \dfrac{\partial y}{\partial t}(t,x)-\Delta y(t,x)+c(t,x)y(t,x)=f(t,x), & (t,x)\in (0,T)...
- 1,154
2
votes
0
answers
165
views
Boundedness for singular parabolic p-Laplace equation
Local boundedness of singular parabolic $p$-Laplace equation
$$\partial_t u - \operatorname{div}(|\nabla u|^{p-2}\nabla u)=0,\,1<p<2,$$
requires additional integrability assumption for the ...
4
votes
1
answer
248
views
Strong positivity of Neumann Laplacian
There are many places in the literature where the positivity of some semigroups is treated. However I did not know anyone which states and proves the strong positivity even for the basic semigroups ...
- 1,154
4
votes
1
answer
139
views
$L^2$ norm for solutions of evolution equations driven by different elliptic operators
Let $u$ be a solution of the heat equation
$$u_t - \Delta u = 0, \qquad t >0, \ x \in \mathbb T^d$$
and $v$ be a solution of the bi-harmonic heat equation
$$v_t +\Delta^2 v = 0, \qquad t >0, \ x ...
- 283
1
vote
0
answers
48
views
Elliptic principal eigenfunction analysis for Langevin dynamics with a varying source term
Consider the Kolmogorov forward equation for a Langevin dynamic:
$$\DeclareMathOperator{\Div}{div}
\begin{cases}
\dfrac{\partial}{\partial t} f = \Delta f + \Div(f\nabla V)\\
\\
\displaystyle\int_{\...
- 11
4
votes
0
answers
92
views
The logistic elliptic equation
Studying the Fisher-KPP evolution equation I came across the steady state elliptic problem which can be written in the following form:
$$
\begin{cases} -d\Delta Y(x)=r(x)Y(x)\left (1-\dfrac{Y(x)}{K(x)}...
- 1,154
3
votes
2
answers
214
views
Change of variables for obtaining a unitary group
Consider the following NLS:
$$i u_t + \Delta u- 2 \operatorname{Re} u = F(u),$$
where $F(u):=(u + \bar{u} + |u|^2)u.$
In Scattering for the Gross–Pitaevskii equation, the authors S. Gustafson, K. ...
- 79
0
votes
0
answers
45
views
What is the deep logic for the resonance function of dispersive nonlinear PDEs
I have been studying some nonlinear dispersive PDEs since some months and I was able to understand some results related to well-posedness. However, I do not feel like I am fully understand the logic ...
- 79