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Questions tagged [parabolic-pde]

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1answer
23 views

How to find locus for given condition?

I've been through all the thinking but i could not get to the answer. Please help me in this. See images for details
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0answers
16 views

Estimates of Hessian of Heat Equation

I am studying the heat equation \begin{align*} u_t - \Delta u = f \end{align*} where $u \in C^\infty(\bar{\Omega} \times (0,1])$ has compact support on $\Omega$ for all $t > 0$. My objective is to ...
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0answers
39 views

Parabolic Partial Diff Equation ( Heat Equation )

Im having a really hard time with this question, I honestly don't even know how to start this...can anyone help me? We are in the Parabolic section of PDE, and we are asked to find the unique ...
2
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0answers
18 views

Frechet Derivative and Convergence of Functionals

Let $\Omega\subset\mathbb{R}$ be a bounded interval, $\{u_{n}(t)\}_{n\in\mathbb{N}}$ be a bounded sequence in $H_{0}^{1}(\Omega)$ and define $J[u_{n}(t)] = \frac{1}{2}||u_{n}(t)||_{H_{0}^{1}(\Omega)}^{...
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0answers
24 views

Meaning of Compactness

Let $\Omega \subset\mathbb{R}$ be a bounded domain (interval) and observe the following problem : \begin{align*} (P) \begin{cases} u_{t} = \Delta u + |u|^{p-1}u\, \quad x\in\Omega, t>0\\ u(0,x) = ...
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0answers
16 views

Trouble understanding this proof: Evans Ch.7

We recently started in class to study about linear evolution equations and the following is a proof on improved regularity presented in PDEs book by Evans. I have trouble deducing the inequality in ...
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0answers
16 views

Decay estimate for inhomogeneous heat equation

Consider a bounded domain $\Omega$ and the problem $$ \begin{cases} u_t(x,t) - \Delta u(x,t) = f(x) & x \in \Omega, t>0\\ u(x,0) = 0 & x \in \Omega \\ u(x,t) = 0 & x \in \partial \Omega,...
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0answers
19 views

Coercivity of mapping A , sequence bounded

in Hess' article I don't understand why the sequence $u_{n\epsilon}$ is bounded (and $\frac{\partial u_{u\epsilon}}{\partial t}$ too) $Q=\Omega \times (0,T)$ (A2) There exist constants $q \quad(1<...
9
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1answer
112 views

Why should we give special attention to at most polynomially growing solutions of PDEs?

The equation \begin{gather} \frac{\partial u}{\partial t} (t,x) = \frac{1}{2} \text{Trace}[\sigma(x) \sigma(x) (\text{Hessian}_x u)(x,t)] + \langle \mu (x) , (\nabla_x u) (t,x) \rangle, \\ u(0,x) = \...
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0answers
11 views

Discretizing a parabolic PDE with finite volume method

I want to discretize the following parabolic PDE: $$u_t = \nabla\cdot(\alpha(x)\nabla u)- \beta u\\ x\in\Omega \subset \mathbb{R}^2\\ \partial_n u = 0\\ u(t,0) = u_0(x)\ge 0, \alpha(x)>0$$ Given ...
2
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0answers
30 views

Parabolic PDE with zero Neumann condition

I'm working with the following parabolic PDE: $$u_t = \nabla\cdot(\alpha(x)\nabla u)- \beta u\\ x\in\Omega \subset \mathbb{R}^2\\ \partial_n u = 0$$ Given the Neumann boundary condition above, if I ...
2
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1answer
29 views

weak convergence and compactness

please how can I prove that if a sequence $u_n \to u$ in $L^{\infty}(\mathbb{R}^+; H^1(\Omega)) $ weak * and $\partial_t u_n \to \partial_t u$ in $L^2(0,T, L^2(\Omega)) $ weak for all $T>0$ ...
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0answers
26 views

Existence of the solution of the 3D Micropolar equations [closed]

Please how to show the local existence for the solution of the 3D micropolar equations in a Besov space setting ? $\left\{ \begin{array}{l} \partial_tu-(\nu+k)\Delta u-2k\nabla\times w+u\nabla u+\...
1
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1answer
29 views

Finite difference stability problem

My apologies for the title I'm not quite sure how to title a problem like this. I need to show the following result: $$u_j^{n+1} = e^{\Delta t\partial/\partial t}u_j^n$$ Where $u_j^n$ is the ...
2
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0answers
27 views

Method of simplifying parabolic PDE

I consider PDE for the function $f(x, t)$ of the form: \begin{equation} \begin{aligned} f_t + Ax^2 f_{xx} + Bx f_{x} + g(x)f + h(x, t) = 0 \\ f(x, T) = 0 \end{aligned} \end{equation} where $f:\...
1
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1answer
30 views

Non-Linear Differential Equation Change of Variable

The function $v(x,t)$ satisfies: $$\frac{\partial v}{\partial t} = \frac{\partial^2v}{\partial x^2} + \left(\frac{\partial v}{\partial x}\right)^2$$ for $0<x<1$, the initial condition $v(x,0)...
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1answer
23 views

Harnack inequality for linear parabolic equations

I am trying to understand the proof of the Harnack inequality using the ideas of J.Nash as proved in the paper "A new proof of Moser's Parabolic Harnack Inequality using the old Ideas of Nash" by E.B....
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0answers
17 views

Equivalence of the Weak Parabolic Maximum Principle

I'm study by myself parabolic PDEs by Avner Friedman's book. Initially, Friedman starts the first section of chapter $2$ with the following: Consider the operator $$(1.1) \ Lu \equiv \sum_{i,j=...
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0answers
10 views

Chapter $2$ - Section $1$ - Lemma $4$ in Friedman's book

I'm studying by myself Parabolic PDEs by Friedman's book and I have a doubt concerning the proof of the lemma $4$ on section $1$ of chapter $2$: $\textbf{Lemma 4.}$ Let $R$ be a rectangle $$ ...
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0answers
31 views

conditions for well-posedness

Im looking for conditions on $f(x,t)$ that make the problem well-posed where $x\in[0,\pi)$ \begin{align} u_t-u_{xx}&=f(x,t) \\ u(0,t)&=0 \\ u_x(0,t)&=0 \\ u_x(\pi,t)&=0 \\ u(x,0)&=...
2
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0answers
37 views

pde with linear coefficients

Is there a general theory for linear, 2nd-order PDE in two dimension, when all coefficients are linear functions of the two variables? If not, how would you proceed with an equation like $$ x f_{xx}...
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0answers
20 views

Infinite differentiability for a solution of the general linear parabolic pde of second order

I'm studying by myself the the chapter of second order parabolic linear equations by Evan's book, which focus in solve $$(11) \ \begin{cases} \begin{eqnarray*} u_t + Lu &=& f \ \...
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0answers
30 views

Question on parabolic smoothing of nonhomogeneous heat equation

Suppose $\Omega $ is a bounded domain in $\mathbb R^n$ and let $u$ be the weak solution of the initial value problem for the nonhomogeneous heat equation: $\begin{cases} \partial_t u-\Delta u=f,\;\...
1
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1answer
45 views

Motivation for the definition of weak solutions to parabolic equations of second order

I'm reading Evans' book about PDEs by myself and I'm trying understand the motivation for the definition of weak solutions to parabolic equations of second order on pages $373$ and $374$. First, we ...
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0answers
6 views

The first initial boundary value problem on parabolic Monge-Ampere equations

What i am concerned equation is $$ -u_t\det D^2u=1\quad \mbox{in } Q, $$ with the first initial boundary value $$ u(x,t)=\varphi(x,t)\quad \mbox{on } \partial Q, $$ where $Q$ is a non-cylinder domain ...
3
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0answers
42 views

$ L^p$ regularity for weak solutions of non homogeneous heat equation

Let $U$ be a compact $C^2-$manifold and suppose $v:U\times (0,T) \to \mathbb R$ is the weak solution of: $\partial_t v= \Delta v+f$ where $f\in L^{\infty}(U\times (0,T))$ I 'm interested in the $L^...
2
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1answer
50 views

Uniqueness of Non-Linear Heat Equation

The following problem comes from an old exam: Consider \begin{cases} u_t - \Delta u + |u_{x_1}| = 0 \text{ in } \mathbb{R}^{n} \times (0,\infty) \\ u(x,0)=g(x) \text{ in } \mathbb{R}^n \end{cases} ...
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0answers
23 views

Discrete Fourier Interpolation Proof

This question is within the context of developing the discrete Fourier interpolation. We begin with an interval $[a,b]$ with a uniform partition $a = x_0 < ··· < x_{N−1} < b$, where each $...
2
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1answer
38 views

What does constant along characteristic mean

When we have linear or quasilinear first order pde $$ a(x,y,u) u_x + b(x,y,u) u_y = c(x,y,u) $$ And suppose we have found characteristics with prescribed Cauchy data $u |_{\text{curve}} = \phi$ I ...
2
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1answer
34 views

Pazy's book: Negation of $\overline{\lim}_{x\uparrow a}|f(x)|<\infty$?

I was reading the proof of Theorem 6.2.2., Pazy, Semigroups of Linear Operators...Equations and confused. Essentially we want to show ${\lim}_{x\uparrow a}|f(x)|=\infty$. I believe by definition this ...
0
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2answers
25 views

$u_{xx} + u_x = 0$ ; parabolic PDE

Find the general solution to the equation $u_{xx} + u_x = 0$, assuming that $u$ is a function of two variables, $x$ and $y$ I know this seems like a simple question and I know it has something to do ...
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0answers
22 views

A parabolic Morrey-Sobolev inequality

Let $B \subset \mathbb R^2$ be the unit ball and $T>0.$ Let $u \in W^{2,1}_p(B \times [0,T]),$ that is $u \in L^p(B \times [0,T])$ and we also have, $$ \partial_t u, \nabla u, \nabla^2 u \in L^p(B \...
0
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1answer
44 views

How to show that the mild solution to this parabolic equation is also a classical solution?

Let $U\subseteq \mathbb{R}^n$ be a smooth bounded domain and consider the following problem: $$ \begin{cases} \partial_t u = \Delta u + u &\text{in } U\times (0, \infty)\\ u = 0 &\text{on } \...
0
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1answer
35 views

In Evans (section on parabolic equations), why can he assume that $u(x_0, t_0)$ is positive?

In Evan's proof for the Weak maximum principle He simply assumes that $u(x_0, t_0) > 0$. Why can he make this assumption? I have spent some time on this but unfortunately I don't see it. Does it ...
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0answers
8 views

Proof of Proposition 5.2.3 An introduction to semilinear evolution equations / Thierry Cazenave and Alain Haraux

I am currently reading the book stated in the title and there is a part I do not understand in the proof. Before I state the proposition, I would like to clarify the general assumptions here. $\...
3
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3answers
55 views

Is it useful to convert a higher order PDE into a 1st order system?

I just learned how a higher order PDE can be converted into a system of PDEs.I am just wondering whether this is a standard way to solve some higher order PDEs which are immune to other methods or is ...
1
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1answer
42 views

Vorticity for the Navier-Stokes equations

The definition that I know of is the vorticity $\omega$ is the curl of the velocity $u$. Now I'm reading a note saying $\omega$ is defined to be the $d\times d$ antisymmetric matrix: $$\omega = \frac{...
3
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1answer
86 views

Continuity of mild solution

This is a question on the proof of Theorem 6.1.2, Pazy's book Semigroups of Linear Operators and Applications to Partial Differential Equations. Also the title might not be too accurate, as my ...
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0answers
19 views

evolution equation satisfied by projection

Consider the linear parabolic problem (as in Evans) $$\begin{cases} \partial_t u+Lu=f\ \: \mbox{in} \:\Omega \times (0,T]\\ u=0\ \: \mbox{on} \:\partial \Omega \times (0,T]\\ u(\cdot,0)=g \ \: \mbox{...
1
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1answer
37 views

Uniform boundedness of weak solution

Let $u_n\in W_{0}^{1,p}(\Omega)$ be a positive weak solution of the equation: $$ -\Delta_p u=\frac{f_n(x)}{(u+\frac{1}{n})^\delta}\text{ in }\Omega. $$ Let $p=N$ and $f\in L^m(\Omega)$ for some $m&...
0
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2answers
124 views

Analytical solution of heat equation with non-homogenous boundary conditions

I am trying to get analytical solution of heat equation with non-homogenous boundary conditions, which i can code in MATLAB and compare with my numerical results. In short, i am unable to reach the ...
1
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0answers
57 views

Physical interpretation of a singular pde

Suppose $\Omega$ is a bounded domain in $\mathbb{R}^N$ and consider the following Dirichlet boundary condition: $$ -\Delta u=\frac{f(x)}{u^\delta}\text{ in }\Omega, u>0\text{ in }\Omega; $$ where $\...
6
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2answers
82 views

Uniqueness of a parabolic-like PDE

Let $\Omega$ be a bounded open set in $\mathbb{R}^n$ and $0 < T < \infty$. Let $\Omega_T = \Omega \times (0, T]$. Given any functions $f, g, h$ show that \begin{equation} u_t - \Delta u + |Du|^2 ...
4
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0answers
48 views

Estimate a solution to the parabolic equation according to its initial value

The following is my question. Let $F$ be smooth, and $u$ a bounded smooth solution in $\mathbb R^+ \times \mathbb R^n$ to the parabolic equation $$u_t+\partial_xF(u)=\Delta u,\ \ \ \ \ u(0)=u_0$$ ...
0
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0answers
19 views

Navier-Stokes eq.

It is well known that the semigroup generated by the Stokes operator has exponential decay in the $L^p$ norm on bounded domain. But I do not know the proof. How do you prove this fact?
0
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0answers
76 views

Parabolic Maximum Principle for weak solutions

Let $B_1(0)\subset\mathbb{R}^d$ be the unit ball of $\mathbb{R}^d$. Let $u: \overline{B_1(0)}\times [0,T]\to \mathbb{R}$ be a function such that: i) $u\in C(\overline{B_1(0)}\times [0,T])$ ii) $u\...
1
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1answer
49 views

Derivation of the weak form for a parabolic PDE - initial-boundary problem

I am reading a paper that seems to provide a solution for the problem I am facing but being unfamiliar with variational calculus I get lost in notation. I am trying to derive the weak form from the ...
-1
votes
1answer
74 views

Weber-Hermite differential equation

I was solving a quantum mechanics problem (harmonic oscillateur) and i need to solve this Weber-Hermite differential equation in an analytic method: $$y"-x^2(y)=0$$ I know the solution of this ...
0
votes
1answer
60 views

Solution of the non-linear Heat Equation

How to find $v$ such that $$u(x,t)=t^{-\alpha}v(xt^{-\beta})$$ is the solution of the non-linear Heat equation : $$u_t-\Delta(u^{\gamma})=0$$where $\frac{n-2}{n}<\gamma<1$ , $x$ $\in R^n$ ...
1
vote
0answers
20 views

Parabolic semilinear Problems and Continuous Dependence on Initial Data

Let $(X, ||\,\cdot\,||)$ be a Banach Space and its norm. We want to look for $T>0$ and $u$ a solution of the following problem : $\begin{cases} u \in C([0,T],D(A))\cap C^{1}([0,T],X) \\ u'(t) = Au(...