# 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.

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### Prove comparison principle for $u_t + f(u)_x = g(t,x)$ with $g \in L^\infty(0,T; L^2(\mathbb R))$

Let us consider $$u_t + f(u)_x = k u_{xx} + g(t,x),$$ with $k>0$, $g \in L^\infty(0,T; L^2(\mathbb R))$ and $f \in W^{1,\infty}(\mathbb R)$ and $f \not \equiv 0$ (possibly, we can also add the ...
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### FEM based solution to parabolic problem

Consider the problem $$\begin{cases} u_t - \Delta u = 0 &\text{ on } \Omega\times (0,T)\\u=0 &\text{ on } \partial \Omega\times (0,T) \\ u(x,0)=g(x) &\text{ on } \Omega \end{cases}$$ ...
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### Looking for references to study $U^p$ and $V^p$ spaces

I am studying some papers in the analysis of nonlinear PDEs and I am encountering the $U^p$ and $V^p$ spaces for the first time. Where can I find references more detailed than papers? Edited The ...
1 vote
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### Is there any class of initial data for which the heat semigroup is increasing in time?

Lee and Ni proved in the paper Global existence, large time behavior and life span of solutions of semi linear parabolic Cauchy problem that the heat semigroup decays as $t$ tends to infinity, that is ...
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### Reference request: continuity of the derivatives of the (fundamental) solution to a parabolic equation

Consider the parabolic equation in $p: \mathbb R^2\to\mathbb R$ $$\partial_t p + b(t)\partial_x p + D(t,x)\partial^2_{xx}p=0,$$ where $b$, $D$ are nice enough functions. I look for the continuity of ...
1 vote
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### Is this generalization of differentiable manifolds to mixed dimensions a known object?

Suppose you are studying the evolution of some electromagnetic quantity in a conductor consisting of objects of several dimensions, i.e. wires, plates and balls. This would amount to studying the ...
1 vote
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### Parabolic/Elliptic equation with nonlinear gradient term

Let $a\in (0,1)$ and $(0,1) \subset \mathbb{R}$, we consider the below equation in $(0,1) \times (0,T)$ $$\partial_t u -\partial_x^2 u - \dfrac{1}{|u|^a} \partial_x u =f(x).$$ And $u(x,0)=x^{1/a}$ ...
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### Set invariance for differential inclusions applied to PDES?

This question is somewhat related to this one that I posted a while back on MSE, but the context has slightly changed since then. My question here relates to the consequences of a result in Weinberger'...
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### A variant to the Fokker–Planck equation

Consider the PDE of $p(t,x)\ge 0$ given as $$\partial_t p = \frac{\partial^2_{xx}p}{(1+m(t))^2} - \partial_x p,\quad \forall t,~x \in (0,\infty)$$ with initial and boundary conditions $p(0,\cdot)=\rho$...
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### Parabolic system with coupling in the diffusion

Let's consider the parabolic system $$\begin{cases} u_t - \Delta u -a\Delta(uv) = 0 \\ v_t - \Delta v - b\Delta(uv) = 0 \end{cases}$$ with $a,b >0$. What is the name of this system? Are there ...
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### Parabolic equation with Cauchy boundary condition

Consider the domain $[0,1] \times [0,T]$ and the uniformly parabolic operator $L -\partial_t$ with smooth coefficient. I would like to obtain the existence of the problem \begin{equation} \left\{\...
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### Gluing of two solutions to the same parabolic equation

Consider the domain $[0,1] \times [0,T]$ and the uniformly parabolic operator $L -\partial_t$ with smooth coefficient. Suppose I have $u_1(x,t) \in C^\infty([0,1] \times [0,T])$ solving \begin{...
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### Solution of parabolic partial differential equation using singular perturbation method

Consider the following parabolic partial differential equation (PDE) \begin{align} \label{eq:42} \left(\cos\psi \frac{\partial}{\partial r} + \frac{\gamma}{r} \sin\psi \frac{\partial}{\partial \psi} + ...