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.

330 questions
Filter by
Sorted by
Tagged with
34 views

How to prove the existence of weak solutions of parabolic PDEs using Rothe's method?

I am reading a textbook on PDE (Introduction to Elliptic and Parabolic Partial Differential Equations by Z. Wu, J. Yin and C. Wang, written in Chinese) where a proof of the existence of weak solutions ...
26 views

Pointwise estimate of solutions to the parabolic equation with a monotonic drift

I wonder for a parabolic equation $$u_t+(a(t,x)u)_x= u_{xx},$$ if we know that $a(t,x)$ is monotonic decreasing in $x$ with $a(t,-\infty)=C_L, a(t,+\infty)=C_R$, $C_L>C_R\geq 0$, are there results ...
79 views

Reaction-diffusion systems treated as dynamical systems

I wonder if there is a good reference on reaction-diffusion systems on $\mathbb{R}^N$, that treats them as dynamical systems. I have the book of Alain Haraux – Systèmes dynamiques dissipatifs et ...
54 views

57 views

While doing Lp estimates, is the constant C monotonically increasing with respect to the parameters it depends on?

For example, consider the third boundary value problem: \begin{align} &\frac{\partial u}{\partial t}-\sum_{i,j=1}^n a_{ij}(x,t) \frac{\partial^2 u}{\partial x_i \partial x_j} + \sum_{i=1}^n ...
28 views

Existence and uniqueness for fractional parabolic equation with transport term

Let us consider the problem \begin{cases} u_t+(-\Delta)^{\sigma}u+\mathrm{div}(a(t,x)u) = 0 & \text{in } \mathbb{R}^n \times [0, T) \\ u(x,0)=u_0(x) & \text{in } \...
67 views

Spectrum of an elliptic operator in divergence form with a reflecting boundary condition

Let $\Omega$ be a bounded open domain and $v:\Omega\to\mathbb{R}^n$. Consider the following elliptic operator in divergence form, defined on smooth functions $u: \Omega \to \mathbb{R}$ \begin{align} L ...
42 views

Rigorous energy estimate for advection-diffusion equation

Let $a \in L^q([0,T];L^p(\mathbb R^N))$ with $2/q + N/p \le 1$ and $q \in [2,\infty), p \in (N,\infty) \text{ if } N \ge 2$ $q \in [2,4], p \in [2,\infty] \text{ if } N = 1$ and consider the ...
71 views

Lower Gaussian estimates for Dirichlet heat kernel on manifolds

Let $(M,g)$ be a Riemannian $n$-manifold with $Ric_g\ge -Kg$, $\Omega\subset M$ be an open subset. We can define Dirichlet heat kernel on $\Omega$, $p_{\Omega}(y,t,y',t')$ as the minimal fundamental ...
147 views

Forwards Feynman–Kac formula

This might be a simple question, but I'm having trouble with it. Consider the Cauchy problem with final condition. \begin{cases} \frac{\partial u}{\partial t}(t,x) + \mathcal{L}u(t,x) ...
106 views

134 views

Looking for a reference or the procedure on how to solve the parabolic equation with $L^2$-weight

Let $\zeta, u_0\in L^2(\Omega)$, with $\zeta \geq 0$ and $\Omega\subset \Bbb R^d$ open and bounded. \label{Star-3.7} \begin{cases} \partial_t u -\Delta u + \zeta u=0 &\mbox{ in }\...
82 views

Wellposedness of semilinear fractional heat equation

Do you have a reference on the wellposedness of the problem \begin{align*} \begin{cases} u_t + (-\Delta)^su = f(u) \qquad & (t,x) \in (0,+\infty) \times (0,1) \\ u(0,\cdot) = u_0 & x \in (0,1)...
45 views

77 views

Equation $u_t - u_{tx} - u_{xx} = 0$

Consider the following heat equation with a perturbation given by a second order mixed derivative: $$u_t - u_{tx} - u_{xx} = 0$$ Does this equation have a name? How can one prove a wellposedness ...
73 views

Estimate in vanishing viscosity for the difference $\Vert u^\epsilon - u^\eta \Vert_{L^2(\mathbb R^N)}$

Consider the following advection-diffusion equation $$\begin{cases} u^\epsilon_t + f(u^\epsilon)_x = \epsilon \Delta u^\epsilon\\ u^\epsilon(0,\cdot) = u_0, \end{cases}$$ How can one prove an ...
73 views

643 views

Modified energy method for transformed Fokker-Planck equation (tricky integration by parts…)

I came across Villani's paper titled "Hypocoercive diffusion operators" and couldn't figure out a computation that is skipped in that paper. Specifically, consider the following transformed ...
54 views

Reference to log-transition-density of a diffusion process

Consider the scalar diffusion $X=(X_t)_{t\geq 0}$ given by $$\mathrm{d}X_t \ = \ b(X_t)\,\mathrm{d}t \ + \ \sigma(X_t)\,\mathrm{d}B_t, \qquad X_0=\xi,$$ with $b$, $\sigma$ smooth, $\xi$ absolutely ...
58 views

Uniqueness of solution to Cauchy problem with quadratic nonlinearity

Consider the non-linear differential operator $$\mathfrak{L}: \ C^2((0,T)\times\mathbb{R}^2)\ni\varphi\equiv\varphi(t,x,y) \, \mapsto \, \partial_x^2\varphi + (\partial_x\varphi)^2.$$ For \$U\subset\...
89 views

78 views

Is there a better reference for existence/regularity for parabolic PDEs (and systems) than the book of Ladyzenskaja, Solonnikov, Uralceva?

The book of Ladyzenskaja, Solonnikov, Uralceva contains almost everything most people need yet the typesetting and notation is disgusting to the eye. Is there any better text that covers the same type ...
Consider the free boundary problem $$\min\{u_t - u_{xx} -1, u \} = 0 \qquad \text{ in } (0,T)\times (-1,1) \\ u(0,\cdot) = 0 \qquad \text{ in } (-1,1)\\ u(\cdot, -1) = u(\cdot, 1) = 0 \qquad \... 0answers 32 views classical solution of nondegenerate HJB equation Let b\in C(\mathbb R) and L \in C_b^2(\mathbb R). Consider an equation$$v_t (x, t) + \inf_{a\in A} \{b(a) v_x(x, t) + a^2 \} + v_{xx}(x, t) + L(x) = 0, \hbox{ on } \mathbb R \times (0, 1)$$with ... 0answers 68 views improved regularization for \lambda-convex gradient flows It is well-known that gradient-flows of convex functionals are "parabolic" in some vague sense, and accordingly solutions tend to regularize instataneously. In the abstract context of gradient flows ... 0answers 68 views Solution existence for two-dimensional parabolic PDEs I am looking for a solution (f,g) \in C^{1,2}([0;T]\times\mathbb R;\mathbb R^2) to the following PDE system$$ f_t(t,x) + a_1(t,x) f_x(t,x) + a_2(t,x) f_{xx}(t,x) - b_1 f(t,x)^2 + c(t) g(t,x) - d(t,...
Consider heat equation with a drift (=reaction-diffusion equation) $$\frac{\partial u}{\partial t}=\frac{\partial^2 u}{\partial x^2}+f(t,u(t,x)), \quad t\ge0,\, x\in [0,1]$$ with periodic or ...