0
votes
0answers
43 views

Parabolic partial differential equation, initial conditions

Let $U\subset\mathbb{R}^n$ be open bounded, $T>0$. Given the parabolic PDE $$\partial_tf+a\partial_xf+b\partial_{xx}f = g \qquad (1)$$ I'm interested in the initial and boundary conditions that ...
1
vote
0answers
107 views

Comparison principle for partial differential equation with singular coefficients

How (or if) a comparison principle works in the case of equations singular at some point? For example, I am analyzing a partial differential equation $$ ...
4
votes
2answers
265 views

Numerical solution to diffusion-like equation with negative diffusion coefficient region?

I am trying to numerically solve the initial value problem (see later discussion for ICs) $$ x \frac{\partial f}{\partial t} = \frac{\partial}{\partial x} (1-x^2) \frac{\partial f}{\partial x} - f$$ ...
3
votes
1answer
123 views

Monotonicity preserving parabolic operators

Let $$ \mathcal{L}\equiv\sum_{i,j}^{n}a_{i,j}\frac{\partial^{2}}{\partial x_{i}\,\partial x_{j}}+\sum_{i}^{n}b_{i}\frac{\partial}{\partial x_{i}}+c $$ be uniformly elliptic on ...
2
votes
1answer
263 views

Uniqueness of classical solution with degenerate boundary

Consider heat equation on the domain $\Omega = (0,1)\times (0,1)$ in the form of $$ \partial_{t} u = \frac 1 2 x^{3} (1-x) \partial_{xx} u, \quad (x,t) \in \Omega$$ with initial data $u(x,0) = x$ for ...
0
votes
1answer
246 views

LINEAR Parabolic equations. Smooth dependence from initial data

I am looking for results that show smooth dependence of a solution to a parabolic equation, from the initial data. More specifically I have the following problem: CONSIDER spaces $P:=\mathbb{R}^k$ ...
5
votes
1answer
805 views

Short time existence on nonlinear parabolic PDE

I saw several papers that without proof accept the fact "Short time existence on nonlinear parabolic PDE" is there any affirmative proof of this fact? in which book we have this fact, the number of ...
0
votes
0answers
362 views

What is the Schauder estimate on usual Hölder space for parabolic type equations

What is the Schauder estimate on usual Holder space for parabolic type equations ?