# Questions tagged [parabolic-pde]

This tag is for questions relating to "Parabolic partial differential equation", are usually time dependent and represent diffusion-like processes. Solutions are smooth in space but may possess singularities. However, information travels at infinite speed in a parabolic system.

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### Existence of solution to a parabolic bvp

I should study some parabolic PDEs, but I'm not an expert, so I would like to ask your advice. First, could you give me some useful references concerning PARABOLIC PDES? I started reading DiBenedetto'...
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### The exact, formal definition of a prabolic problem/equation

While searching for information about properties of parabolic problems, I stumbled upon a publication titled "Study of Nonlinear Parabolic Problems". This made me wonder why the writers ...
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### numerically compute eigenfunctions of $a(u,v)=\langle f(\nabla u)\nabla u,\nabla v\rangle_{L^2}$

Let $D:=(0,1)^2$ and consider the nonnegative form $$a(u,v):=\langle f(\nabla u)\nabla u,\nabla v\rangle_{L^2(D;\:\mathbb R^2)}$$ for $u,v\in L^2(D)$ where $f:\mathbb R^2\to(0,\infty)$ is a smooth ...
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### A non-separable heat-convection differential equation with 1 space-variable

There is a problem I have much difficulties to solve. It is about a temperature difference induced by pressure variation. It is this equation solved by kieransquared, but this time with specific ...
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### Time Discretisation of a System of PDEs

Suppose we have an arbitrary system of PDEs $$\partial_t u - D_1 \Delta u + a(v) u = f(t)$$ $$\partial_t v - D_2 \Delta v + b(u) v = g(t)$$ We want to discretise the system in time. We use the ...
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1 vote
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### Inequality for a parabolic pde

I am trying to solve the following exercise. Consider the linear parabolic equation \begin{equation} \partial _t u -\nabla (A(x,t)\nabla u)=f(x,t), x \in \Omega,t>0, \end{equation} subject to the ...
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### Inequality for parabolic pde

I am trying to solve the following exercise. Consider the linear parabolic equation \begin{equation} \partial _t u -\nabla (A(x,t)\nabla u)=f(x,t), x \in \Omega,t>0, \end{equation} subject to the ...
1 vote
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### Is there any functional f(s) such that this system of differential equations has an analytical solution?

I am dealing with the following system of partial differential equations that describes the effects of random motility on bacteria that consume a diffusible substrate: \begin{cases} b_{t} = \mu b_{xx} ...
1 vote
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### Parabolic PDE whose solution is a product of lower dimensional solutions?

Consider the partial differential equation: $$\bigg(\sum_{i=1}^n s_i\bigg) \Delta \Phi=nu \Phi_u$$ for $\Phi(u,s_1,s_2,\cdot\cdot\cdot,s_n).$ This is a linear parabolic partial differential equation ...
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### (Evans) Incomplete proof of the existence and uniqueness of weak solutions to a reaction-diffusion system

In Example 1 in section 9.2 (Fixed Point Methods), Evans employs Banach's Fixed Point Theorem to prove the existence and uniqueness of a weak solution to a system of reaction-diffusion equations. A ...
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### Reversal of Variable Transformation in the Solution of Inhomogeneous Dirichlet PDE

Initially, we are dealing with a non-homogeneous Dirichlet problem: \begin{equation} \begin{cases} q_t(z,t)-D(z,t)q_{zz}(z,t) = 0 & \text{for $0<z<L,t>0$} \\ q(0,t)=a &...