It has long been known that the Cauchy initial-value problem for the classical heat equation on $\mathbb{R}$ (or $\mathbb{R}^n$) doesn't have unique solutions, without additional a …

Let $M$ be a smooth manifold with Riemannian metric $g$, which is not smooth but only continuous. Question: Is there still an asymptotic expansion of the heat kernel of the form $ …

This is crosspost from math.stackexchange http://math.stackexchange.com/questions/311213/heat-kernel-asymptotics-on-manifold-with-boundary where the question did not yield any answ …

My stackexchange post [http://math.stackexchange.com/questions/275830/schrodinger-kernels-on-manifolds] was somewhat unsatisfactory (also because I may not have stated clear enough …

I am familiar with Witten's proof of the Morse inequalities by semiclassical analysis. Hey uses semiclassical expansions of the first eigenfunctions to construct the Morse complex …

Consider a uniform bar of length L which has been insulated both ends. If the inital temperature is given by u(x,0)=f(x) (some general distribution), and there is a constant, nonze …

Let $(M,g^{TM})$ a Riemannian manifold of dimension $n$ and $\Delta$ the Laplace–Beltrami operator. I would like to find a reference (analytic or probabilistic) for the following c …

Determine the equilibrium temperature distribution, if it exists, for each of the folowing problems. Specify values/ranges for A if necessary. $\frac{\partial u}{\partial t} = \fr …

Consider the heat equation $u_t=\Delta u$ with Neumann boundary condition and initial condition $u(x,0)=u^0(x)$ in a bounded domain $\Omega$ with smooth boundary. Is this true: A …

Consider the initial value problem $$ \partial_t u = \partial_{xx} u$$ $$ u(0,x) = u_0(x)$$ for the heat equation in one dimension, where $u_0: {\bf R} \to {\bf R}$ is a smooth ini …

For the heat equation $(\partial_t-\partial_x^2)f(t,x)=0$ defined on $[0,T)\times(-\infty,\infty)$, to obtain uniqueness of the initial value problem, usually it is required to lim …

Consider the heat equation $$\dot{u} = \Delta u$$ with initial conditions $$u_0 = \delta(x)$$ for some point $x$ in the domain $\Omega$ of the problem. If $\Omega$ is $\mathbb …

I want to run water through a black hose to increase the water's temperature via the sun. Is there a hose placement pattern that would maximize the surface area available to the s …