Reaction-Diffusion-Advection Equations

Suppose that \(U\subset \mathbb {R}^n\) is a bounded domain and \(b:U\to \mathbb {R}^n\) is a smooth velocity field. Suppose that \(u(x, t)\) solves the reaction-diffusion- advection equation with a logistic reaction term

Show that if \(u(x, 0) < 1\) in \(U\) then \(u(x, t) < 1\) for all \(t > 0\) and all \(x \in U\).

1. Why does the heat/diffusion equation imply infinite propagation speed?
2. Why does the wave equation imply finite propagation speed?

Formulate and prove the maximum principle for the heat/diffusion equation \(u_t=u_{xx}\).

Show that the heat/diffusion equation with the wrong sign \(u_t=-u_{xx}\) is extremely unstable (and so, the Cauchy problem for this equation is ill posed in the sense of Hadamard).

Consider the global Cauchy problem \[ u_t=Du_{xx}+bu_x+cu,\qquad x\in \mathbb {R},\ t>0,\qquad u(x,0)=g(x), \] where \(D>0\), \(b\) and \(c\) are constant coefficients.

1. Convert to a diffusion equation by setting \(v(x,t)=u(x,t)\exp (hx+kt)\) for appropriately chosen \(h,k\).
2. Solve the diffusion equation for \(v\) using a transform method.
3. Hence, show that if \(c<0\) and \(g\) is bounded, then \(u(x,t)\to 0\) as \(t\to \infty \).

1. Derive the “diffusion” or “heat” equation in several spatial dimensions.
2. In the derivation of this equation, it is routinely assumed that \[ \frac {d}{dt}\int _\Omega u(x,t)\,dx=\int _\Omega \frac {\partial u}{\partial t}\,dx. \] Under what conditions is this interchange of order valid? Prove your result. Hint: Examine the function \[ w(t)=\int _0^t\left (\int _\Omega \frac {\partial u(x,t)}{\partial t}\,dx\right )\,dt. \]

Describe quantitatively the diffusion of 3 units of a diffusing material which is placed at the initial moment \(t=0\) at the origin of an infinite two-dimensional plane. Assume that the plane has unit conductivity.