Parabolic Equations

Consider the Cauchy problem for the Heat Operator in $\mathbb {R}^{1}$:

u_t = u_xx (-< x < , t > 0) u(x,0) = f(x) (-< x < ),

where $f$ is bounded, continuous, and satisfies

Show that there exists a constant $C$ such that

for all $-\infty < x < \infty$, $t > 0$.

Let $\Omega \subset \mathbb {R}^{n}$ be an open set with smooth boundary and suppose $u \in C^{\infty }(\Omega \times [0,\infty ))$ is a solution of the equation

with $u = 0$ on $\partial \Omega \times [0,\infty )$. Assume that

Let

Prove that $\psi (t) \to 0$ as $t \to +\infty$.

Consider the Cauchy problem

u_t = u_xx (-< x < , t > 0), u(x,0) = f(x),

where $f$ is bounded and continuous.

Consider the problem

u_t = u_xx (x R, t > 0) u(x,0) = u_0(x) L^2(R).

Find a bound on $\int_{-\infty }^{\infty } \left|u_x(x,t)\right|^2\, dx$ in $t > 0$.

For the indicated domain $\Omega$ (interior of the parabola) let $\Omega_T$ denote the points $(x,t)$ with $x \in \Omega$ and $t < T$. Let $B_1$ denote the open line-segment forming the top of $\partial \Omega_T$, and let $B_2 = \partial \Omega_T \setminus B_1$. Let $u(x,t) \in C^0(\overline{\Omega_T})$ with $u_{xx}, u_t \in C^0(\Omega_T \cup B_1)$ be a solution of

with $a, f \in C^0(\overline{\Omega })$. Show that if $a < 0$, $f \leq 0$ in $\Omega_T$, then every non-constant solution $u$ assumes its negative minimum (if one exists) on $B_2$. What can you say if $a < 0$, $f \geq 0$ in $\Omega_T$?

Consider the boundary-value problem

u_xx + u_t = 0 (0 x and t > 0), B.C. u(0,t) = 0, u_x(,t) = 0, I.C. u(x,0) = f(x).

Use the sequence $\left\{ f_n(x) = \dfrac {2}{n+1} \sin \left(\dfrac {n+1}{2}x\right)\right\}$ and an appropriate space of continuous functions to decide whether this problem is well-posed.

In Probability Theory one encounters the Ornstein-Uhlenbeck process, in which the particles of Brownian motion are subjected to an elastic force. One is required to solve

u_t = u_xx - xu_x (x R, t > 0) u(x,0) = u_0(x).

Assume $u_0 \in S$ and find $u$. (You should be able to find $u$ explicitly; if not, leave your result in the form of a one-dimensional integral.)

Let $L$ be the usual heat operator

let $\phi \in D(\mathbb {R}^{n})$, and let $h$ be a fixed point of $\mathbb {R}^{n}$. Consider the problem

(Lu)(x,t) = u(x-h,t), x R^n, t > 0, u(x,0) = (x), x R^n

for an unknown function $u(x,t)$.

Let $\Omega \subset \mathbb {R}^{n}$ be a bounded domain and assume $u(x,t) \geq 0$ is a function with $u \in C^2(\overline{\Omega } \times [0,\infty ))$, which solves the equation

(heat conduction with heat loss due to radiation) with the boundary condition $u|_{\partial \Omega } = 0$. Prove that we can find a constant $C$ such that

regardless of the initial value $u(x,0)$.

Let

satisfy

with $g,h$ continuous and $g,h < 1$. Prove that $\max u < 1$ on $\overline{B(0,r)} \times [0,T]$.

Let $u \in C^2(\Omega \times (0,T)) \cap C^0(\overline{\Omega } \times [0,T])$ be a solution to the problem

where $\Omega$ is a smooth bounded domain in $\mathbb {R}^{n}$.

For each $p \in \mathbb {R}^{n}$, let $A(p)$ be a real positive definite $n \times n$ matrix with entries $a_{ij}(p)$. Assume $A \in C^1(\mathbb {R}^{n})$. Let $\Omega \subset \mathbb {R}^{n}$ be a bounded open set. Prove (from first principles) the following comparison principle:

Let $u$ and $v$ be $C^2$ solutions to the inequalities

and

respectively. Suppose, furthermore, that $u$ and $v$ are continuous on $\overline{\Omega } \times [0,\infty )$ and satisfy the condition $u \leq v$ on the parabolic boundary

Then $u \leq v$ on $\overline{\Omega } \times [0,\infty )$.

For $\Omega \subset \mathbb {R}^{n}$ bounded and open, let $u \in C^2(\overline{\Omega } \times [0,\infty ))$. Assume $a_{ij}(x,t,z,p)$ and $b_i(x,t,z)$ are continuous functions of their arguments and assume that

where $\lambda$ is a positive function. Suppose $u$ satisfies the inequality

Prove that for any $T > 0$

where $Q = (\Omega \times \left\{ 0\right\} ) \cup (\partial \Omega \times [0,T])$.

Let $\Omega$ be a bounded open set of $\mathbb {R}^{n}$ with smooth boundary $\partial \Omega$. Let $u \in C^2(\Omega \times (0,\infty ))$, with $u$ continuous on $\Omega \times [0,\infty )$, be a solution of the initial-boundary problem

u_t - u = (u) for x , t > 0 u(x,t) = 0 for x u(x,0) = f(x) for x .

Show that if $f(x) \leq 1$, then $u(x,t) \leq e^t$ for all $x \in \Omega , t > 0$.

Consider the Cauchy problem for Burger's equation:

u_t - u_xx + uu_x = 0, R (0,), u(x,0) = g(x)

where $g \in L^{\infty } \cap L^1(-\infty ,\infty )$. Derive the explicit formula of the solution for the Cauchy problem.

Consider the rectangle $Q = [0,a] \times [0,b] \subset \mathbb {R}^{2}$.