Exercise 1 (Properties of the fundamental solution to the Heat equation). Let \[\Phi(x,t) = \frac{1}{\left( 4\pi kt\right)^{\frac{n}{2}}}e^{-\frac{\left\lvert x\right\rvert^2}{4kt}},\] where \(k>0\) is given, be the fundamental solution to the Heat equation.
Show that for any \(p\geq 1\) \[\left\lVert\Phi(x,t)\right\rVert_{L^p\left( \mathbb{R}^n\right)}=\frac{\left( 4\pi kt\right)^{\frac{n(1-p)}{2p}}}{p^{\frac{n}{2p}}}\] You may use the fact that \(\int_{\mathbb{R}^n}e^{-\frac{\left\lvert y\right\rvert^2}{2}}dy=\left( 2\pi\right)^{\frac{n}{2}}\).
In particular, \[\int_{\mathbb{R}^n} \Phi(x,t)dx=\left\lVert\Phi\left( \cdot,t\right)\right\rVert_{L^1\left( \mathbb{R}^n\right)}=1\] for all \(t>0\).
Young’s convolution inequality states that if \(f\in L^p\left( \mathbb{R}^n\right)\) and \(g\in L^q\left( \mathbb{R}^n\right)\) where \(p,q\in [1,\infty]\) are such that \[\frac{1}{p}+\frac{1}{q}=1+\frac{1}{r}\] for some \(r\in [1,\infty]\) then \(f\ast g \in L^r\left( \mathbb{R}^n\right)\) and \[\left\lVert f\ast g\right\rVert_{L^r\left( \mathbb{R}^n\right)} \leq \left\lVert f\right\rVert_{L^p\left( \mathbb{R}^n\right)}\left\lVert g\right\rVert_{L^q\left( \mathbb{R}^n\right)}.\] Use this to show that for any \(g\in C_c\left( \mathbb{R}^n\right)\) we have that for any \(r\geq 1\) and any \(1\leq p\leq r\) \[\left\lVert\Phi(\cdot,t)\ast g\right\rVert_{L^r} \leq \frac{C_{p,n,k}\left\lVert g\right\rVert_{L^{\frac{rp}{rp+p-r}}\left( \mathbb{R}^n\right)}}{t^{\frac{n(p-1)}{2p}}}\] where \(C_{p,n,k}\) is an explicit constant that depends only on \(p\), \(n\), and \(k\).
Exercise 2 (Bonus – additional bounds on \(u=\Phi\ast g\)). Consider the solution to the heat equation \[\nonumber \begin{split} &u_t(x,t)-k\Delta u(x,t) = 0,\qquad x\in\mathbb{R}^n,\;t>0,\\ &u(x,0)=g(x), \qquad\qquad\qquad \; x\in\mathbb{R}^n, \end{split}\] where \(k>0\) and \(g\in L^1\left( \mathbb{R}^n\right)\), given by \[u(x,t) = \int_{\mathbb{R}^n}\Phi(x-y,t)g(y)dy\] with \[\Phi(x,t) = \frac{1}{\left( 4\pi kt\right)^{\frac{n}{2}}}e^{-\frac{\left\lvert x\right\rvert^2}{4kt}}.\]
Show that if \(g\in L^\infty\left( \mathbb{R}^n\right)\) then so is \(u\). Moreover \[\left\lVert u\right\rVert_{L^\infty\left( \mathbb{R}^n\times (0,\infty)\right)} \leq \left\lVert g\right\rVert_{L^\infty\left( \mathbb{R}^n\right)}.\]
If \(g\in L^2\left( \mathbb{R}^n\right)\) one can show that \(u(\cdot,t)\in L^2\left( \mathbb{R}^n\right)\) for all \(t>0\) (follows from the previous exercise!). Show that for any \(t>0\) \[\left\lVert u\left( \cdot,t\right)\right\rVert_{L^2\left( \mathbb{R}^n\right)} \leq \left\lVert g\right\rVert_{L^2\left( \mathbb{R}^n\right)}.\]
Hint: You may use the following: \[\Phi\geq0,\qquad \text{and}\qquad\int_{\mathbb{R}^n}\Phi(z,t)dz=1,\;\;\forall t>0.\] \[\widehat{\Phi}(\xi,t) = \frac{1}{\left( 2\pi\right)^{\frac{n}{2}}}e^{-kt\left\lvert\xi\right\rvert^2}.\]
Exercise 3 (The energy method: Uniqueness for the heat equation in a time dependent domain). Let \(k>0\), \(T>0\) be given. Let \(a,b:[0,T]\to \mathbb{R}\) be smooth functions such that \(a(t)<b(t)\) for all \(t\in [0,T]\). Let \(U\subset \mathbb{R}\times (0,T]\) be the non-cylindrical domain \[U=\left\lbrace (x,t)\in \mathbb{R}\times (0,T]\;|\;a(t)<x<b(t)\right\rbrace.\] Consider the heat equation \[\nonumber \left\{ \begin{array}{ll} u_t-ku_{xx} =f(x,t)& \left( x,t\right)\in U,\\ u(a(t),t)=g_1(t) & t\in [0,T],\\ u(b(t),t)=g_2(t) & t\in [0,T],\\ u(x,0)=u_0(x) & x\in \left( a(0),b(0)\right). \end{array} \right.\] Use the energy method to prove that the equation has at most one smooth solution.
Exercise 4 (Grönwall’s inequality). Another
important inequality in the study of PDEs, in particular in the study of
long time behaviour of solutions, is the so-called Grönwall’s inequality
which state that if \(y:[0,T]\to
\mathbb{R}\) is continuous and differentiable on \((0,T)\), and if there exists \(\lambda \in \mathbb{R}\) such that \[y^\prime (t) \leq \lambda y(t)\] then we
have that \[y(t) \leq y(0) e^{\lambda
t}.\] Prove Grönwall’s inequality.
Remark: The above inequality can be generalised to show
that if \[y^\prime (t) \leq \lambda
(t)y(t)\] then \(y(t) \leq
y(0)e^{\int_{0}^t \lambda(s)ds}\). There are additional important
Grönwall inequalities which we won’t mention at this point.
Exercise 5. Let \(T>0\) be given and define \[\Omega_T= \left( a,b\right)\times (0,T]\] for a given \(-\infty<a<b<\infty\).
Show that there exists at most one solution \(u\in C_1^2\left( \Omega_T\right)\cap C\left( \overline{\Omega_T}\right)\) to the problem \[\begin{equation} \label{eq:heat} \left\{ \begin{array}{ll} u_t-u_{xx} =1& \left( x,t\right)\in \Omega_T,\\ u=0 & \left( x,t\right)\in \Gamma_T, \end{array} \right. \end{equation}\] where \(\Gamma_T = (a,b)\times \left\lbrace 0\right\rbrace \cup \left\lbrace a\right\rbrace\times [0,T] \cup \left\lbrace b\right\rbrace\times [0,T]\).
Assume that \(u\) is a \(C_1^2\left( \Omega_T\right)\cap C\left( \overline{\Omega_T}\right)\) solution to \(\eqref{eq:heat}\). Show that for any \(\left( x,t\right)\in \Omega_T\) \[0\leq u(x,t) \leq t.\]