Department of Mathematics University of Utah Ph. D. Preliminary Exam in Differential Equations Autumn 1998

Consider the problem \[ \begin {aligned} \frac {dx}{dt} &= f(x,t),\\ x(t_0) &= x_0 \end {aligned} \tag {1.1} \] where \(f\in C(D,\mathbb {R}^n)\) and \(D\subset \mathbb {R}^n\times \mathbb {R}\) is open. Suppose that \(f\) is Lipschitz continuous in \(x\) for \((x,t)\in D\).

1. Prove the existence of a local (in \(t\)) solution to (1.1) using successive approximations.
2. Prove the uniqueness of this solution and show that it can be extended until it meets \(\partial D\).
3. Prove that the solution is continuous in \((x_0,t_0)\).

Consider \[ \begin {aligned} \frac {dx}{dt} &= Ax+f(x),\\ x(0) &= x_0 \end {aligned} \tag {2.1} \] where \(x\in \mathbb {R}^n\) and \(A\) is a constant \(n\times n\) matrix. Suppose that \(A\) is semisimple, that \(f\in C^2\) and that \(f(0)=0\), \(D_xf(0)=0\).

1. Define what is meant by the stable manifold, the center manifold and the unstable manifold at the origin for (2.1).
2. Give a representation of the solution of (2.1) when \(f=0\), in terms of the eigenvalues of \(A\) and the associated projections and nilpotents. Indicate how to compute the projections.
3. Discuss the asymptotic behavior of the solution in (b) for \(t\to \infty \). Characterize the manifolds defined in (a) as precisely as possible.
4. Give a representation of the solution of (2.1) for \(f\ne 0\). Under what conditions, if any, on \(A\) and the initial data can one conclude that this solution exists for all \(t\in \mathbb {R}^+\)? You need not prove all the assertions for the last part, but should at least state your reasons.
5. State the Hartman-Grobman theorem for \[ x'=F(x) \] and state clearly what further conditions, if any, are needed on (2.1) to apply the Hartman-Grobman theorem to (2.1).

Consider \[ x'=f(x) \tag {3.1} \] where \(x\in \mathbb {R}^n\) and \(f\in C^1(\mathbb {R}^n,\mathbb {R}^n)\). Suppose that the solution through any point \(x_0\in \mathbb {R}^n\) exists for all \(t\in \mathbb {R}\).

1. Define the positive and negative semiorbits of (3.1) through \(x_0\in \mathbb {R}^n\).
2. Define the \(\alpha \)- and \(\omega \)-limit sets of a point \(x_0\in \mathbb {R}^n\).
3. Consider the nonlinear spring equation \[ x''+g(x)=0 \tag {3.2} \] where \(g(x)\) is an odd function of \(x\) and \(g'(x)>0\). Describe the \(\alpha \)- and \(\omega \)-limit set for any point on the positive \(x\)-axis.
4. Consider \[ x''+x'+g(x)=0 \tag {3.3} \] where \(g\) is as in (c). Define what is meant by a Liapunov function for this equation and derive one (verify that it is a Liapunov function). Describe the \(\omega \)-limit set for any point in the plane.

Consider the system \[ x'=f(x) \tag {4.1} \] where \(f\in C^1(\mathbb {R}^n,\mathbb {R}^n)\), and suppose that (4.1) has a periodic solution \(\gamma (t)\) of least period \(T\).

1. Define what it means to say that \(\gamma (t)\) is orbitally asymptotically stable with asymptotic phase.
2. Let \[ \xi '=A(t)\xi \tag {4.2} \] be the variational equation (the linearization of (4.1)) relative to \(\gamma (t)\). State the Floquet representation for the fundamental matrix solution of (4.2) and discuss the stability of \(\gamma (t)\) in terms of properties of the fundamental matrix solution.
3. Consider the system \[ \begin {aligned} x' &= x-y-x(x^2+y^2),\\ y' &= x+y-y(x^2+y^2). \end {aligned} \] Find a nontrivial periodic solution of this equation, and prove (in the easiest way possible) that this periodic solution is orbitally asymptotically stable with asymptotic phase. Compute the asymptotic phase explicitly.
4. Compute the Poincare map associated with the periodic orbit you found in (c).

Consider the problem \[ \begin {aligned} Lu &\equiv -\frac {d}{dx}\left (p\frac {du}{dx}\right )+qu = \lambda u,\\ \alpha u(0)+\beta \frac {du}{dx}(0) &= 0,\\ \gamma u(1)+\delta \frac {du}{dx}(1) &= 0 \end {aligned} \tag {5.1} \] where \(x\in [0,1]\), \(p\in C^1[0,1]\), \(q\in C[0,1]\), and \(\alpha ,\beta ,\gamma ,\delta \) are constants.

1. Show that (5.1) is self-adjoint if and only if \(p\) and \(q\) are real, \(\gamma \bar {\delta }=\bar {\gamma }\delta \), and \(\alpha \bar {\beta }=\bar {\alpha }\beta \).
2. Suppose hereafter that \(p=q=1\), \(\alpha =1\), \(\beta =0\), \(\gamma =0\), \(\delta =1\). Compute the eigenvalues and eigenfunctions for \(L\) with the given boundary conditions.
3. Compute the Green’s function for \(L\) (with coefficients and boundary conditions as in (b)).
4. Show how to solve the nonhomogeneous problem \[ Lu=f(x) \] with \(L\) and the boundary conditions as in (c). What conditions on \(f\) suffice to make \(u\) a classical solution?

Consider the linear partial differential equation \[ Lu\equiv \sum _{\lvert \alpha \rvert \le k} a_\alpha (x)\partial ^\alpha u=f \tag {6.1} \] where \(\alpha =(\alpha _1,\ldots ,\alpha _n)\) is a multi-index. Let \(\Omega \subset \mathbb {R}^n\) be a bounded domain with piecewise smooth boundary and suppose throughout this problem that the coefficients in the equations are smooth.

1. Define what is meant by a weak solution of (6.1).
2. Define the principal part of \(L\), the characteristic variety of \(L\), and a characteristic surface for \(L\).

3. Obtain conditions on \(a\), \(b\) and \(c\) under which \[ Lu=a(x,y)u_{xx}+2b(x,y)u_{xy}+c(x,y)u_{yy}=0 \] is

   1. hyperbolic
   2. elliptic
   3. parabolic.
4. Define what is meant by a well-posed problem for a partial differential equation with boundary and/or initial data specified. Give an example of a well-posed problem for one of the three types of equation in (c) and briefly justify your assertions.

Consider the problem \[ \begin {aligned} u_{tt}-u_{xx} &= 0,\\ u(x,0) &= F(x),\\ u_t(x,0) &= 0,\\ u(0,t)=u(\pi ,t) &= 0 \end {aligned} \tag {7.1} \]

1. Use separation of variables to formally solve (7.1) when \(F\in C[0,\pi ]\).
2. Suppose that \(F\in C^4[0,\pi ]\) and that \(F''(0)=F''(\pi )=0\). Prove that the series you obtain in (a) defines a classical solution of (7.1) under the foregoing conditions on \(F\).

Consider the equation \[ a(x,y)u_x+b(x,y)u_y=c(x,y,u) \tag {8.1} \] where \(a\), \(b\) and \(c\) are suitably smooth.

1. Define what is meant by an integral surface for (8.1).
2. Write down the characteristic equations for (8.1) and show how they can be used to construct an integral surface of (8.1) that contains the curve \[ x=\varphi (s)\qquad y=\psi (s)\qquad u=\eta (s), \] where \(\varphi ,\psi \) and \(\eta \) are \(C^1\).
3. What additional conditions, if any, must be placed on the data to guarantee a locally unique solution? Explain carefully.
4. Consider the equation \[ u_t+uu_x=0 \tag {8.2} \] with initial data given by \[ u(x,0)=u_0(x)= \begin {cases} 1, & x\le 0,\\ 1-x, & x\in (0,1),\\ 0, & x\ge 1. \end {cases} \tag {8.3} \] Does (8.2)–(8.3) have a unique classical solution for \(t\ge 0\)? A unique weak solution for \(t\ge 0\)? Explain.

1. Let \(\Omega \subset \mathbb {R}^n\) be open and let \(u\in C(\Omega )\). Define what it means to say that \(u\) is subharmonic in \(\Omega \).
2. Let \(\Omega \) be open, bounded and connected and let \(u\in C^2(\Omega )\cap C^0(\bar {\Omega })\) be subharmonic. State the weak form of the maximum principle for \(u\).
3. Consider the Dirichlet problem \[ \begin {aligned} \Delta u &= f && \text {in }\Omega ,\\ u &= 0 && \text {on }\partial \Omega \end {aligned} \tag {9.1} \] where \(f\in C^0(\Omega )\) and \(\Omega \) is as in (b). Describe in as much detail as you can the major steps in proving the existence of a classical solution of (9.1) via Perron’s Method.

Let \(\Omega \subset \mathbb {R}\) be open, and let \(C_0^\infty (\Omega )\) be the space of \(C^\infty \) functions with support in \(\Omega \). Define convergence of a sequence \(\varphi _n\) as follows: \(\varphi _n\to 0\) iff \(\exists \) a compact set \(K\subset \Omega \) such that

1. Support \(\varphi _n\subset K\) for all \(n\)
2. \(\displaystyle \lim _{n\to \infty }\sup _{x\in K}\lvert \varphi _n^{(p)}(x)\rvert =0\) for \(p\in Z^+\).

1. Define what is meant by a distribution on \(C_0^\infty (\Omega )\).
2. Define \[ \mu _f(\varphi )=\int _{-\infty }^{\infty } f(x)\bar {\varphi }(x)\,dx \] where \(f\in L^1_{\mathrm {loc}}(\Omega )\) and \(\varphi \in C_0^\infty (\Omega )\). Does this define a distribution on \(C_0^\infty (\Omega )\)? Give reasons.
3. Give a concrete example of a regular distribution and verify that is is a distribution.
4. Show how to define the derivatives of a regular distribution.