Differential Equations Preliminary Examination Department of Mathematics University of Utah Salt Lake City, Utah 84112 September 1996

Give brief arguments that \(E\) and \(X\) are Banach spaces.

Define \[ f:E\to X \] by \[ u\mapsto f(u)=u''+\lambda u+u^3, \tag {2.1} \] where \(\lambda \in \mathbb {R}\). Show that \(f\) is a \(C^1\) mapping.

Let \[ E_0=E\cap \{u:u(0)=u(\pi )=0\}, \] and restrict \(f\) to \(E_0\). Show that, except for a countable set of values of \(\lambda \) (describe this set !), the following holds: There exists \(\delta =\delta (\lambda )\) such that for all \(h\in X\), \(\norm {h}_X<\delta \), there exists a unique \(u\in E_0\) such that \[ f(u)=h, \tag {2.2} \] i.e. the problem \[ \begin {cases} u''+\lambda u+u^3=h,\\ u(0)=0=u(\pi ) \end {cases} \tag {2.3} \] has a unique solution.

Define the mapping \[ y\in \mathbb {R}^N\mapsto F_t(y)\in \mathbb {R}^N \tag {2.5} \] by \[ F_t(y)=x(t,y). \tag {2.6} \] where \(x(t,y)\) solves \[ \begin {aligned} x'&=f(t,x),\\ x(0,y)&=y. \end {aligned} \tag {2.7} \] Give reasons that \(F_t\) is well-defined for each small \(t\) and well-defined for all \(t\in \mathbb {R}\), whenever \(\lvert y\rvert \le R\).

Show that for each \(T\ge 0\) there exists \(y_T\), \(\lvert y_T\rvert \le R\) such that \[ F_T(y_T)=y_T. \tag {2.8} \]

What can one say if the condition (2.4) is replaced by \[ x\cdot f(t,x)>0,\quad t\in \mathbb {R},\quad \lvert x\rvert =R. \tag {2.9} \]

Let \(f\) have the special form \[ f(t,x)=Ax+h(t), \tag {2.10} \] where \(h:\mathbb {R}\to \mathbb {R}^N\) is a continuous function of period \(T\) and \(A\) is a constant \(N\times N\) matrix which is positive (negative) definite (i.e. \(Ax\cdot x\ge \mu \lvert x\rvert ^2\), resp. \(Ax\cdot x\le -\mu \lvert x\rvert ^2\), \(\mu >0\)). Use the result proved above to show that the equation \[ x'=Ax+h(t) \tag {2.11} \] has a unique \(T\)-periodic solution.

Convert problem (2.13) into an equivalent fixed point problem in an appropriate Banach space.

Use Gronwall’s inequality (or a similar argument) to establish the existence of a priori bounds on \(\lvert x(t)\rvert ^2\) for a solution \(x\) of (2.13).

Use properties of the Leray-Schauder degree to deduce that (2.13) has a \(T\)-periodic solution for all \(\epsilon \).

Give a brief sketch of the proof of the following theorem:

Theorem: Let \[ E=\{x:\phi '(x)=0\} \] and let \(M\) be the largest invariant set of (2.14) in \(E\). Then every bounded (for \(t\ge 0\)) semiorbit of (2.14) tends to \(M\) as \(t\to \infty \).

Apply this theorem to the system \[ \begin {aligned} u'&=-v\\ v'&=-v+g(u)\\ x&=\begin {pmatrix}u\\v\end {pmatrix}, \end {aligned} \] where \(g:\mathbb {R}\to \mathbb {R}\) is a \(C^1\) function such that \[ ug(u)>0,\quad u\ne 0, \] using \[ \phi (u,v)=\frac {v^2}{2}+\int _0^u g(\xi )\,d\xi . \] What is your conclusion?

Let \(G\subset \mathbb {R}^N\) be an open set and let \(m\ge 0\) be an integer. Let \(f\in C^m(G)\) be real or complex valued function. Show that \(f\) may be identified with a distribution \(T_f\) and compute the distributional derivatives of \(T_f\) up to order \(m\).

Let \[ r(x)= \begin {cases} x^2, & x\ge 0,\\ 0, & x\le 0. \end {cases} \] Compute all distributional derivatives of \(r\).

Let \(G\) be an open set. Define the Sobolev spaces \(H^m(G)\), \(W^{m,2}(G)\), and \(W_0^{m,2}(G)\) and show that \(H^m(G)\) is a Hilbert space. Give conditions under which any two of the three spaces are equal.

Let \(G\) be a bounded open set. State and prove Poincaré’s inequality for functions \(u\in W_0^{1,2}(G)\) and thus verify that \(W_0^{1,2}(G)\) is also a Hilbert space with respect to the inner product \[ (u,v)_{W_0^{1,2}(G)}=\int _G \nabla u\cdot \overline {\nabla v}\,dx. \]

Show that in the sense of distributions there exists a unique solution \(u\in W_0^{1,2}(G)\) of the equation \[ -\Delta u+u=f. \]

Let \(G\) be bounded. Show that in the sense of distributions there exists a unique solution \(u\in W_0^{1,2}(G)\) of the equation \[ -\Delta u=f. \]

Characterize the orthogonal complement of \(W_0^{1,2}(G)\) in the space \(W^{1,2}(G)\).

Let \(H\) be a Hilbert space and let \[ A:D(A)\to H \] be a linear operator with domain \(D(A)\). State the Hille-Yosida-Phillips theorem giving necessary and sufficient conditions for the unique solvability of the problem \[ \begin {aligned} u'+Au&=0\\ u(0)&=u_0\in D(A). \end {aligned} \] Be sure to define pertinent terms.

Apply the above result to the initial boundary value problem \[ \begin {aligned} u_t-u_{xx}&=0,\quad t>0,\quad x\in (0,\pi )\\ u(0,x)&=g(x),\quad x\in (0,\pi )\\ u(t,0)&=0=u(t,\pi ),\quad t\ge 0. \end {aligned} \]