Small perturbations for a Duffing-like evolution equation involving non-commuting operators

Abstract

We consider an abstract evolution equation with linear damping, a nonlinear term of Duffing type, and a small forcing term. The abstract problem is inspired by some models for damped oscillations of a beam subject to external loads or magnetic fields, and shaken by a transversal force.The main feature is that very natural choices of the boundary conditions lead to equations whose linear part involves two operators that do not commute.We extend to this setting the results that are known in the commutative case, namely that for asymptotically small forcing terms all solutions are eventually close to the three equilibrium points of the unforced equation, two stable and one unstable.

A Appendix

In the second paragraph of Sect. 2 we described in finite dimension the role of the eigenvalues of $A^{-1}B^{2}$ in the study of the negative inertia index of $B^{2}-\lambda A$ as a function of $\lambda$. In the rest of the paper we developed our theory in the infinite dimensional case without mentioning $A^{-1}B^{2}$ explicitly.

In this appendix we present a possible functional setting in which the spectral theory can be applied to the operator $A^{-1}B^{2}$, both in the general and in the concrete case.

1.1 A.1 The correct framework for $A^{-1}B^2$ in general

Let H be a Hilbert space, and let A and B be two coercive self-adjoint unbounded operators on H with dense domains $D(B)\subseteq D(A)$. Then a reasonable definition of $A^{-1}B^{2}$ seems to be the following.

Let us consider the Hilbert spaces $V:=D(A^{1/2})$ and $W:=D(B)$. If we identify H with its dual space $H'$, then we have the inclusions

$$\begin{aligned} W\subseteq V\subseteq H=H'\subseteq V'\subseteq W'. \end{aligned}$$

With these notations we can consider A as a bounded operator ${\widehat{A}}:V\rightarrow V'$, and represent the scalar product in V in terms of the duality pairing as

$$\begin{aligned} \langle u,v\rangle _{V}=\langle {\widehat{A}}u,v\rangle _{V',V} \qquad \forall (u,v)\in V^{2}. \end{aligned}$$

Similarly, we con consider B as a bounded operator ${\widehat{B}}:W\rightarrow H$, whose adjoint is a bounded operator ${\widehat{B}}^{*}:H'\rightarrow W'$ with ${\widehat{B}}^{*}u=Bu$ if $u\in W\subseteq H'$.

Now we can consider the unbounded operator C in V with domain

$$\begin{aligned} D(C):=\left\{ u\in W:{\widehat{B}}^{*}Bu\in V'\right\} , \end{aligned}$$

and defined by

$$\begin{aligned} Cu:={\widehat{A}}^{-1}{\widehat{B}}^{*}Bu \qquad \forall u\in D(C). \end{aligned}$$

We claim that C is an extension of $A^{-1}B^{2}$ that is symmetric and maximal monotone as an unbounded operator in V.

To begin with, for every u and v in D(C) it turns out that

$$\begin{aligned} \langle v,Cu\rangle _{V}=\langle v,{\widehat{A}}Cu\rangle _{V',V}=\langle v,{\widehat{B}}^{*}Bu\rangle _{V',V}=\langle {\widehat{B}}v,Bu\rangle _{H}=\langle Bv,Bu\rangle _{H}, \end{aligned}$$

which proves that C is symmetric and monotone.

It remain to show that C is maximal, namely that, for every $f\in V$, the equation $u+Cu=f$ has a (unique) solution $u\in D(C)$. Applying ${\widehat{A}}$ to both sides, this equation becomes

$$\begin{aligned} {\widehat{A}}u+{\widehat{B}}^{*}Bu={\widehat{A}}f\in V'. \end{aligned}$$

Now the operator ${\widehat{A}}+{\widehat{B}}^{*}B$ is coercive from W to $W'$, and hence surjective (see for example [2, Corollary 14]). Since ${\widehat{A}}f\in V'$, the solution belongs to D(C).

1.2 A.2 The operator $A^{-1}B^2$ in the concrete case

Instead of fussing with generalities, we give an explicit description of the operator $A^{-1}B^{2}$ in the case where A and B are as in Proposition 2.7.

Let us consider the Hilbert space $V=H^{1}_{0}((0,1))$ with scalar product

$$\begin{aligned} \langle u,v\rangle _{V}:=\int _{0}^{1}u_{x}(x)v_{x}(x)\,dx \qquad \forall (u,v)\in H^{1}_{0}((0,1))^{2}. \end{aligned}$$

(A.1)

Let us consider the unbounded linear operator C in V with domain

$$\begin{aligned} D(C):=H^{3}((0,1))\cap H^{2}_{0}((0,1)), \end{aligned}$$

and defined by

$$\begin{aligned}{}[Cu](x):=-u_{xx}(x)+u_{xx}(0)+[u_{xx}(1)-u_{xx}(0)]x \qquad \forall u\in D(C). \end{aligned}$$

(A.2)

We observe that

$$\begin{aligned} Cu=A^{-1}B^{2}u \qquad \forall u\in D(B^{2})=H^{4}((0,1))\cap H^{2}_{0}((0,1)), \end{aligned}$$

and therefore C is a natural extension of $A^{-1}B^{2}$. Indeed $B^{2}u=u_{xxxx}\in L^{2}((0,1))$, and hence $A^{-1}B^{2}u$ is the solution v(x) to equation $-v_{xx}=u_{xxxx}$ in (0, 1), with Dirichlet boundary conditions in $x=0$ and $x=1$. The solution is exactly the function Cu defined in (A.2).

We claim that C is a symmetric positive operator in V with domain D(C), and compact inverse. If we prove this claim, then the eigenfunctions of C are a basis of $H^{1}_{0}((0,1))$, and hence also a basis of $L^{2}((0,1))$, but orthogonal with respect to the scalar product (A.1). These eigenfunctions are exactly the solutions to (7.1) that we characterized in Sect. 7.

To begin with, for every u and v in D(C) it turns out that

$$\begin{aligned} \langle v,Cu\rangle _{V}= & {} \int _{0}^{1}v_{x}(x)\cdot [Cu]_{x}(x)\,dx \\= & {} \int _{0}^{1}v_{x}(x)\cdot \{-u_{xxx}(x)+u_{xx}(1)-u_{xx}(0)\}\,dx \\= & {} -\int _{0}^{1}v_{x}(x)\cdot u_{xxx}(x)\,dx \\= & {} \int _{0}^{1}v_{xx}(x)\cdot u_{xx}(x)\,dx, \end{aligned}$$

which is enough to conclude that C is both symmetric and positive.

It remains to show that the inverse is compact, namely that for every $f\in H^{1}_{0}((0,1))$ the equation

$$\begin{aligned} -u_{xx}(x)+u_{xx}(0)+[u_{xx}(1)-u_{xx}(0)]x=f(x) \qquad \forall x\in (0,1) \end{aligned}$$

(A.3)

has a unique solution $Tf\in D(C)$ (note that lying in this domain entails four boundary conditions), and T is compact as an operator $T:H^{1}_{0}((0,1))\rightarrow H^{1}_{0}((0,1))$.

Uniqueness follows from the positivity of C. Existence follows from the explicit formula for the solution. Indeed, if we set

$$\begin{aligned} F(x):=\int _{0}^{x}f(t)\,dt, \qquad \qquad {\widehat{F}}(x):=\int _{0}^{x}F(t)\,dt, \end{aligned}$$

then a standard computation shows that the solution to (A.3) is

$$\begin{aligned} u(x):=-{\widehat{F}}(x)+\left( 3{\widehat{F}}(1)-F(1)\right) x^{2}+\left( F(1)-2{\widehat{F}}(1)\right) x^{3}. \end{aligned}$$

The same formula reveals that if a sequence $\{f_{n}\}$ is bounded in $H^{1}_{0}((0,1))$, then the sequence of corresponding solutions is bounded in $H^{3}((0,1))$, and therefore relatively compact in $H^{1}_{0}((0,1))$.