Abstract
We study the stabilization and the wellposedness of solutions of the wave equation with subcritical semilinearities and locally distributed nonlinear dissipation. The novelty of this paper is that we deal with the difficulty that the main equation does not have good nonlinear structure amenable to a direct proof of a priori bounds and a desirable observability inequality. It is well known that observability inequalities play a critical role in characterizing the long time behaviour of solutions of evolution equations, which is the main goal of this study. In order to address this, we truncate the nonlinearities, and thereby construct approximate solutions for which it is possible to obtain a priori bounds and prove the essential observability inequality. The treatment of these approximate solutions is still a challenging task and requires the use of Strichartz estimates and some microlocal analysis tools such as microlocal defect measures. We include an appendix on the latter topic here to make the article self contained and supplement details to proofs of some of the theorems which can be already be found in the lecture notes of Burq and Gérard (http://www.math.u-psud.fr/~burq/articles/coursX.pdf, 2001). Once we establish essential observability properties for the approximate solutions, it is not difficult to prove that the solution of the original problem also possesses a similar feature via a delicate passage to limit. In the last part of the paper, we establish various decay rate estimates for different growth conditions on the nonlinear dissipative effect. We in particular generalize the known results on the subject to a considerably larger class of dissipative effects.
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Research of Marcelo M. Cavalcanti is partially supported by the CNPq Grant 300631/2003-0. Research of Valéria N. Domingos Cavalcanti is partially supported by the CNPq Grant 304895/2003-2. Research of Victor Hugo Gonzalez Martinez is partially supported by CAPES Grant 88882.449176/2019-01. Research of Türker Özsarı is supported by the Science Academy’s Young Scientist Award in Mathematics (BAGEP 2020).
Appendix A: Preliminaries on Microlocal Analysis
Appendix A: Preliminaries on Microlocal Analysis
In this section, we supplement details to proofs of some of the theorems on pseudo-differential operators and microlocal defect measures, whose original versions in French can be found in the elegant lecture notes of Burq and Gérard [7].
1.1 A.1. Pseudo-differential Operators
Let \(\Omega \) be an open and nonempty subset of \({\mathbb {R}}^d\), \(d\ge 1\). A differential operator on \(\Omega \) is a linear map \(P:{\mathcal {D}}'(\Omega ) \rightarrow {\mathcal {D}}'(\Omega )\) of the form
where \(a_\alpha \in C^\infty (\Omega )\) are complex valued functions. The greatest integer m such that the functions \(a_\alpha \), \(|\alpha |=m\) are not all zero is called the order of P. The map \(p:\Omega \times {\mathbb {R}}^d \rightarrow {\mathbb {C}}\), defined by
is called the symbol of P.
We observe that P is characterized by the identity
Adopting to the notation
the operator P can be rewritten as
The formula (A.2) can be generalized as follows: for all \((x,\xi )\in \Omega \times {\mathbb {R}}^d\) and for all \(u\in {\mathcal {D}}(\Omega )\),
where the previous sum is finite once p is a polynomial in the variable \(\xi \).
If P is a differential operator of order m and symbol p, then the principal symbol of order m, denoted by \(\sigma _m(P)\), is the homogeneous part of degree m in \(\xi \) of the polynomial function \(p(x,\xi )\), namely
Definition A.1
Let \(m\in {\mathbb {R}}\). Then a symbol of order at most m in \(\Omega \) is said to be a function \(a:\Omega \times {\mathbb {R}}^d \rightarrow {\mathbb {C}}\) of class \(C^\infty \), with support in \(K \times {\mathbb {R}}^d\), where K is a compact subset of \(\Omega \), such that for all \(\alpha \in {\mathbb {N}}^d\), \(\beta \in {\mathbb {N}}^d\), there exists a constant \(C_{\alpha ,\beta }\) with
We shall denote the vectorial space of all symbols of order at most m in \(\Omega \) by \({\mathcal {S}}_c^m(\Omega )\) .
Proposition A.1
If \(a\in {\mathcal {S}}_c^m(\Omega )\), the formula
defines, for all \(u\in C_0^\infty (\Omega )\), an element Au of \(C_0^\infty (\Omega )\).
The formula (A.7) defines a linear map \(A:C_0^\infty (\Omega ) \longrightarrow C_0^\infty (\Omega )\), which is called the pseudo-differential operator of order m and symbol a. We will often denote the map A by a(x, D).
The set of all pseudo-differential operators of order m on \(\Omega \) will be denoted by \(\Psi ^m_c(\Omega )\).
Definition A.2
An operator \(A\in \Psi ^m_c(\Omega )\) is essentially homogeneous if there exists a function \(a_m=a_m(x,\xi )\) with \(\mathrm{supp}a_m \subset K \times ({\mathbb {R}}^{d}\setminus \{0\})\), homogeneous of order m in \(\xi \) and smooth except at \(\xi =0\) and a function \(\chi \in C^\infty ({\mathbb {R}}^d)\) being zero near 0 and 1 in the infinity such that
for some \(r\in {\mathcal {S}}^{m-1}_{c}(\Omega )\).
Proposition A.2
Let \(A\in \Psi ^m_c(\Omega )\) be essentially homogeneous. Then, for all \(u\in C_0^\infty (\Omega )\), \(\xi \in {\mathbb {R}}^d\backslash \{0\}\), and \(x\in \Omega \),
Definition A.3
Under the conditions of Proposition A.2, we say that A admits a principal symbol of order m. The function \(a_m\) characterized by (A.9) is called the principal symbol of order m of A and is denoted by \(\sigma _m(A)\).
Theorem A.1
Let A be a pseudo-differential operator of symbol \(a\in {\mathcal {S}}_{c}^{m}(\Omega )\) with \(\mathrm{supp}a \subset K \times {\mathbb {R}}^{d}\), and let \(\chi \in C_0^\infty (\Omega )\) satisfy \(\chi =1\) in a neighborhood of K. Then, there exists a pseudo-differential operator \(A_{\chi }^*\) on \(\Omega \) such that, for all \(u,v \in C_0^\infty (\Omega )\),
In addition, \(A_{\chi }^*\) admits a symbol \(a_\chi ^*\in {\mathcal {S}}_{c}^{m}(\Omega )\) verifying, for all \(N\in {\mathbb {N}}\),
In particular, if A admits a principal symbol of order m, it is the same of \(A^*\), and
Theorem A.2
Let A and B be pseudo-differential operators of symbols \(a\in {\mathcal {S}}^{m}_{c}(\Omega )\), \(b\in {\mathcal {S}}_{c}^{n}(\Omega )\), respectively. Then, the composition AB is a pseudo-differential operator admitting a symbol \(a\#b \in S_c^{m+n}(\Omega )\) which satisfies
for all \(N \in {\mathbb {N}}\).
In addition, if A admits a principal symbol of order m and B admits a principal symbol of order n, then AB admits a principal symbol of order \(m+n\) and [A, B] admits a principal symbol of order \(m+n-1\), given by
Definition A.4
For any compact set K contained in \(\Omega \) and for all \(s\in {\mathbb {R}}\), \(H_K^s(\Omega )\) shall denote the space of distributions with compact support in K whose extensions by zero belong to \(H^s({\mathbb {R}}^d)\), i.e.,
We set \(H_{comp}^s(\Omega )=\underset{K}{\bigcup }H_K^s(\Omega )\) where K ranges over all compact subsets of \(\Omega \). We equip \(H_{comp}^s(\Omega )\) with the finest locally convex topology such that all the inclusion maps
are continuous.
Theorem A.3
Let \(a\in {\mathcal {S}}_{c}^{m} (\Omega )\) and let K be the projection on \(\Omega \) of \(\mathrm{supp}(a)\). Thus, for all \(s\in {\mathbb {R}}\), the operator defined in (A.7) admits a unique extension to a linear and continuous map from \(H_{comp}^s(\Omega )\) in \(H_K^{s-m}(\Omega )\).
Remark A.1
If \(A \in \Psi _{c}^{m}(\Omega )\) is a pseudo-differential operator with \(m < 0\) then \(A: L_{comp}^{2}(\Omega ) \rightarrow L^{2}(\Omega )\) is a compact operator. Indeed, from Rellich’s Theorem the inclusion \(H_{comp}^{-m} \hookrightarrow L^2(\Omega )\) is compact.
Remark A.2
Let \(P \in \Psi ^m(\Omega )\) be a compactly supported operator, then P extends continuously from \(H_{loc}^{s}(\Omega )\) to \(H^{s-m}_{comp}(\Omega ).\)
Theorem A.4
(A Gårding type inequality) Let A be a pseudo-differential operator of order 0 on \(\Omega \) whose principal symbol \(\sigma _0(A)\) exists and is a positive function in \({\mathcal {S}}_{c}^{0}(\Omega )\). Then, for all \(\delta >0\) there exists \(C_\delta \) such that
Furthermore, there exists \(C>0\) such that
1.2 A.2. Microlocal Defect Measures
Let \(\{u_n\}_{n\in {\mathbb {N}}}\) be a bounded sequence in \(L^2_{loc}(\Omega )\), i.e.,
for any compact set K contained in \(\Omega \).
We shall say that \(u_n\) converges weakly to \(u\in L^2_{loc}(\Omega )\) if one has
for each \(f\in L_{comp}^2(\Omega )\).
We describe the loss of strong convergence of the sequence \(u_n\) to 0 in \(L^2_{loc}(\Omega )\), by means of a positive Radon measure on \(\Omega \times S^{d-1}\), that is, \(u_n \rightarrow 0\) in \(L^2_{loc}(\Omega )\) if, and only if, \(\mu =0\).
Lemma A.1
Let \(\{u_n\}_{n\in {\mathbb {N}}}\) be a bounded sequence in \(L_{loc}^2(\Omega )\) weakly converging to 0. Let A be a pseudo-differential operator of order 0 on \(\Omega \) that admits a principal symbol \(\sigma _0(A)\) of order 0. If \(\sigma _0(A) \ge 0\), then one has
Proof
First, note that since \(u_n\) is bounded in \(L_{loc}^2(\Omega )\) and converges weakly to 0, we have
Indeed, recall that
where
tends to 0 for all \(\xi \in {\mathbb {R}}^d\) and remains uniformly bounded in \(\xi \) and n. Indeed, from the Cauchy–Schwarz inequality we have
Applying the dominated convergence theorem, we have, for all \(R>0\),
On the other hand, the Plancherel theorem yields
so,
which proves (A.17), since R is arbitrary. Applying (A.15) and (A.16) to \(v=\chi u_n\), letting \(n\rightarrow +\infty \), and using (A.17), we obtain
In addition,
which proves the lemma since \(\delta >0\) is arbitrary. \(\square \)
Theorem A.5
Let \((u_n)_{n\in {\mathbb {N}}}\) be a bounded sequence in \(L_{loc}^2(\Omega )\) weakly converging to 0. Then, there exists a subsequence \((u_n)_{n \in {\mathbb {N}}'}\) and a positive Radon measure \(\mu \) on \(\Omega \times S^{d-1}\) such that for any essentially homogeneous pseudo-differential operators \(A \in \Psi ^{0}_{c}(\Omega )\) with principal symbol \(\sigma _{0}(A)\), one has
for all \(\chi \in C_{0}^{\infty }(\Omega )\) such that \(\chi =1\) in \(\pi _x(\mathrm{supp}(\sigma _0(A)))\).
Proof
From the assumption of the theorem, one has
Indeed, first observe that since \(A\in \Psi _c^0(\Omega )\), it maps \(H_{comp}^s(\Omega )\) into \(H_K^s(\Omega )\) for all \(s\in {\mathbb {R}}\) and some compact set \(K\subset \Omega \). In particular, A maps \(L_{comp}^2(\Omega )\) continuously into \(L_K^2(\Omega )\). Since \((u_n)_{n\in {\mathbb {N}}}\) is bounded in \(L^2_{loc}(\Omega )\), it follows that \((A (\chi u_n))\) is bounded in \(L_{K}^2(\Omega )\).
Now, from the Cauchy–Schwarz inequality,
The principal symbol of the operator \(A_{\chi }^{*}A\) is \(|\sigma _0(A)|^2\). Hence, \(A_{\chi }^{*}A= b(x,D) +r(x,D)\), where \(r(x,D)\in \Psi _{c}^{-1}(\Omega )\) and b(x, D) is an operator with symbol \(\gamma (\xi )|\sigma _{0}(A)|^{2}\), in which \(\gamma \) is 0 near the origin and 1 at infinity.
Then,
Observe that \(\chi u_{n} \rightharpoonup 0\) weakly in \(L_{comp}^{2}(\Omega )\). Indeed, let \(g \in L_{loc}^{2}(\Omega )\). Then, \(\chi g \in L_{comp}^{2}(\Omega )\). Thus
since \(u_n \rightharpoonup 0\) in \(L_{loc}^{2}(\Omega )\).
From Remark A.1, \(r(x, D)(\chi u_{n}) \rightarrow 0\) strongly in \(L^{2}(\Omega )\), and hence
On the other hand, since \(\chi =1\) in \(K = \pi _{x}(\mathrm{supp}(\sigma _{0}(A)))\),
We estimate the term \(\displaystyle \limsup _{n \rightarrow \infty } \Vert A(\chi u_{n}) \Vert _{L^{2}(\Omega )}^{2}\) as follows:
where the last inequality follows from Lemma A.1.
Therefore, from the properties of \(\limsup \), it follows that
Thus, from (A.20) we obtain
For any compact set K of \(\Omega \), \(C_K^\infty (\Omega \times S^{d-1})\) will denote the vectorial space of functions \(C^\infty \) on \(\Omega \times S^{d-1}\), with compact support in \(K\times S^{d-1}\), endowed with the \(L^\infty \) norm. The space \(C_K^\infty (\Omega \times S^{d-1})\) is separable, since it is isometric to a subspace of the separable space \(C(K \times S^{d - 1})\).
So, let \(D := \mathrm{span}\{ a_i : i\in {\mathbb {N}}\}\) be a countable and dense subset of \(C_K^\infty (\Omega \times S^{d-1})\). For all \(i \in {\mathbb {N}}\), let \(A_i\) be a pseudo-differential operator such that \(\sigma _0(A_i)=a_i\).
Taking (A.21) into account, for each fixed i there exists a constant
such that
By virtue of Cantor’s diagonal argument there exists a infinite subset \({\mathbb {N}}'\) of \({\mathbb {N}}\) such that the quantity \((A_i(\chi u_{n}),\chi u_n)_{n \in {\mathbb {N}}'}\) has a limit for all i.
In fact, the sequence \(\{(A_1(\chi u_n),\chi u_n)\}_{n \in {\mathbb {N}}}\), being bounded, has a convergent subsequence. Thus there exist an infinite subset \({\mathbb {N}}_{1} \subset {\mathbb {N}}\) and a number \(\alpha _1\) for which \(\underset{n \in {\mathbb {N}}_1}{\lim } (A_1(\chi u_n),\chi u_n)=\alpha _1\). The sequence \((A_2(\chi u_n),\chi u_n)_{n \in {\mathbb {N}}_1}\) is also bounded. So there exist an infinite subset \({\mathbb {N}}_2 \subset {\mathbb {N}}_1\) and a number \(\alpha _2\) such that \(\underset{n \in {\mathbb {N}}_2}{\lim } (A_1(\chi u_n),\chi u_n)=\alpha _2\). Proceeding in the same fashion, we obtain, for each \(i \in {\mathbb {N}}\), an infinite subset \({\mathbb {N}}_i \subset {\mathbb {N}}\), such that \({\mathbb {N}}_1 \supset {\mathbb {N}}_2 \supset \cdots \supset {\mathbb {N}}_i \supset \cdots \) and a number \(\alpha _i\) such that \(\underset{n \in {\mathbb {N}}_i}{\lim } (A_i(\chi u_n),\chi u_n)=\alpha _i\). Let us then define an infinite subset \({\mathbb {N}}' \subset {\mathbb {N}}\), taking the i-th element of \({\mathbb {N}}'\) as the i-th element of \({\mathbb {N}}_i\). For each \(i \in {\mathbb {N}}\), the sequence \((A_i(\chi u_n),\chi u_n)_{n \in {\mathbb {N}}'}\) is, from its i-th element, a subsequence of \((A_i(\chi u_n),\chi u_n)_{n \in {\mathbb {N}}_i}\) and therefore converges. Thus, \((A_i(\chi u_n),\chi u_n)_{n \in {\mathbb {N}}'} \rightarrow \alpha _i\) for all \(i \in {\mathbb {N}}\).
Define \(L: D \subset C_{K}^{\infty }(\Omega \times S^{d-1}) \longrightarrow {\mathbb {C}}\) by setting \(L(a_k)=\alpha _k\). From (A.21), it follows that
Note that \(\alpha _k\) does not depend on the choice of the operator \(A_k\) satisfying \(\sigma _0(A_k)=a_k\). In fact, let \(B_k\) be another operator with \(\sigma _0(B_k)=a_k\), then from (A.21), it follows that
Therefore,
Thus, the mapping L is well defined and densely defined. Hence, it uniquely extends to a bounded linear functional \(\tilde{L}: C_{K}^{\infty }(\Omega \times S^{d-1}) \rightarrow {\mathbb {C}}\) satisfying the estimate
By construction, given \(\sigma _0(A) \in C_{K}^{\infty }(\Omega \times S^{d-1})\), we have
In fact, let \((a_i) \subset D\) with \(a_i \rightarrow \sigma _0(A)\) as \(i\rightarrow +\infty \); that is, given \(\varepsilon > 0,\) there exists \(i_{0} \in {\mathbb {N}}\) such that
Moreover, there is \(n_{0} \in {\mathbb {N}},\) such that for all \(n \ge n_{0}\),
On the other hand, there exists \(n_{1} \in {\mathbb {N}}\) such that for all \(n \ge n_{1}\),
So, taking \(n_{2} = \max \{n_{0}, n_{1}\}\), from (A.23), (A.24) and (A.25), it follows that for all \(n \ge n_{2},\)
Then,
Note that L does not depend on the choice of \(\chi \). Indeed, let \(\chi , \tilde{\chi } \in C_{0}^{\infty }(\Omega )\) with \(\chi \sigma _{0}(A) = \sigma _{0}(A) = \tilde{ \chi }\sigma _{0}(A)\). For \(\eta \in C_{0}^{\infty }(\Omega )\) such that \(\eta =1\) in a neighborhood of \(\mathrm{supp}\chi \cup \mathrm{supp}\tilde{\chi }\), we have
since \(A=b(x,D) + r(x,D)\), where \(b_{\eta }^{*}(x,\xi )=\overline{b(x,\xi )}+r_1\) with \(r_1 \in {\mathcal {S}}_{c}^{-1}(\Omega )\). Since \(\chi - \tilde{\chi } =0\) in \(\pi _{x}(\mathrm{supp}\sigma _{0}(A))\), we have
Therefore,
So far we have extracted a subsequence of \((u_n)\) such that
for all \(A \in \Psi _{c}^{0}(\Omega )\) with \(\mathrm{supp}\sigma _0(A) \subset K \times {\mathbb {R}}^{d}\) in some fixed compact K. We need to remove the dependence on K. To this end, let \(K_i \subset \overset{\circ }{K}_{i+1}\) be a monotone sequence of compact subsets of \(\Omega \) such that \(\Omega = \overset{\infty }{\underset{n=1}{\bigcup }}K_n\). From the construction, there exists an infinite subset \({\mathbb {N}}_1 \subset {\mathbb {N}}\) and a continuous linear form \(\tilde{L}_1\) in \(C_{K_1}^{\infty }(\Omega \times S^{d-1})\) such that
for all \(A_1 \in \Psi _{c}^{0}(\Omega )\) with \(\mathrm{supp}\sigma _0(A_1) \subset K_1 \times \Omega \). The sequence \((u_n)_{n \in {\mathbb {N}}_1}\) is still a bounded sequence in \(L_{loc}^2(\Omega )\) weakly converging to 0. Thus we can obtain an infinite subset \({\mathbb {N}}_{2} \subset {\mathbb {N}}_1\) and a continuous linear form \(\tilde{L}_2\) in \(C_{K_2}^{\infty }(\Omega \times S^{d-1})\) such that
for all \(A_2 \in \Psi _{c}^{0}(\Omega )\) with \(\mathrm{supp}\sigma _0(A_2)\subset K_2 \times \Omega \).
Proceeding in the same fashion, we obtain, for each \(i \in {\mathbb {N}}\), an infinite subset \({\mathbb {N}}_i \subset {\mathbb {N}}\) and a continuous linear form \(\tilde{L}_i\) in \(C_{K_i}^{\infty }(\Omega \times S^{d-1})\) such that \({\mathbb {N}}_1 \supset {\mathbb {N}}_2 \supset \cdots \supset {\mathbb {N}}_i \supset \cdots \) and, for all \(i\in {\mathbb {N}}\),
for all \(A_i \in \Psi _{c}^{0}(\Omega )\) with \(\mathrm{supp}\sigma _0(A_i)\subset K_i \times \Omega \). Let us then define an infinite subset \({\mathbb {N}}' \subset {\mathbb {N}}\), whose i-th element is the i-th element of \({\mathbb {N}}_i\). In this way, for each \(i \in {\mathbb {N}}\), the sequence \((u_n)_{n \in {\mathbb {N}}'}\) is, starting from its i-th element, a subsequence of \((u_n)_{n \in {\mathbb {N}}_i}\).
We want to define a linear functional \(L:C_{0}^{\infty }(\Omega \times S^{d-1}) \longrightarrow {\mathbb {C}}\) satisfying the limit
for all \(A \in \Psi _{c}^{0}(\Omega )\) with \(\sigma _0(A)\) having compact support in the variable x.
Now, if \(a \in C_{0}^{\infty }(\Omega \times S^{d-1})\), \(\mathrm{supp}a \subset K \times S^{d-1}\), and \(a=\sigma _0(A)\), with \(K \subset K_{i_0}\), the same argument used to show the independence of the cut-off functions implies
where \(\chi =1\) in a neighborhood of K, \(\chi _{i_0}=1\) in a neighborhood of \(K_{i_0}\) and that the following inequality holds:
Defining \(L(\sigma _0(A))=\tilde{L}_{i_0}(\sigma _0(A))\), we obtain
From Lemma A.1 it follows that, L extends to a Radon measure \(\mu \ge 0\) on \(\Omega \times S^{d-1}\), which finishes the proof. \(\square \)
Definition A.5
\(\mu \) is said to be the microlocal defect measure of the sequence \(\{u_n\}_{n\in {\mathbb {N}}'}\) in Theorem A.5.
Remark A.3
Theorem A.5 assures that any bounded sequence \((u_n)_{n\in {\mathbb {N}}}\) in \(L_{loc}^2({\mathcal {O}})\) that weakly converges to zero has a subsequence associated with a microlocal defect measure (in short, m.d.m.). We observe from (A.18) that if \(A=f\in C_0^\infty ({\mathcal {O}})\), then in particular
so that \((u_n)_{{\mathbb {N}}'}\) strongly converges to 0 if and only if \(\mu =0\).
Theorem A.6
Let P be a differential operator of order m on \(\Omega \) with \(P^*=P\), and let \((u_n)_{n\in {\mathbb {N}}}\) be a bounded sequence in \(L_{loc}^2(\Omega )\) weakly converging to 0 associated with a microlocal defect measure \(\mu \). Let us assume that \(P u_n \underset{n\rightarrow +\infty }{\longrightarrow }0\) strongly in \(H_{loc}^{1-m}\). Then, for any \(a \in C^\infty (\Omega \times ({\mathbb {R}}^d)\backslash \{0\})\) homogeneous of degree \(1-m\) in the second variable and with compact support in the first variable,
Proof
Let \(\chi , \chi _1 \in C_{0}^{\infty }(\Omega )\) such that \(\chi =1\) in a neighborhood of \(\pi _1(\mathrm{supp}(a))\) and \(\chi _1=1\) in a neighborhood of the support of the function \(\chi \). Consider \(\gamma \in C^\infty (\Omega )\) with \(\gamma =0\) in a neighborhood of 0 and \(\gamma =1\) at infinity. Let A be the operator defined by the symbol \(b(x,\xi )=\gamma (\xi )a(x,\xi )\). The symbol of the adjoint operator \(A_{\chi _1}^{*}\) is given by \(b^*(x,\xi )=\overline{b(x,\xi )}+r(x,\xi )\), where \(r \in S_{c}^{-m}(\Omega )\). For a sufficiently regular function u, we have
Note that
Therefore, \([P,\chi ]\) is a differential operator of order \(m-1\) involving the partial derivatives of \(\chi \). We note that \(A(\chi u)(x)=0\) for \(x \not \in \pi _1(\mathrm{supp}(a))\). Thus, we can write
since \(\chi =1\) in a neighborhood of this set. Therefore, \(\partial _{x}^{\beta } \chi =0\) for \(\beta >0\).
Regarding the term \((P(\chi u),A_{\chi _1}^{*}(\chi u)_{L^2(\Omega )}\), we have
By the same argument above,
With respect to the term \((P(\chi u),r(x,D)(\chi u))_{L^2(\Omega )}\), we have,
From the hypothesis, \(\chi P u_n \rightarrow 0\) in \(H_{comp}^{1-m}(\Omega )\). Then, considering \(r(x,D)_{\chi _1}^{*}:H_{comp}^{1-m}(\Omega )\longrightarrow H_{K_1}^{1}\), \(K_1=\mathrm{supp}\chi _1\), we have \(r(x,D)_{\chi _1}^{*}(\chi P u_n) \rightarrow 0\) in \(L^2(\Omega )\). Thus,
Note that \([P,\chi ]\) has order \(m-1\) and is compactly supported, that is, has compact kernel. Thus, \([P,\chi ]:L_{loc}^{2}(\Omega )\longrightarrow H_{comp}^{1-m}\). By Rellich’s theorem,
is a compact operator. Therefore, \(r(x,D)_{\chi _1}^{*} \circ [P,\chi ]\) maps weakly convergent sequences to strongly convergent sequences. Hence,
Hence,
From (A.28), (A.29) and (A.30), we obtain
Observing that
the second term of (A.27) is
Combining the fact that \(\chi P u_n \rightarrow 0\) in \(H^{1-m}_{comp}(\Omega )\) with \(\overline{b}(x,D)_{\chi _1}^{*}:H_{comp}^{1-m}(\Omega ) \longrightarrow L^2(\Omega )\), we obtain
Due to (A.32), it is necessary to prove that the term \(|(A(\chi u_n),\chi Pu_n)_{L^2(\Omega )}|\) converges to 0. In fact, \(A:L_{comp}^{2}(\Omega ) \longrightarrow H_{K}^{m-1}(\Omega )\) is a linear continuous operator. Therefore, A maps bounded sets to bounded sets. In particular, \(\{A(\chi u_n)\}\) is a bounded set of \(H_{K}^{m-1}(\Omega )\). Therefore,
Hence,
which concludes the proof. \(\square \)
Theorem A.7
Let X be a locally compact Hausdorff space and \(\mu \) be a positive Radon measure on X.
-
(a)
Let N be the union of all open \(U \subset X\) such that \(\mu (U)=0\). Then N is open and \(\mu (N)=0\). The complement of N is called the support of \(\mu \).
-
(b)
\(x \in \mathrm{supp}\mu \) if and only if \(\int _{X} f d\mu >0\) for every \(f \in C_{0}(X)\) with \(f(x)>0\).
Proof
a) clearly follows from \(\mu (N)=\sup \{\mu (K):K \subset N, K \hbox { compact}\}.\)
To prove b), suppose \(x \in \mathrm{supp}\mu =N^c\). Let \(f \in C_c(X,[0,1])\) such that \(f(x)>0\). By continuity, there is an open neighborhood U of x such that \(f(y)>\frac{1}{2}f(x)\) for all \(y \in U\). If \(\mu (U)=0\), then since U is open, \(U \subset N\), which contradicts \(x \in N^c\). Thus, \(\mu (U)>0\). Hence, \(\int _{X}fd\,\mu \ge \int _U f\, d\mu \ge \frac{1}{2}f(x)\mu (U)>0\).
Now, suppose \(x \notin \mathrm{supp}\mu \) and let \(F=\mathrm{supp}\mu \). Then, there exists an open neighbourhood U of x such that \(\mu (U)=0\) and \(U \cap F=\emptyset \). Let K be a compact set so that \(x \in K \subset U\). From Urysohn’s theorem, there exists a function \(f \in C_0(X,[0,1])\) such that \(f=1\) in K and \(f=0\) outside of a compact subset of U. In particular, \(f(x)=1>0\) and
\(\square \)
Theorem A.8
Let \(\{u_n\}_n\) be a bounded sequence in \(L_{loc}^2(\Omega )\) that weakly converges to zero and admits a microlocal defect measurement \(\mu \). Then, \((x_0,\xi _0) \notin \mathrm{supp}\mu \) if and only if there exists \(A \in \Psi _{c}^{0}(\Omega )\) essentially homogeneous such that \(\sigma _0(A)(x_0,\xi _0)\ne 0\) and \(A(\chi u_n) \rightarrow 0\) in \(L^2(\Omega )\) for all \(\chi \in C_{0}^{\infty }(\Omega )\).
Proof
If \((x_0,\xi _0) \notin \mathrm{supp}\mu \), then there exist open subsets \(U \subset \Omega \), \(V \subset S^{d-1}\) such that \((x_0,\xi _0) \in U \times V\) and \(U \times V \cap \mathrm{supp}\mu = \emptyset \). Let K be a compact neighborhood of \(\xi _0\), such that \(K \subset V\). Let \(\psi \in C^\infty (S^{d-1})\) such that \(\mathrm{supp}\psi \subset K\) and \(\psi =1\) in a neighborhood of \(\xi _0\) contained in K. Consider \(\phi \in C_{0}^{\infty }(\Omega )\) such that \(\phi =1\) in a neighborhood of \(x_0\) and \(\mathrm{supp}\phi \subset U\). Pick \(\eta \in C^\infty (\Omega )\) such that \(\eta =0\) near 0 and \(\eta =1\) at infinity. Define \(a(x,\xi )= \eta (\xi )\phi (x)\psi \left( \frac{\xi }{|\xi |}\right) \). Let A be the operator defined by \(a(\cdot ,\cdot )\). Note that \(\sigma _0(A)(x_0,\xi _0)=1\) and for all \(\chi \in C_{0}^{\infty }(\Omega )\)
where \(\chi _1 =1\) in a neighborhood of \(\mathrm{supp}\chi \). Since \(\mathrm{supp}\sigma _0(A) \subset \mathrm{supp}\phi \times \mathrm{supp}\psi \subset \Omega \times S^{d-1} \setminus \mathrm{supp}\mu \), we have
Therefore, \(\Vert A(\chi u_n)\Vert _{L^{2}(\Omega )}\rightarrow 0\) in \(L^2(\Omega )\).
Conversely, if there exists \(A \in \Psi _{c}^{0}(\Omega )\) essentially homogeneous such that \(\sigma _0(A)(x_0,\xi _0)\ne 0\) and \(A(\chi u_n) \rightarrow 0\) in \(L^2(\Omega )\), from (A.33) and by the uniqueness of the limit, we obtain
By using Theorem A.7, we deduce that \((x_0,\xi _0) \notin \mathrm{supp}\mu \). \(\square \)
We will call the vector field defined in \(\Omega \times {{\mathbb {R}}}^d\backslash \{0\}\) and given by
a Hamiltonian field of p, denoted by \(H_p\).
The Lie derivative of a function f with respect to the Hamiltonian field \(H_p\) is given by \(H_p (f)=\{p,f\}\), where
A Hamiltonian curve of p is an integrable curve of the vector field \(H_p\), which is a maximal solution \(s\in I \mapsto (x(s),~\xi (s))\) to the Hamilton-Jacobi equations
where I is an open interval of \({\mathbb {R}}\). By null bicharacteristics we mean those integral curves of \(H_p\) along which \(p=0\).
Remark A.4
Since \(H_p p=0\), p is constant along the integral curves of \(H_p\). Indeed, let \(\alpha (t)=(H_p)_t(x_0,\xi _0)\) be an integral curve of \(H_p\) starting at \((x_0,\xi _0)\). Then,
Now, from the connectedness of the domain of \(\alpha \), the remark follows.
Lemma A.2
Assume \(p(x_0,\xi _0)=0\) and \(\lambda \) is a \(C^\infty \) function on \(T^0\Omega \) with real values that never vanishes. Then, for all \(t \in (T_0,T_1)\) there exists \(s_t \in {\mathbb {R}}\) such that \((H_p)_t(x_0,\xi _0)=(H_{\lambda p})_{s_t}(x_0,\xi _0)\). Moreover, there exists \(S_0<0<S_1\) for which the mapping \(t \mapsto s_t\) is a \(C^\infty \) diffeomorphism from \((T_0,T_1)\) onto \((S_0,S_1)\).
Proof
Let us write \(b=\frac{1}{\lambda }\) and \(a=\lambda p\). Since \(p=ab\), using Remark A.4 we have \((ab)((H_p)_t(x_0,\xi _0)=0\). Since b never vanishes, we obtain \(a((H_p)_t(x_0,\xi _0))=0\) for all \(t \in (T_0,T_1)\). Thus,
Writing \(f(t)=b(x(t),\xi (t))\), we have
with \(x(0)=x_0\) and \(\xi (0)=\xi _0\). Now, let \((y(t),\eta (t))=(H_a)_t(x_0,\xi _0)=(H_{\lambda p})_t(x_0,\xi _0)\) and define
By virtue of the mean value theorem and the fact that f never vanishes, we have either \(f>0\) or \(f<0\). Let’s see that s is injective. Indeed, let \(t_1,t_2 \in (T_0,T_1)\) such that \(t_1 \ne t_2 \) and suppose by contradiction that \(s(t_1)=s(t_2)\). Without loss of generality, we can suppose \(t_1<t_2\). Then
Therefore,
In the same fashion, if \(f>0\), then \(\int _{t_1}^{t_2}f(s)ds>0\) and if \(f<0\), then \(\int _{t_1}^{t_2}f(s)ds<0\), which yields a contradiction in both cases. Therefore, \(s(t_1)=s(t_2)\). Moreover, we have \(s'(t)=f(t)\ne 0\) for all \(t\in (T_0,T_1)\). Hence, s is a \(C^\infty \) injective local diffeomorphism. Since \(s(0)=0\), there exists an open neighbourhood of 0, \((S_0,S_1)\), such that s is a diffeomorphism from \((T_0,T_1)\) onto \((S_0,S_1)\). Let \((z(t),\gamma (t))=(y(s(t)),\eta (s(t)))\) and note that
with \(z(0)=y(s(0))=y(0)=x_0\) and \(\gamma (0)=\eta (s(0))=\eta (0)=\xi _0\). By virtue of uniqueness of solutions, we obtain \((x(t),\xi (t))=(y(s(t)),\eta (s(t))\). Therefore, the bicharacteristics of \(\lambda p\) and p coincide modulo a reparametrization. \(\square \)
Theorem A.9
Let P be a differential operator of order m on \(\Omega \) and let \((u_n)_{n\in {\mathbb {N}}}\) be a bounded sequence in \(L_{loc}^2(\Omega )\) weakly converging to 0, associated with a microlocal defect measure \(\mu \). Then, the following statements are equivalent:
-
(i)
\(Pu_n \underset{n \rightarrow +\infty }{\longrightarrow }0 \hbox { strongly in } H_{loc}^{-m}(\Omega )~(m>0)\),
-
(ii)
\(\mathrm{supp}(\mu ) \subset \{(x,\xi )\in \Omega \times S^{d-1}: \sigma _m(P)(x,\xi )=0\}.\)
Proof
Observe that (i) is equivalent to \(\chi Pu_{n} \rightarrow 0\) in \(H^{-m}(\Omega )\) for every \(\chi \in C_{0}^{\infty }(\Omega )\). However, P is a properly supported operator, since it is a differential operator, so that there exists \(\psi \in C_{0}^{\infty }(\Omega )\) such that \(\chi Pu_{n} = \chi P(\psi u_{n})\) for every \(n \in {\mathbb {N}}\). Therefore, (i) is also equivalent to
If we set \(B = P^{*}\overline{\chi }\Lambda ^{-m}\chi P \in \Psi _{c}^{0}(\Omega )\), then the principal symbol of B is
and
Employing Theorem A.7 we conclude that the condition
for every \(\chi \in C_{0}^{\infty }(\Omega )\) is equivalent to (ii), which completes the proof. \(\square \)
Theorem A.10
Let P be a self-adjoint differential operator of order m on \(\Omega \) and let \((u_n)_{n\in {\mathbb {N}}}\) be a bounded sequence in \(L_{loc}^2(\Omega )\) that weakly converges to 0, with a microlocal defect measure \(\mu \). Suppose that \(P u_n\) converges to 0 in \(H_{loc}^{-(m-1)}\). Then the support of \(\mu \), \(\mathrm{supp}(\mu )\), is a union of curves like \(s\in I \mapsto \left( x(s), \frac{\xi (s)}{|\xi (s)|}\right) \), where \(s\in I \mapsto (x(s),\xi (s))\) is a null-bicharacteristic of p, where p is the principal symbol of P.
Proof
We first consider the function \(q(x,\xi )=|\xi |^{1-m}p(x,\xi ),\) which is smooth on \(\Omega \times ({\mathbb {R}}^d\setminus \{0\})\) and homogeneous of degree 1 in the variable \(\xi .\)
We have already noticed that the null-bicharacteristics of q are reparametrizations of the null-bicharacteristics of p. Hence, it is enough to prove that \(\mathrm{supp}(\mu )\) is a union of curves like \(s\in I \mapsto \left( x(s), \frac{\xi (s)}{|\xi (s)|}\right) \), where \(s\in I \mapsto (x(s),\xi (s))\) is a null-bicharacteristic of q.
Notice that \(\Omega \times S^{d-1}\) is covered by the bicharacteristics of q (that is, the integral curves of the Hamiltonian \(H_q\)). Since \(H^{1-m}_{loc}(\Omega )\) is continuously included in \(H^{-m}_{loc}(\Omega ),\) a previous result implies that \(\mathrm{supp}(\mu )\) is a subset of \(\{(x,\xi )\in \Omega \times S^{d-1}:p(x,\xi )=0\}=\{(x,\xi )\in \Omega \times S^{d-1}:q(x,\xi )=0\}.\)
If \(s\in I \mapsto (x(s),\xi (s))\) is a bicharacteristic in \(\Omega \times ({\mathbb {R}}^d\setminus \{0\}),\) in which q never vanishes, then the homogeneity of q in the second variable implies that \(q(x(s),\xi (s)/|\xi (s)|)\ne 0\) for all s. Hence, the curve \(s\in I \mapsto (x(s),\xi (s)/|\xi (s)|)\) never touches \(\mathrm{supp}(\mu ).\) It follows that \(\mathrm{supp}(\mu )\) is a subset of the union of curves like \(s\in I \mapsto \left( x(s), \frac{\xi (s)}{|\xi (s)|}\right) \), where \(s\in I \mapsto (x(s),\xi (s))\) is a null-bicharacteristic of q.
To complete the proof, we must show that, for a null-bicharacteristic of q, \((x(s),\xi (s)),\) defined on an interval I, such that \(\left( x(s), \frac{\xi (s)}{|\xi (s)|}\right) \in \mathrm{supp}(\mu )\) for some \(s_0\in I,\) we have \(\left( x(s), \frac{\xi (s)}{|\xi (s)|}\right) \in \mathrm{supp}(\mu )\) for all \(s\in I.\)
We first notice that it is enough to consider a local version of the above assertion. Indeed, the set \(A=\left\{ t\in I: \left( x(s), \frac{\xi (s)}{|\xi (s)|}\right) \in \mathrm{supp}(\mu )\right\} \) is closed in I. Moreover, A is open in I if for each \(s_0\in A,\) there exists \(\varepsilon >0\) such that \(\left( x(s), \frac{\xi (s)}{|\xi (s)|}\right) \in \mathrm{supp}(\mu ),\) for all \(s\in (s_0-\varepsilon ,s_0+\varepsilon )\cap I.\) In this case, we have \(A=I\) whenever \(A\ne \emptyset .\)
By the remarks in the above paragraphs, the proof reduces to show that for each \(s_0\in I\) such that \(\left( x(s_0), \frac{\xi (s_0)}{|\xi (s_0)|}\right) \in \mathrm{supp}(\mu ),\) there exists \(\varepsilon >0\) such that \(\left( x(s), \frac{\xi (s)}{|\xi (s)|}\right) \in \mathrm{supp}(\mu )\) for all \(s\in (s_0-\varepsilon ,s_0+\varepsilon )\cap I.\)
We will prove this by contradiction.
Assume that \(s_0\in I\) is such that \(\left( x(s_0), \frac{\xi (s_0)}{|\xi (s_0)|}\right) \in \mathrm{supp}(\mu )\) and for all \(\varepsilon >0\) we find \(s^{*}\in (s_0-\varepsilon ,s_0+\varepsilon )\cap I\) such that \(\left( x(s^{*}), \frac{\xi (s^{*})}{|\xi (s^{*})|}\right) \not \in \mathrm{supp}(\mu ).\) Without loss of generality, we may assume \(s^*\in (s_0,s_0+\varepsilon ).\)
Let \((x_0,\xi _0)=(x(s_0),\xi (s_0))\) and \((x_0^{*},\xi _0^*)=(x(s^*),\xi (s^*)).\) By semigroup properties, we have \((x_0^{*},\xi _0^*)=(x(s^*-s_0,x_0,\xi _0),\xi (s^*-s_0,x_0,\xi _0)),\) where \((x(s,x,\xi ),\xi (s,x,\xi ))\) denote the bicharacteristics that satisfy \((x(0,x,\xi ),\xi (0,x,\xi ))=(x,\xi ).\) Notice that we also have \((x_0,\xi _0)=(x(s_0-s^*,x_0^*,\xi _0^*),\xi (s_0-s^*,x_0^*,\xi _0^*)).\)
Since we are working only locally, we can assume that we are working on a connected component of \(\Omega .\) Moreover, by choosing \(\varepsilon >0\) sufficiently small, we may assume that \(x_0^*\in B(x_0,r)\subset B[x_0,4r]\subset \Omega \) and that the bicharacteristics through the points on a compact set \(B[x_0,4r]\times S^{d-1}\subset \Omega \times ({\mathbb {R}}^d\setminus \{0\})\) are all defined on a same interval \((-\varepsilon ,\varepsilon ).\) Such interval contains \(s^*-s_0.\)
We claim that, by the continuous dependence on initial data, we may choose \(\varepsilon >0\) so that, for all \(s\in (-\varepsilon ,\varepsilon )\) with \(r\le 1/6\) sufficiently small and for all \((x,\xi )\in [B[x_0,4r]\setminus B(x_0,3r)]\times S^{d-1}\) we have \((x(s,x,\xi ),\xi (s,x,\xi ))\in (\Omega \setminus B[x_0,r])\times ({\mathbb {R}}^d\setminus \{0\}).\)
Indeed, for each \((\tilde{x},\tilde{\xi })\in [B[x_0,4r]\setminus B(x_0,3r)]\times S^{d-1},\) the solution \((x(s,\tilde{x},\tilde{\xi }),\xi (s,\tilde{x},\tilde{\xi }))\) is contained in \((\Omega \setminus B(x_0,11r/4))\times ({\mathbb {R}}^d\setminus B(0,3r))\) for s belonging to a sufficiently small interval \(I_{\tilde{x},\tilde{\xi }} =[-\delta _{\tilde{x},\tilde{\xi }},\delta _{\tilde{x},\tilde{\xi }}].\) Recall that the continuous dependence on initial data in the system \(\dot{x}=\partial q/\partial \xi \) and \(\dot{\xi }=-\partial q/\partial x\) implies that for each \(\varepsilon '>0,\) there exists \(\delta =\delta (\delta _{\tilde{x},\tilde{\xi }},\varepsilon ')>0\) such that for all \((x,\xi )\) satisfying \(\Vert (x,\xi )-(\tilde{x},\tilde{\xi })\Vert _{\max }<\delta ,\) we have \((x(s,x,\xi ),\xi (s,x,\xi ))\) defined on \(I_{\tilde{x},\tilde{\xi }}\) and
for all \(s\in I_{\tilde{x},\tilde{\xi }}.\) Hereafter, we assume that \(\varepsilon _1<r/2\). For each \(s \in I_{\tilde{x},\tilde{\xi }}\) we define
where \(\eta <\min \{\delta ,r/4\}\). We also set
and
Note that the union of the sets \(A_{\tilde{x},\tilde{\xi }}\) with \((\tilde{x},\tilde{\xi }) \in B[x_0,4r]\setminus B(x_0,3r) \times S^{d-1}\) does not intersect \(B(x_0,5r/2)\times B(0,11r/4)\). Hence by virtue of the compactness of \(B[x_0,4r]\setminus B(x_0,3r) \times S^{d-1}\) there exist \((\tilde{x}_1,\tilde{\xi }_1),\dots , (\tilde{x}_N,\tilde{\xi }_N) \in B[x_0,4r]\setminus B(x_0,3r) \times S^{d-1}\) such that
Therefore, given \((x_1,\xi _1) \in B[x_0,4r]\setminus B(x_0,3r) \times S^{d-1}\), there exists \(i_0 \in \{1,\dots , N\}\) such that
Thus, there exists \(s_0 \in I_{\tilde{x}_{i_0},\tilde{\xi }_{i_0}}\) such that
Using the continuous dependence on data we deduce that
for all \(s \in I_{\tilde{x}_{i_0},\tilde{\xi }_{i_0}}\) such that \(s+s_0 \in I_{\tilde{x}_{i_0},\tilde{\xi }_{i_0}}\). Thus, for all \((a,b)\in B[x_0,r] \times B[0,r]\), we obtain
where I denotes the smallest of the intervals \(I_{\tilde{x}_i, \tilde{\xi }_i}\), for \(i \in \{1,\dots , N\}\). The situation described above is illustrated in Figure 1.
Since \(\left( x_{0}^{*}, \frac{\xi _{0}^{*}}{|\xi _0^{*}|}\right) \not \in \mathrm{supp}(\mu ),\) it follows from Theorem A.7 that there exists \(g\in C^{\infty }_{0}(\Omega \times S^{d-1},[0,1])\) such that \(g\left( x_0^{*}, \frac{\xi _0^{*}}{|\xi _0^{*}|}\right) >0\) and \(\int _{\Omega \times S^{d-1}}gd\mu =0.\)
Let \(\phi \in C^{\infty }_0(\Omega )\) be such that \(\mathrm{supp}(\phi )\subset B[x_0^*,r_2]\subset B(x_0,r),\) \(\phi (x_0^*)=1\) and \(0\le \phi \le 1.\)
We then define \(\tilde{g}(x,\xi )=|\xi |^{1-m}\phi (x)g \left( x,\xi /|\xi |\right) ,\) \((x,\xi )\in \Omega \times ({\mathbb {R}}^{d}\setminus \{0\}).\) It follows that \(\tilde{g}\) is smooth on \(\Omega \times ({\mathbb {R}}^{d}\setminus \{0\}),\) homogeneous of degree \(1-m\) in the variable \(\xi ,\) and \(\tilde{g}\) has compact support in the variable x. Notice that
Hence, \(\int _{\Omega \times S^{d-1}}\tilde{g}d\mu =0.\)
Next, we use the bicharacteristics to bring the information given by g in a neighborhood of \(\left( x_{0}^{*}, \frac{\xi _{0}^{*}}{|\xi _0^{*}|}\right) \) to a neighborhood of \(\left( x_0, \frac{\xi _0}{|\xi _0|}\right) \).
For \((x,\xi )\in B(x_0,4r)\times S^{d-1},\) we define \(f(x,\xi )=\tilde{g}(x(s^*-s_0,x,\xi ),\xi (s^*-s_0,x,\xi )).\) It follows that f is well-defined since we previously established that the bicharacteristics \((x(s,x,\xi ),\xi (s,x,\xi ))\) are defined on a same interval which contains \(s^*-s_0,\) for all \((x,\xi )\in B(x_0,4r)\times S^{d-1}.\) Notice that f extends to a continuous function on \(\Omega \times S^{d-1}\) since \(f(x,\xi )=0\) for all \((x,\xi )\in (\Omega \setminus B[x_0,3r])\times S^{d-1}\) (recall that \((x(s^*-s_0,x,\xi ),\xi (s^*-s_0,x,\xi ))\) belongs to \((\Omega \setminus B[x_0,r])\times ({\mathbb {R}}^d\setminus \{0\}),\) for all \((x,\xi )\in (\Omega \setminus B[x_0,3r])\times S^{d-1}\)). It follows that \(f\in C_0(\Omega \times S^{d-1},[0,1]).\) In order to complete the proof, we will show that \(f(x_0,\xi _0/|\xi _0|)>0\) and \(\int _{\Omega \times S^{d-1}}fd\mu =0.\) Note that, by Theorem A.7, this implies a contradiction, since \((x_0,\xi _0/|\xi _0|)\in \mathrm{supp}(\mu )\).
Before we proceed, we note that for \(\lambda >0\) we have
for all \((x,\xi )\in \Omega \times ({\mathbb {R}}^{d}\setminus \{0\}).\) Indeed, since q is homogeneous of degree one in the variable \(\xi ,\) it follows that \(\partial q/\partial \xi \) is homogeneous of degree zero and \(\partial q/\partial x\) is homogeneous of degree one. Hence,
and
It follows that both \((x(s,x,\xi ),\lambda \xi (s,x,\xi ))\) and \((x(s,x,\lambda \xi ),\xi (s,x,\lambda \xi ))\) are solutions through the point \((x,\lambda \xi ).\) By uniqueness we conclude that (A.35) holds.
It follows that
In order to compute \(\int _{\Omega \times S^{d-1}}fd\mu ,\) we use the notation
for \(s\in (-\varepsilon ,\varepsilon )\) and \((x,\xi )\in B(x_0,4r)\times S^{d-1}\). In particular,
and
where \(\chi \in C^\infty _0(B(x_0,4r))\) is such that \(0\le \chi \le 1\) and \(\chi \equiv 1\) on a neighborhood of \(B[x_0,3r].\)
We complete the proof by showing that
since
For \(s\in (-\varepsilon ,\varepsilon ),\) we have
Observing that \(\mathrm{supp}(H_q(\chi )(x,\xi ))\subset B(x_0,4r)\setminus B[x_0,3r]\), it follows that the function \((\tilde{g}\circ \phi _s)(x,\xi )H_q(\chi )(x,\xi )\) vanishes identically on \(\Omega \times S^{d-1}\). Thus, we also have
Hence,
since \(H_q=|\xi |^{1-m}H_p\) on the support of \(\mu .\)
Applying Theorem A.6, it follows that
Therefore,
is constant on \((-\varepsilon ,\varepsilon ).\) Using \(s=0\) and \(s=s^{*}-s_0,\) we obtain
which completes the proof. \(\square \)
We finish this section by examining the case of the wave equation in an inhomogeneous medium:
whose principal symbol is given by
\(t \in {\mathbb {R}}\), \(x \in \Omega \subset {\mathbb {R}}^{d}\), \((\tau ,\xi ) \in {\mathbb {R}}^{d+1}\), \(\rho \in C^\infty (\Omega )\), \(0<\alpha \le \rho (x)\le \beta <\infty \), and \(K(x)=(k_{ij}(x))_{1 \le i,j \le d}\) is a positive-definite matrix satisfying
for \(0<a<b<\infty \).
Let’s describe the bicharacteristics of p. From Lemma A.2 the bicharacteristics do not change if we multiply p by a non-zero function. So we can study the Hamiltonian curves of
Proposition A.3
Up to a change of variables, the bicharacteristics of (A.38) are curves of the form
where \(s \mapsto y(s)\) is a geodesic of the metric \(G=\left( \frac{K}{\rho }\right) ^{-1}\) on \(\Omega \), parameterized by the arc length.
Proof
Let’s define a curve
such that
From (A.38) and (A.39)\(_1\), we have
Also, from (A.38) and (A.39)\(_2\) it follows that
Equation (A.38) and (A.39)\(_3\) ensure that
Finally, from (A.39)\(_3\) we obtain
Thus, the sought after curve must satisfy the equations
Introducing the matrix \(G(x(s)) = \left( \frac{K(x(s))}{\rho (x(s))}\right) ^{-1},\) from the second equation of (A.41) we obtain
Let \(f:\mathrm{GL}(n,{\mathbb {R}}) \longrightarrow \mathrm{GL} (n,{\mathbb {R}})\) be defined by \(f(X) = X^{-1}\) so that its derivative is given by \(d (f)_{A} (V) = - A^{-1} \cdot V \cdot A^{-1}.\) So,
That is,
Therefore, from (A.42) and (A.43), we obtain
Thus, from (A.42) and (A.44), we have
Since \(\tilde{p}\) is 0 over each null bicharacteristic, it follows that
And since \(\dot{\tau }(s) = 0,\) we have \(\tau (s) = \tau ,\) for some \(\tau \in {\mathbb {R}}.\) Thus,
On the other hand, from (A.40)
that is, \(\dot{x}(s)^{\top } \cdot G(x(s)) \cdot x(s)\) over each null bicharacteristic of \(\tilde{p}\). By (A.45) and (A.46) we can write
Defining the arc length functional L by
from (A.47) we obtain
that is, L satisfies the Euler–Lagrange equations, so x is a geodesic.
Therefore, the bicharacteristics are curves of the form
Consider the function \(\alpha (s) = - \frac{s}{\tau },\) thus \(\alpha \) is the arc length parameter for x. In fact, it suffices to show that for \(y(s) = (x \circ \alpha )(s)\) we have
Indeed,
Note that, \((x \circ \alpha )^{\varvec{\cdot }}(s) = \dot{x}(\alpha (s)) \dot{\alpha }(s) = - \frac{\dot{x}(\alpha (s))}{\tau }.\) So, \(\dot{x}(\alpha (s)) = - \tau (x \circ \alpha )^{\varvec{\cdot }}(s).\) Thus,
Therefore, less than one reparametrization, the bicharacteristics of p are curves of the form
where y(s) is a geodesic of the G metric parameterized by the arc length.
Conversely, consider \(s \mapsto y(s)\) a geodesic of the G metric parameterized by the arc length. Let us show that, less than one reparameterization of the curve \(\gamma \) given by
is a bicharacteristics of p. For this, we must prove that the component functions satisfy the Hamilton-Jacobi equations.
Let \( \beta (s) = - \tau s, \) then defining \(x(s) = y(\beta (s)),\) we have
Let \(t(s) = - \tau s\) and \(\xi (s) = G(x(s)) \dot{x}(s).\) Thus,
It remains to show that
which is equivalent to
Since y(s) is parameterized by arc length, we have
that is,
From the Euler–Lagrange equations,
Hence, taking \(f(s) = G(y(s)) \dot{y}(s)\), we have
Thus, from (A.49) and recalling that \(\dot{\beta }(s) = - \tau \), we obtain
Therefore,
which ends the proof. \(\square \)
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Cavalcanti, M.M., Domingos Cavalcanti, V.N., Gonzalez Martinez, V.H. et al. Decay Rate Estimates for the Wave Equation with Subcritical Semilinearities and Locally Distributed Nonlinear Dissipation. Appl Math Optim 87, 2 (2023). https://doi.org/10.1007/s00245-022-09918-4
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DOI: https://doi.org/10.1007/s00245-022-09918-4