Generic Transversality of Heteroclinic and Homoclinic Orbits for Scalar Parabolic Equations

Abstract

In this paper, we consider the scalar reaction–diffusion equations $\partial _t u= \Delta u+f(x,u, \nabla u)$ on a bounded domain $\Omega \subset \mathbb {R}^d$ of class $\mathcal {C}^{2,\gamma }$. We show that the heteroclinic and homoclinic orbits connecting hyperbolic equilibria and hyperbolic periodic orbits are transverse, generically with respect to f. One of the main ingredients of the proof is an accurate study of the singular nodal set of solutions of linear parabolic equations. Our main result is a first step for proving the genericity of Kupka–Smale property, the generic hyperbolicity of periodic orbits remaining unproved.

Appendices

Appendix A: The Whitney Topology

If we want to prove generic properties for the parabolic equation (1.1) with respect to the non-linearity f, we need to equip the space of nonlinear functions f with a topology. Let $E\subset \mathbb {R}^n$, $n\geqslant 1$, by $f\in \mathcal {C}^r(E,\mathbb {R})$, we mean that f is r times differentiable in the set E and that these derivatives are continuous. We do not a priori endow $\mathcal {C}^r(E,\mathbb {R})$ with any topology and we do not assume that f or its derivatives are bounded.

In this article, we consider $E={{\overline{\Omega }}}\times \mathbb {R}\times \mathbb {R}^d$ which is unbounded in $\mathbb {R}^{2d+1}$. Since we do not want to exclude unbounded non-linearities, we cannot equip $\mathcal {C}^r(E,\mathbb {R})$ with the classical $\mathcal {C}^r$-topology.

Definition A.1

For any $r\in \mathbb {N}$, we denote by $\mathfrak {C}^r \equiv \mathfrak {C}^r(E,\mathbb {R})$ the space $\mathcal {C}^r(E,\mathbb {R})$ endowed with the Whitney topology, that is the topology generated by the neighborhoods

$$\begin{aligned} \{g\in \mathcal {C}^r(E,\mathbb {R})~|~|D^i f(y)-D^i g(y)|\leqslant \delta (y),~~\forall i\in \{0,1,\ldots ,r\},~~\forall y\in E\}, \end{aligned}$$

where f is any function in $\mathcal {C}^r(E,\mathbb {R})$ and $\delta$ is any positive continuous function.

We emphasize that, if E is bounded, then the Whitney topology coincides with the classical $\mathcal {C}^r$-topology and thus $\mathfrak {C}^r(E,\mathbb {R})$ is a Banach space equipped with the classical norm $\Vert f\Vert = \sup _{i=0, 1,\ldots ,r}\Vert f^{(i)}\Vert _{L^{\infty }}$. However, if $E={{\overline{\Omega }}}\times \mathbb {R}\times \mathbb {R}^d$, the neighborhoods of a function f in the Whitney topology cannot be generated by a countable number of them. As a consequence, this topology is not metrizable and open or closed sets cannot be characterized by sequences. In order to give an idea about the uncountable conditions imposed by the Whitney topology, we recall that a sequence of functions $(f_n)$ converges to a function f in the Whitney topology if and only if there is a compact set $K \subset E$ such that $f_n \equiv f$ in $E\setminus K$ for any $n\in \mathbb {N}$, but for a finite number of them, and such that $(f_n)$ converges to f in the space $\mathcal {C}^r (K,\mathbb {R})$, equipped with the classical topology of uniform convergence of the functions together with their derivatives up to order r. This means that the Whitney topology imposes an uncountable number of conditions of proximity outside compact sets and thus a sequence has to be constant there in order to be convergent.

As already written in Sect. 7, we could have chosen a simpler topology, but the Whitney topology seems to be the most usual one. In order to overcome several technical problems due to this topology, we make more precise some arguments in this appendix. We omit the corresponding problems during the main proofs of this paper to avoid too heavy proofs. However, if all the technical details are written, the interested reader will notice that we easily deal with the fact that the Whitney topology does not generate a Banach space as follows.

Genericity and Baire Property The main purpose of this paper is to obtain the genericity of the transversality of heteroclinic and homoclinic orbits. The notion of generic sets, that are sets containing a countable intersection of dense open sets, is important because it provides a nice notion of large subset. However, the acceptance of this notion is mainly related to the Baire property, that is the fact that the countable intersection of generic sets is generic. A space satisfying the Baire property is called a Baire space. Complete spaces, and in particular Banach ones, are Baire spaces. But when E is unbounded, $\mathfrak {C}^r(E,\mathbb {R})$ with its Whitney topology is even not metrizable. Thus, it is important to emphasize that it is at least a Baire space, implying that the genericity is still a meaningful concept (see [18] or [32] for example).

Smooth Dependences, Open or Dense Subsets and Other Abuses of Notations When E is unbounded, since $\mathfrak {C}^r(E,\mathbb {R})$ is not metrizable, we can speak about continuous dependence on $f\in \mathfrak {C}^r(E,\mathbb {R})$ but not about smooth dependence, even not about derivatives with respect to f. We sometimes use the following abuse of notation. Consider K a compact subset of E and define P as the canonical projection from $\mathfrak {C}^r(E,\mathbb {R})$ onto $\mathfrak {C}^r(K,\mathbb {R})$, that is $Pf:=f_{|K}$ is the restriction of f to K. Now, as already noticed, $\mathfrak {C}^r(K,\mathbb {R})$ endowed with the Whitney topology is equivalent to the Banach space $\mathcal {C}^r(K,\mathbb {R})$ endowed with the classical $\mathcal {C}^r-$norm. Consider a function $\Phi$ depending on f via the values in K only. We may thus associate with $\Phi$ defined in $\mathfrak {C}^r(E,\mathbb {R})$ a function ${\tilde{\Phi }}$ defined in $\mathfrak {C}^r(K,\mathbb {R})$ and then it is relevant to say that ${\tilde{\Phi }}$ depends smoothly on Pf. In this case, we may use an abuse of notations by saying that $\Phi$ depends smoothly on f instead of saying that ${\tilde{\Phi }}$ depends smoothly on Pf (notice that, rigorously, we should not even say that Pf depends smoothly on f).

At this point, it is important to notice that, the restriction operator

$$\begin{aligned} P:\mathfrak {C}^r(E,\mathbb {R})\rightarrow \mathfrak {C}^r(K,\mathbb {R})~\text { with }K\subset E \text { compact and }E\subset \mathbb {R}^n \end{aligned}$$

is continuous, open and surjective. Continuity is clear and surjectivity follows from the Whitney extension theorem (see [1]), or a simpler result if $r=0$ or K is a regular subdomain for which the extension is easily constructed. Openness follows from the following argument: consider $g\in \mathfrak {C}^r(K,\mathbb {R})$ close to 0, extend g to $f\in \mathfrak {C}^r(E,\mathbb {R})$ and truncate f by multiplying it by a smooth function $\chi$ with $0\leqslant \chi \leqslant 1$, $\chi _{|K}\equiv 1$ and $\chi \equiv 0$ outside a small neighborhood of K. This provides a function $\chi f\in \mathfrak {C}^r(E,\mathbb {R})$ with $P(\chi f)=g$ and $\chi f$ as close to 0 in $\mathfrak {C}^r(E,\mathbb {R})$ as wanted as soon as g is small enough. Thus, the image by P of any neighborhood of 0 contains a neighborhood of 0.

The surjectivity of P enables to define the above functional ${\tilde{\Phi }}$ in $\mathfrak {C}^r(K,\mathbb {R})$ because to each function $g\in \mathfrak {C}^r(K,\mathbb {R})$ indeed corresponds a class of equivalence of functions $f\in \mathfrak {C}^r(E,\mathbb {R})$ with $Pf=g$. The openness is useful to show that a property is open in $\mathfrak {C}^r(E,\mathbb {R})$ if this property depends on the value of f in K only: if the property is open in $\mathfrak {C}^r(K,\mathbb {R})$ with the above abuse of notation, then it is open in $\mathfrak {C}^r(E,\mathbb {R})$. Together, these properties show that, with the abuse of notation, if a property is open and dense (resp. generic) in $\mathfrak {C}^r(K,\mathbb {R})$ then it is open and dense (resp. generic) in $\mathfrak {C}^r(E,\mathbb {R})$.

Notice that the above tricks have already been widely used in previous articles (see [7] for instance). Finally, for a further study of the Whitney topology and the comparison with the weak topology, we refer the reader to [18] or [32] for example.

Appendix B: Sard Theorem and Sard–Smale Transversality Theorems

The Sard theorem [57] and the transversality theory (which goes back to Thom [67]) are very useful tools for proving the genericity of a given property in finite dimension. In [63], Smale has shown how to use Fredholm theory to generalize the transversality theorems to infinite-dimensional Banach spaces. There exist different versions of this kind of transversality theorems (often called Sard–Smale theorems or Thom theorems) with slight changes in the hypotheses, depending on the framework, in which they are used. We recall here the general framework and the version used in this paper.

Let $\mathcal {M}$ and $\mathcal {N}$ be two differentiable Banach manifolds and let $f:\mathcal {M}\longrightarrow \mathcal {N}$ be a differentiable map. We say that $x\in \mathcal {M}$ is a regular point of f if $Df(x):T_x\mathcal {M}\rightarrow T_{f(x)}\mathcal {N}$ is surjective and its kernel splits (that is, has a closed complement in $T_x\mathcal {M}$). A point $y\in \mathcal {N}$ is a regular value of f if any $x\in \mathcal {M}$ such that $f(x)=y$ is a regular point of f. The points of $\mathcal {N}$ which are not regular values are said critical values. The classical theorem of Sard is as follows.

Theorem B.1

If U is an open set of $\mathbb {R}^p$ and if $f:U\longrightarrow \mathbb {R}^q$ is of class $\mathcal {C}^s$ with $s>max(p-q,0)$, then, the set of critical values of f in $\mathbb {R}^q$ is of Lebesgue measure zero.

Using Fredholm operators and a Lyapounov–Schmidt method, Smale has generalized Sard Theorem to infinite-dimensional spaces (for introduction to Fredholm operators, see [6] for example). As a consequence of Smale theorem in [63], many versions of Sard–Smale theorems can be obtained, see [1, 31] for example and Fig. 3 for illustration. The versions involving a functional formulation have been used since the pioneer work of Robbin [55] and are very useful in the PDE context where the geometrical arguments may be too difficult to perform, see Theorem B.4 below and [7, 8, 35,36,37]. In this article, the transversality of connecting orbits may be proved with a more geometrical version of Sard–Smale theorems. Indeed, we only need to perturb an unstable manifold, which is finite-dimensional, and we may do it far from the periodic orbit, so that the basic framework does not depend on the parameter (see Sect. 6). This kind of geometrical setting is more difficult to use if we want to prove generic hyperbolicity as discussed in Appendix C below.

We recall the following definition (see [1] for more details).

Definition B.2

Let $\mathcal {M}$ and $\mathcal {N}$ be two $\mathcal {C}^1$ Banach manifolds and let $f\in \mathcal {C}^1(\mathcal {M},\mathcal {N})$. Let $\mathcal {W}$ be a $\mathcal {C}^1$ submanifold of $\mathcal {N}$. The function f is said to be transversal to $\mathcal {W}$ at a point $x\in \mathcal {M}$ if either $f(x)\not \in \mathcal {W}$ or $f(x)\in \mathcal {N}$ and

• (i) $D_xf^{-1}(T_{f(x)}\mathcal {W})$ is a closed subspace of $T_x\mathcal {M}$ which admits a closed complementary space,

• (ii) $D_x f(T_x \mathcal {M})$ contains a closed complement to $T_{f(x)}\mathcal {W}$ in $T_{f(x)}\mathcal {N}$.

We need in this article a slight improvement of Theorem 19.1 of [1]. The idea of replacing the condition on $\Lambda$ by a condition on a dense subset ${\hat{\Lambda }}$ only has been already used in [7, 8, 35] for example.

Theorem B.3

Let $r\geqslant 1$. Let $\mathcal {M}$ be a $\mathcal {C}^r$ separable manifold of dimension n. Let $\mathcal {W}$ be a $\mathcal {C}^r$ manifold of codimension m in a Banach space Y. Let $\Lambda$ be an open subset of a separable Banach space and let ${\hat{\Lambda }}$ be a dense subset of $\Lambda$. Let $\Phi \in \mathcal {C}^r(\mathcal {M}\times \Lambda ,Y)$. Assume that

• (i) $r>n-m$,

• (ii) $\Phi$ is transversal to $\mathcal {W}$ at any point $(x,\lambda )\in \mathcal {M}\times {\hat{\Lambda }}$.

Then, there is a generic set of parameters $\lambda \in \Lambda$ such that the map $x\mapsto \Phi (x,\lambda )$ is everywhere transversal to $\mathcal {W}$.

Proof

Theorem B.3 is proved as Theorem 19.1 of [1]. The only difference is that hypothesis (ii) is assumed here only for a dense set of parameters $\lambda$. To obtain this improvement from the classical version where (ii) is assumed everywhere, we argue as follows. Since $\mathcal {M}$ is separable and finite dimensional, we can find a countable sequence of open subsets $(\mathcal {M}_k)$ such that $\mathcal {M}=\cup \mathcal {M}_k$ and ${{\overline{\mathcal {M}}}}_k$ is contained in $\mathcal {M}$ and is compact. Let $\lambda _0\in {\hat{\Lambda }}$. Let $(\lambda _p)$ be a sequence converging to $\lambda _0$. Assume that there is a point $x_p\in {{\overline{\mathcal {M}}}}_k$ such that $\Phi$ is not transversal to $\mathcal {W}$ at $(x_p,\lambda _p)$. By the compactness property, one may assume that $(x_p)$ converges to $x_0\in {{\overline{\mathcal {M}}}}_k$. Since $\Phi$ is $\mathcal {C}^1$, $\Phi$ is not transversal to $\mathcal {W}$ at $(x_0,\lambda _0)$ which is absurd. Thus, there exists a neighborhood $\mathcal {U}$ of $\lambda _0$ such that ii) holds for any $(x,\lambda )\in \mathcal {M}_k \times \mathcal {U}$. By applying [1, Theorem 19.1], we obtain a generic subset $\mathcal {U}_k\subset \mathcal {U}$ such that for any $\lambda \in \mathcal {U}_k$, the map $x\mapsto \Phi (x,\lambda )$ is transversal to $\mathcal {W}$ for any $x\in \mathcal {M}_k$. Since ${\hat{\Lambda }}$ is dense in $\Lambda$, we have a generic subset ${\tilde{\mathcal {U}}}_k\subset \Lambda$ such that for any $\lambda \in {\tilde{\mathcal {U}}}_k$, the map $x\mapsto \Phi (x,\lambda )$ is transversal to $\mathcal {W}$ for any $x\in \mathcal {M}_k$. The generic set of parameters appearing in the conclusion of Theorem B.3 is then $\cap _k {\tilde{\mathcal {U}}}_k$. $\square$

Fig. 3

The geometric idea behind Sard–Smale theorems as Theorem B.3: if perturbing the parameter $\lambda$ provides enough freedom, a non-transversal intersection between $\Phi (\mathcal {M},\lambda )$ and $\mathcal {W}$ is generically perturbed into either an empty, and thus transversal, intersection or a non-empty transversal intersection

For brief discussions in Appendix C and for the curious reader, we finish by a brief recall of one of the simplest version of Sard–Smale theorem with a functional formulation (see for example [31] for other versions or proofs). Let us recall that a continuous linear map $f:E\longrightarrow F$ between two Banach spaces is a Fredholm map if its image is closed and if the dimension of its kernel and the codimension of its image are finite.

Theorem B.4

Let $k\geqslant 1$ and let $\mathcal {M}$, $\mathcal {N}$ and $\Lambda$ be three $\mathcal {C}^k$ Banach manifolds. Let $y\in \mathcal {N}$ and let $\Phi \in \mathcal {C}^k(\mathcal {M}\times \Lambda ,\mathcal {N})$. Assume that:

• (i) for any $(x,\lambda )\in \Phi ^{-1}(\{y\})$, $D_x \Phi (x,\lambda ):T_x\mathcal {M}\rightarrow T_y\mathcal {N}$ is a Fredholm map of index i strictly less than k,

• (ii) for any $(x,\lambda )\in \Phi ^{-1}(\{y\})$, $D\Phi (x,\lambda ):T_x\mathcal {M}\times T_\lambda \Lambda \rightarrow T_y\mathcal {N}$ is surjective,

• (iii) $\mathcal {M}$ is separable.

Then, there is a generic set of parameters $\lambda \in \Lambda$ such that for all $x\in \mathcal {M}$ such that $(x,\lambda )\in \Phi ^{-1}(\{y\})$, $D_x\Phi (x,\lambda )$ is surjective.

As in Theorem B.3, a similar result holds if $\Lambda$ is replaced by a dense subset ${\hat{\Lambda }} \subset \Lambda$ and if $\Lambda$ is separable (see [7]).

Appendix C: Discussion About Proving the Generic Hyperbolicity of Periodic Orbits

The purpose of this section is unusual. To obtain the genericity of the Kupka–Smale property for the parabolic equation (1.1), it remains to prove the genericity of hyperbolicity of equilibrium points and periodic orbits. The generic hyperbolicity of equilibrium points is proved in [36]. We tried to obtain the generic hyperbolicity of periodic orbits but failed to get a complete proof. In this section, we would like to present some ideas and to point out where there is still a gap in the proof. Maybe this discussion could inspire a motivated reader.

The first proofs of generic hyperbolicity of periodic orbits appeared in [40, 61]. Peixoto in [49] introduced a nice recursion argument, which has been modified in [1, 41]. Basically, the recursion is as follows. We introduce the sets

$$\begin{aligned} {{{\mathcal {G}}}}_1(K)=\{f \in \mathfrak {C}^r \, | \,\hbox { any equilibrium point }e\hbox { of } (1.1)\hbox { with }\Vert e\Vert _{X^\alpha }\leqslant K \hbox { is hyperbolic}\} \\ \begin{aligned} {{{\mathcal {G}}}}_{3/2}&(A,K) = \{f \in {{{\mathcal {G}}}}_1(K)\, | \, \hbox { any non-constant periodic solution } p(t) \hbox { of }(1.1) \\&\text {with period }T \in (0,A] \hbox { such that sup}_{t\in \mathbb {R}} \Vert p(t)\Vert _{X^\alpha } \leqslant K \hbox { is non-degenerate} \}. \end{aligned} \end{aligned}$$

and

$$\begin{aligned} \begin{aligned} {{{\mathcal {G}}}}_{2}(A,K) =&\{f \in {{{\mathcal {G}}}}_1(K) \, | \, \hbox { any non-constant periodic solution } p(t) \hbox { of }(1.1) \\&\text {with period } T \in (0,A] \hbox { such that sup}_{t\in \mathbb {R}} \Vert p(t)\Vert _{X^\alpha } \leqslant K \hbox { is hyperbolic} \}. \end{aligned} \end{aligned}$$

The slightly strange above notation comes from the fact that $\mathcal{G}_1$ and ${{{\mathcal {G}}}}_2$ are the sets originally introduced by Peixoto, whereas the set ${{{\mathcal {G}}}}_{3/2}$ has been introduced later.

We know from the arguments of the second part of Sect. 3 of [36] that $\mathcal {G}_1(K)$ is a dense open subset of $\mathfrak {C}^r$. The idea of the recursion argument is that there exists $\varepsilon >0$ small enough, such that ${{{\mathcal {G}}}}_{2}(\varepsilon ,K)={{{\mathcal {G}}}}_1(K)$ due to the absence of periodic orbits of small period. Then, the method of Peixoto would consist in proving, like in [41], that ${\mathcal {G}}_2(A,K) \cap {\mathcal {G}}_{3/2}(3A/2,K)$ is dense in ${\mathcal {G}}_2(A,K)$ and that ${\mathcal {G}}_{2}(3A/2,K)$ is dense in ${\mathcal {G}}_{3/2}(3A/2,K)$. By this way, we obtain a chain of dense inclusions

$$\begin{aligned} \ldots {\mathcal {G}}_{2}(9\varepsilon /4,K) \underset{\text {dense}}{{\subset }} {\mathcal {G}}_{3/2}({9\varepsilon }/4,K) \underset{\text {dense}}{{\subset }} {\mathcal {G}}_{2}({3\varepsilon }/2,K) \underset{\text {dense}}{{\subset }} {\mathcal {G}}_{3/2}({3\varepsilon }/2,K) \underset{\text {dense}}{{\subset }} {{{\mathcal {G}}}}_{2}(\varepsilon ,K)={{{\mathcal {G}}}}_1(K) \end{aligned}$$

which shows the density of the hyperbolicity of periodic orbits in ${{{\mathcal {G}}}}_1$. The openness of these sets is rather simple and similar to the finite-dimensional case considered in [49]. This scheme of proof has been exactly performed in [1, 41]. The difficulty lies in the proofs of density.

We claim that the following density holds.

Proposition C.1

For any positive A and K, ${{{\mathcal {G}}}}_{3/2}(3A/2,K)\cap \mathcal{G}_{2}(A,K)$ is dense in ${{{\mathcal {G}}}}_{2}(A,K)$.

Proof

We give here very brief arguments since this proposition is only an auxiliary result in the whole proof of generic hyperbolicity, which is unfortunately not yet completed.

The proof of Proposition C.1 is very similar to the one of Proposition 6.2. We apply a suitable version of Sard–Smale theorem (similar to Theorem B.4) to the map

$$\begin{aligned} \Phi ~:~(T,u_0,g) ~ \longmapsto ~ S_{f_0+g}(T)u_0 - u_0. \end{aligned}$$

As usual, the main difficulty is to obtain a surjectivity as required by Hypothesis ii) of Theorem B.4. We skip the details, but simply notice that checking this property is very similar to the end of the proof of Proposition 6.2: we have to find for any solution $\varphi ^*$ of the adjoint equation along a periodic orbit p, a perturbation g of f such that

$$\begin{aligned} \int _{\Omega }\int _{0}^{T} g(x, p(x,s),{\nabla }p(x,s)) \varphi ^*(x,s) ds dx \ne 0. \end{aligned}$$

This is achieved by constructing a function as in Proposition 6.1 by using Proposition 5.1. $\square$

The proof of the genericity of the Kupka–Smale property would be obtained if we could prove the following result.

Conjecture C.2

For any $A>0$ and K, ${{{\mathcal {G}}}}_{2}(3A/2,K)$ is dense in $\mathcal{G}_{3/2}(3A/2,K)\cap {{{\mathcal {G}}}}_{2}(A,K)$.

To prove this conjecture, we only need to know how to make hyperbolic a given simple periodic orbit in the following sense.

Conjecture C.3

Let $f \in \mathcal {C}^\infty (\Omega \times \mathbb {R}\times \mathbb {R}^d,\mathbb {R})$ and let $\mathcal {N}$ be any small open neighborhood of f in $\mathcal {C}^r$. Let p be a simple periodic solution of (1.1) with minimal period $\omega >0$ and such that $\sup _{t \in [0,\omega ]} \Vert p(t)\Vert _{X^\alpha } \leqslant {\tilde{K}}$, where ${\tilde{K}}>0$. Then, there exists a function ${\tilde{f}} \in \mathcal {N}$ such that p is a hyperbolic periodic solution of (1.1) with non-linearity ${\tilde{f}}$.

Once again, the usual strategy would be to apply a Sard–Smale theorem (similar to Theorem B.4) to an appropriate functional $\Phi$ and then to check a surjectivity hypothesis as (ii) of Theorem B.3. If we try the most natural way, we will have to find a perturbation g of f satisfying

$$\begin{aligned} {\text {Re}}\int _0^\omega \int _\Omega \,(D_u g,D_{{\nabla }u} g)(x,p(x,t),{\nabla }p(x,t))\,.\,\psi ^*(x,t)(\phi ,{\nabla }\phi )(x,t)~dx dt ~\ne ~ 0 \end{aligned}$$

(C.1)

where p is the considered simple periodic orbit, $\phi$ a solution of the linearized equation associated to an eigenvalue $\lambda$ with modulus $|\lambda |=1$ and $\psi ^*$ a solution of the adjoint equation. Notice in (C.1) the presence of the real part ${\text {Re}}$ since the spectrum of a periodic orbit has complex eigenvalues. To obtain this perturbation g, we may use a construction as follows.

Proposition C.4

Let $f\in \mathcal {C}^\infty ({{\overline{\Omega }}}\times \mathbb {R}\times \mathbb {R}^d,\mathbb {R})$ and let $p \in \mathcal {C}^\infty (\Omega \times \mathbb {R},\mathbb {R})$ be a periodic solution of (1.1) with minimal period $\omega$. Let $V\in \mathcal {C}^\infty (\Omega \times [0,\omega ],\mathbb {R}^{d+1})$ be a function, which is not everywhere colinear to $(p_t(x,t),{\nabla }p_t(x,t))$. Then, there exists a function $g\in \mathcal {C}^\infty ({{\overline{\Omega }}}\times \mathbb {R}\times \mathbb {R}^d,\mathbb {R})$ such that

$$\begin{aligned} \mathrm{(i)}~~&g(x,p(x,t),{\nabla }p(x,t))=0~~~\forall (x,t)\in \Omega \times \mathbb {R},\\ \mathrm{(ii)}~~&\int _0^\omega \int _\Omega (D_ug,D_{\nabla u}g)(x,p(x,t),{\nabla }p(x,t)).V(x,t)~dxdt\ne 0. \end{aligned}$$

Proof

To simplify the notations, we denote by U the variable $(u,{\nabla }u)\in \mathbb {R}^{d+1}$ and we set $P(x,t)=(p(x,t),{\nabla }p(x,t))\in \mathbb {R}^{d+1}$.

By assumption, there is an open set $\mathcal {U}$ with ${\overline{\mathcal {U}}} \subset \Omega \times (0,\omega )$ such that V is never colinear to $P_t$ on $\mathcal {U}$. Notice that, in particular $P_t(x,t)\ne 0$ for all $(x,t) \in \mathcal {U}$. Due to Proposition 5.1, restricting $\mathcal {U}$, we can assume that, for all $(x_0,t_0)\in \mathcal {U}$, the map $(x,t) \in \Omega \times [0,\omega ) \mapsto (x,P(x,t))\in \Omega \times \mathbb {R}^{d+1}$ reaches the value $(x_0,P(x_0,t_0))$ at $(x_0,t_0)$ only.

Let $(x_0,t_0)\in \mathcal {U}$. We complete $(P_t,V)$ to a basis of $\mathbb {R}^{d+1}$: let $W_1,\ldots ,W_{d-1}$ be $d-1$ vectors of $\mathbb {R}^{d+1}$ such that ($P_t(x_0,t_0)$, $V(x_0,t_0)$, $W_1$, $\ldots$, $W_{d-1}$) is a basis of $\mathbb {R}^{d+1}$. Restricting again $\mathcal {U}$, we can assume that ($P_t(x,t)$, V(x, t), $W_1$, $\ldots$, $W_{d-1}$) is a basis of $\mathbb {R}^{d+1}$ for all $(x,t)\in \mathcal {U}$. Let $\mathcal {V}=\mathcal {U}\times \mathcal {W}$ where $\mathcal {W}\subset \mathbb {R}^d$ is a neighborhood of 0. We define $h:\mathcal {V}\rightarrow \Omega \times \mathbb {R}^{d+1}$ by

$$\begin{aligned} h(x,t,\tau ,s_1,\ldots ,s_{d-1})= \left( x,P(x,t) + \tau V(x,t) + s_1W_1+\cdots +s_{d-1}W_{d-1}\right) . \end{aligned}$$

Up to choosing $\mathcal {V}$ smaller, the local inversion theorem shows that h is a $\mathcal {C}^\infty$-diffeomorphism into its image. We recall that for all $(x_0,t_0)\in \mathcal {U}$, the map $\Omega \times [0,\omega )\ni (x,t)\mapsto (x,P(x,t))\in \Omega \times \mathbb {R}^{d+1}$ takes the value $(x_0,P(x_0,t_0))$ at $(x_0,t_0)$ only. Due to the compactness of the graph of this map, we can restrict $\mathcal {W}$ such that (x, P(x, t)) belongs to $h(\mathcal {V})$ if and only if (x, t) belongs to $\mathcal {U}$. Let $\chi \in \mathcal {C}^\infty (\Omega \times \mathbb {R}^{d+1},\mathbb {R})$ be a function with compact support in $\mathcal {V}$, which will be made more precise later. We set $\theta (x,t,\tau ,s_1,\ldots ,,s_{d-1})=\chi (x,t,\tau ,s_1,\ldots ,,s_{d-1})\tau$. We define the function $g: h(\mathcal {V})\rightarrow \mathbb {R}$ by $g(x,u,{\nabla }u)=g(x,U)=\theta \circ h^{-1}(x,U)$. We can extend g by 0 outside $h(\mathcal {V})$ to obtain a function in $\mathcal {C}^\infty ({{\overline{\Omega }}}\times \mathbb {R}^{d+1})$. By construction, for all $(x,t) \notin \mathcal {U}$, $g(x,P(x,t))=0$ and $D_U g(x,P(x,t))=0$. Moreover, for all $(x,t)\in \mathcal {U}$, $g(x,P(x,t))=\theta (x,t,0,0,\ldots ,0)=0$ and

$$\begin{aligned} \partial _U g(x,P(x,t)).V(x,t)&=D\theta (h^{-1}(x,P(x,t))).\left( \partial _U h^{-1}(x,P(x,t)).V(x,t)\right) \\&=D\theta (x,t,0,\ldots ,0).\left( \partial _U h^{-1}(h(x,t,0,\ldots ,0)).\partial _\tau h(x,t,0,\ldots ,0)\right) \\&=D\theta (x,t,0,\ldots ,0).\partial _\tau (h^{-1}\circ h) (x,t,0,\ldots ,0)\\&=\partial _\tau \theta (x,t,0,\ldots ,0)\\&=\chi (x,t,0,\ldots ,0) \end{aligned}$$

Thus, Property i) of Proposition C.4 holds and moreover

$$\begin{aligned} \int _0^\omega \int _\Omega \partial _U g(x,P(x,t)).V(x,t)~dxdt=\int _\mathcal {U}\chi (x,t,0,\ldots ,0)~dxdt. \end{aligned}$$

Therefore, we can easily choose $\chi$ such that Property ii) of Proposition C.4 also holds. $\square$

The final problem lies in checking that the real part of $\psi ^*(x,t)(\phi ,{\nabla }\phi )$ in (C.1) is not everywhere colinear to $(p_t,{\nabla }p_t)$. This is true if we only consider real functions (see Proposition C.5 below), but we consider here complex solutions $\psi ^*$ and $\phi$ and thus the real part of $\psi ^*(\phi ,{\nabla }\phi )$ correspond to a combination of two real solutions of the linearized equation: the real and the imaginary parts of $\phi$. Even if this colinearity would be very strange and holds surely in very rare cases only (remember that we may break potential symmetries by perturbing f), we found no rigorous argument to avoid it.

We finish with a statement of non-colinearity which could be inspiring.

Proposition C.5

Let I be an open interval of $\mathbb {R}$ and $\Omega$ and open subset of $\mathbb {R}^d$. Let $a\in \mathcal {C}^\infty (\Omega \times I,\mathbb {R})$ and $b\in \mathcal {C}^\infty (\Omega \times I,\mathbb {R}^d)$ be bounded coefficients. Let $v_1$ and $v_2$ be two solutions of the real equation

$$\begin{aligned} \partial _t v(x,t)=\Delta v(x,t) + a(x,t)v(x,t)+b(x,t). \nabla _x v(x,t). \end{aligned}$$

(C.2)

Assume that $(v_1,{\nabla }v_1)$ is colinear to $(v_2,{\nabla }v_2)$ at each points (x, t), meaning that there exist non-simultaneously zero real values $\alpha (x,t)$ and $\beta (x,t)$ such that for all $(x,t)\in \Omega \times I$,

$$\begin{aligned} \alpha (x,t)(v_1,{\nabla }v_1)(x,t)+\beta (x,t)(v_2,{\nabla }v_2)(x,t)=0. \end{aligned}$$

(C.3)

Then $v_1$ and $v_2$ are colinear as solutions, that is that (C.3) holds with real constants $\alpha$ and $\beta$.

Proof

If $v_i\equiv 0$ for $i=1$ or $i=2$ the conclusion is trivial. By the unique continuation properties of Sect. 2, up to choose I and $\Omega$ smaller, we may thus assume that $(v_i,{\nabla }v_i)$ are not zero and thus that $\alpha (x,t)$ and $\beta (x,t)$ are smooth non-zero functions. Moreover, we may fix the normalization $\alpha ^2(x,t)+\beta ^2(x,t)=1$. Fix $(x_0,t_0)$ and set $({\tilde{\alpha }},{\tilde{\beta }})=(\alpha (x_0,t_0),\beta (x_0,t_0))$. We notice that the value $({\tilde{\alpha }},{\tilde{\beta }})$ is taken by $(\alpha (x,t),\beta (x,t))$ in a submanifold $\mathcal {M}$ of dimension $d'\geqslant d$ of $\Omega \times I$ because the possible values of the function lie in the circle $S^1$ which is one-dimensional. The function $w={\tilde{\alpha }} v_1+{\tilde{\beta }} v_2$ is also a solution of (C.2) and by construction $(w,{\nabla }w)$ vanishes in the submanifold $\mathcal {M}$ of dimension $d'$. We now apply Theorem 4.1 with families independent of $\tau \in J=\mathbb {R}$. The singular nodal set of $w(x,t,\tau )$ is $\mathcal {M}\times J$ of dimension $d'+1\geqslant d+1$. Thus $w\equiv 0$ which concludes the proof. $\square$