Pulsating solutions for multidimensional bistable and multistable equations

Abstract

We investigate the existence of pulsating front-like solutions for spatially periodic heterogeneous reaction–diffusion equations in arbitrary dimension, in both bistable and more general multistable frameworks. In the multistable case, the notion of a single front is not sufficient to understand the dynamics of solutions, and we instead observe the appearance of a so-called propagating terrace. This roughly refers to a finite family of stacked fronts connecting intermediate stable steady states whose speeds are ordered. Surprisingly, for a given equation, the shape of this terrace (i.e., the involved intermediate states or even the cardinality of the family of fronts) may depend on the direction of propagation.

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Acknowledgements

This study was funded by European Research Council (No. 321186) and Agence Nationale de la Recherche (No. ANR-14-CE25-0013).

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Correspondence to Luca Rossi.

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Communicated by Y. Giga.

Appendix

Appendix

Here we recall the order interval trichotomy of Dancer and Hess [5]; see also [15].

Theorem A.1

[5] Let \(p<p'\) be two periodic steady states of (1.1). Then one of the following situations occurs:

  1. (a)

    there is a periodic steady state \({\tilde{p}}\) satisfying \(p<{\tilde{p}}<p'\),

  2. (b)

    there exists an entire solution u to (1.1) such that \((u(k,\cdot ))_{k\in \mathbb {Z}}\) is an increasing family of periodic functions satisfying

    $$\begin{aligned} u(-k,\cdot )\searrow p,\qquad u(k,\cdot )\nearrow p',\qquad \text {as } k\rightarrow +\infty , \text { uniformly in }[0,1]^N, \end{aligned}$$
  3. (c)

    there exists an entire solution u to (1.1) such that \((u(k,\cdot ))_{k\in \mathbb {Z}}\) is a decreasing family of periodic functions satisfying

    $$\begin{aligned} u(-k,\cdot )\nearrow p',\qquad u(k,\cdot )\searrow p,\qquad \text {as } k\rightarrow +\infty , \text { uniformly in }[0,1]^N. \end{aligned}$$

This trichotomy plays a crucial role in our proofs, as it allows us to look at multistable equations as juxtapositions of monostable problems. Owing to Theorem 1.3 quoted from Weinberger [19], we infer the existence of the minimal speeds of fronts above and below any unstable steady state q. In Assumption 1.3 we require that such speeds are strictly ordered. In the next proposition we show that a sufficient condition guaranteering this hypothesis is that q is linearly unstable. We also point out for completeness that the order between the speeds is always true in the large sense.

Proposition A.2

Assume that \(u \mapsto f (x,u)\) is of class \(C^1\).

Under either Assumption 1.1 or 1.2, and with the notation of Assumption 1.3, for any unstable periodic steady state q between 0 and \({\bar{p}}\) and any \(e\in \mathbb {S}^{N-1}\), there holds that

$$\begin{aligned} \overline{c}_q \ge 0 \ge \underline{c}_q. \end{aligned}$$

Moreover, if q is linearly unstable, then

$$\begin{aligned} \overline{c}_q> 0 > \underline{c}_q. \end{aligned}$$

Proof

We show the inequalities for \(\overline{c}_q\), the ones for \(\underline{c}_q\) follow by considering the nonlinear term \(-f(x,-u)\) and the direction \(-e\).

We recall that \(\overline{c}_q\) is the minimal speed of fronts in the direction e connecting \(p_{i_1}\) to q, where \(p_{i_1}\) is the smallest stable periodic steady state lying above q. Let \(\lambda _0\) denote the periodic principal eigenvalue of the linearized operator

$$\begin{aligned} \mathcal {L}_0w := \text {div} (A(x)\nabla w ) + \partial _u f (x, q(x)) w. \end{aligned}$$

The instability of q implies that \(\lambda _0\ge 0\). We distinguish two cases.

Linearly unstable case: \(\lambda _0>0\).

Because the operator \(\mathcal {L}_0\) is self-adjoint, it is well-known that \(\lambda _0\) can be approximated by the Dirichlet principal eigenvalue of \(\mathcal {L}_0\) in a large ball (see, e.g., [4, Lemma 3.6]). Namely, calling \(\lambda (r)\) the principal eigenvalue of \(\mathcal {L}_0\) in \(B_r\) with Dirichlet boundary condition, there holds that \(\lambda (r)\rightarrow \lambda _0\) as \(r\rightarrow +\infty \). Then we can find r large enough so that \(\lambda (r)>0\). Let \(\varphi \) be the associated principal eigenfunction. The function \(\psi \) defined by

$$\begin{aligned} \psi (t,x):=q(x)+\varphi (x)e^{\frac{1}{2}\lambda (r)t}, \end{aligned}$$

satisfies for \(t\in \mathbb {R}\), \(x\in B_R\),

$$\begin{aligned} \partial _t\psi - \text {div} (A(x) \nabla \psi )= f (x,q) +\left( \partial _u f(x,q)-\frac{1}{2}\lambda (r)\right) \varphi (x)e^{\frac{1}{2}\lambda (r)t}. \end{aligned}$$

Hence, by the \(C^1\) regularity of \(u\mapsto f(x,u)\), there exists \(T\in \mathbb {R}\) such that \(\psi \) is a subsolution of (1.1) for \(t\le T\), \(x\in B_r\). Up to reducing T, we further have that \(\psi < p_{i_1}\) for all \(t \le T\).

Assume by way of contradiction that (1.1) admits a pulsating front \(U ( x , x\cdot e -ct)\) connecting \(p_{i_1}\) to q with a speed \(c\le 0\). Let \(\xi \in \mathbb {Z}^N\) be such that \(U ( \xi , \xi \cdot e-cT)<\psi (T,0)\). Observe that \(U ( x , x\cdot e -ct)\) is bounded from below away from q for \(t\le T\) and \(x\in {\overline{B}}_r(\xi )\), because \(c\le 0\) and \(U(\cdot ,-\infty )\equiv p_{i_1}>q\). We can then find \(T'<T\) such that

$$\begin{aligned} \forall x\in {\overline{B}}_r(\xi ),\quad U ( x , x\cdot e -cT')>\psi (T',x-\xi ). \end{aligned}$$

Because \(\psi (t,x-\xi )\) is a subsolution of (1.1) for \(t<T\) and \(x\in B_r(\xi )\), which is equal to q(x) for \(x\in \partial B_r(\xi )\), the comparison principle eventually yields

$$\begin{aligned} \forall \,T'\le t\le T,\ x\in {\overline{B}}_r(\xi ),\quad U ( x , x\cdot e -ct)>\psi (t,x-\xi ), \end{aligned}$$

contradicting \(U ( \xi , \xi \cdot e-cT)<\psi (T,0)\). This shows that \(\overline{c}_q > 0\) in this case.

Case \(\lambda _0=0\).

The definition of \(p_{i_1}\), together with either Assumption 1.1 or 1.2, imply that the case (b) is the only possible one in Theorem A.1 with \(p=q\) and \(p'=p_{i_1}\). Let u be the corresponding entire solution. For \(\sigma \in \mathbb {R}\), let \(\lambda _{\sigma }\) and \(\varphi _{\sigma }\) denote the periodic principal eigenvalue and eigenfunction of the operator

$$\begin{aligned} \mathcal {L}_{\sigma } w:= & {} \text {div} (A(x)\nabla w ) + 2\sigma e A(x)\nabla w \\&+\,\left( \sigma ^2 e A(x) e + \sigma \text {div} (A(x)e) + \partial _u f (x, q(x)) \right) w. \end{aligned}$$

Fix \(\varepsilon >0\). We define the following function:

$$\begin{aligned} \psi (t,x):=u(t,x)-\varphi _{\sigma }(x) e^{\sigma (x\cdot e+\varepsilon t)}. \end{aligned}$$

We compute

$$\begin{aligned} \partial _t \psi - \text {div} (A(x) \nabla \psi ) = f (x,u) -[\sigma \varepsilon +\partial _u f(x,q)-\lambda _{\sigma }]\varphi _{\sigma }(x) e^{\sigma (x\cdot e+\varepsilon t)}. \end{aligned}$$

For \(\sigma >0\), there exists \(\delta >0\) depending on \(\varepsilon ,\sigma \) such that \(q+\delta <p_{i_1}\) and moreover, for \(0\le s_1\le s_2\le \delta \), there holds that

Then take \(k\in \mathbb {Z}\), also depending on \(\varepsilon ,\sigma \), in such a way that

$$\begin{aligned} \forall t\le k,\ x\in [0,1]^N,\quad u(t,x)\le q(x)+\delta . \end{aligned}$$

We deduce that, for \(t<k\) and \(x\in \mathbb {R}^N\) such that \(\psi (t,x)>q(x)\), the following holds:

$$\begin{aligned} \partial _t \psi - \text {div} (A(x) \nabla \psi )\le f (x,\psi )- \left[ \frac{1}{2}\sigma \varepsilon -\lambda _{\sigma }\right] \varphi _{\sigma }(x) e^{\sigma (x\cdot e+\varepsilon t)}. \end{aligned}$$

Now, for \(r>0\), call as before \(\lambda (r)\) and \(\varphi \) the Dirichlet principal eigenvalue and eigenfunction of \(\mathcal {L}_0\) in \(B_r\). Direct computation shows that for \(\sigma \in \mathbb {R}\), \(\varphi (x) e^{-\sigma x\cdot e}\) is the Dirichlet principal eigenfunction of \(\mathcal {L}_{\sigma }\) in \(B_r\), with eigenvalue \(\lambda (r)\). It follows that \(\lambda (r)<\lambda _\sigma \), because otherwise \(\varphi _{\sigma }\) would contradict the properties of this principal eigenvalue. Because \(\lambda (r)\rightarrow \lambda _0=0\) as \(r\rightarrow +\infty \), we deduce that \(\lambda _\sigma \ge \lambda _0=0\). Namely, \(\sigma \mapsto \lambda _\sigma \) attains its minimal value 0 at \(\sigma =0\) and thus, being regular (see [13]) it satisfies \(\lambda _{\sigma }\le C\sigma ^2\) for some \(C>0\) and, say, \(|\sigma |\le 1\) (this inequality can also be derived using the min-max formula of [16, Theorem 2.1]). As a consequence, taking \(\sigma =\varepsilon ^2\) we find that, for \(\varepsilon \) smaller than some \(\varepsilon _0\), the function \(\psi \) is a subsolution of (1.1) for the values (tx) such that \(t<k\) and \(\psi (t,x)>q(x)\).

Assume now by contradiction that there is a pulsating front \(U ( x , x\cdot e -ct)\) connecting \(p_{i_1}\) to q with a speed \(c<-\varepsilon \) and \(\varepsilon <\varepsilon _0\). Up to translation in time, it is not restrictive to assume that \(U ( 0 , -ck)<u(k,0)\). Let \(R\in \mathbb {R}\) be such that \(U(x,z)>q+\delta \) for \(x\in \mathbb {R}^N\) and \(z\le R\). It follows that \(U ( x , x\cdot e -ct)\ge \psi (t,x)\) for \(t\le k\) and \(x\cdot e-ct\le R\). On the other hand, we see that

$$\begin{aligned} \forall t<k,\ x\cdot e-ct\ge R,\quad \psi (t,x)\le \delta -\left( \min \varphi _{\sigma }\right) e^{\varepsilon ^2(R+ct+\varepsilon t)}. \end{aligned}$$

The right-hand side goes to \(-\infty \) as \(t\rightarrow -\infty \) because \(c+\varepsilon <0\). We can then find \(T<k\) such that \(U ( x , x\cdot e -ct)\ge \psi (t,x)\) for all \(t\le T\) and \(x\in \mathbb {R}^N\). Hence, because \(U>q\), we can apply the comparison principle and infer that \(U ( 0 , -ck)\ge u(k,0)\), which is a contradiction. We have shown that fronts cannot have a speed smaller than \(-\varepsilon \), for \(\varepsilon \) sufficiently small, whence \(\overline{c}_q \ge 0\). \(\square \)

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Giletti, T., Rossi, L. Pulsating solutions for multidimensional bistable and multistable equations. Math. Ann. 378, 1555–1611 (2020). https://doi.org/10.1007/s00208-019-01919-z

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