Rates of Convergence to Non-degenerate Asymptotic Profiles for Fast Diffusion via Energy Methods

Abstract

This paper is concerned with a quantitative analysis of asymptotic behaviors of (possibly sign-changing) solutions to the Cauchy–Dirichlet problem for the fast diffusion equation posed on bounded domains with Sobolev subcritical exponents. More precisely, rates of convergence to non-degenerate asymptotic profiles are revealed via an energy method. The sharp rate of convergence to positive ones was recently discussed by Bonforte and Figalli (Commun Pure Appl Math 74:744-789, 2021) based on an entropy method. An alternative proof for their result is also provided. Furthermore, the dynamics of fast diffusion flows with changing signs is discussed more specifically under concrete settings; in particular, exponential stability of some sign-changing asymptotic profiles is proved in dumbbell domains for initial data with certain symmetry.

1 Introduction

Let $\Omega$ be a bounded $C^{1,1}$ domain of $\mathbb R^N$ with boundary $\partial \Omega$. We are concerned with the Cauchy–Dirichlet problem for the fast diffusion equation of the form

$$\begin{aligned} \partial _t \left( |u|^{q-2}u \right)&= \Delta u \quad{} & {} \text{ in } \Omega \times (0, \infty ), \end{aligned}$$

(1.1)

$$\begin{aligned} u&= 0 \qquad \ {}{} & {} \text{ on } \partial \Omega \times (0, \infty ), \end{aligned}$$

(1.2)

$$\begin{aligned} u&= u_0\quad{} & {} \text{ on } \Omega \times \{0\}, \end{aligned}$$

(1.3)

where $\partial _t = \partial /\partial t$, under the assumptions that

$$\begin{aligned} u_0 \in H^1_0(\Omega ), \quad 2< q < 2^* := \dfrac{2N}{(N-2)_+}. \end{aligned}$$

The Cauchy–Dirichlet problem (1.1)–(1.3) arises from the Okuda–Dawson model (see [40]), which describes an anomalous diffusion of plasma (see also [9, 11]). We refer the reader to [4, Section 2] for the definition of weak solutions concerned in the present paper and their existence and regularity along with a couple of energy estimates (see also [46, 47] as a general reference).

It is well known that every weak solution $u = u(x,t)$ of (1.1)–(1.3) vanishes at a finite time $t_*$, which is uniquely determined by the initial datum $u_0$ [12, 25, 36, 43]; hence, we may write $t_* = t_*(u_0)$. Moreover, Berryman and Holland [10] proved that the rate of finite-time extinction of $u(\cdot ,t)$ is just $(t_*-t)_+^{1/(q-2)}$ as $t \nearrow t_*$, that is,

$$\begin{aligned} c_1 (t_* - t)_+^{1/(q-2)} \le \Vert u(\cdot ,t)\Vert _{H^1_0(\Omega )} \le c_2 (t_* - t)_+^{1/(q-2)} \quad \text{ for } \text{ all } \ t \ge 0 \end{aligned}$$

with $c_1, c_2 > 0$, provided that $u_0 \not \equiv 0$ (see also [16, 21, 26, 39, 44]). Then we define the asymptotic profile $\phi (x)$ of u(x, t) as

$$\begin{aligned} \phi (x) = \lim _{t\nearrow t_*} (t_*-t)^{-1/(q-2)} u(x,t) \not \equiv 0 \ \text{ in } H^1_0(\Omega ). \end{aligned}$$

(1.4)

Apply the change of variables,

$$\begin{aligned} v(x,s) = (t_* - t)^{-1/(q-2)} u(x,t) \quad \text{ with } \ s = \log (t_*/(t_* - t)). \end{aligned}$$

(1.5)

Then $v=v(x,s)$ solves the rescaled problem

$$\begin{aligned} \partial _s \left( |v|^{q-2}v \right)&= \Delta v + \lambda _q|v|^{q-2}v \quad{} & {} \text{ in } \Omega \times (0, \infty ), \end{aligned}$$

(1.6)

$$\begin{aligned} v&= 0 \qquad{} & {} \text{ on } \partial \Omega \times (0, \infty ), \end{aligned}$$

(1.7)

$$\begin{aligned} v&= v_0 \qquad{} & {} \text{ on } \Omega \times \{0\} \end{aligned}$$

(1.8)

with $\lambda _q:= (q-1)/(q-2) > 0$ and the initial datum

$$\begin{aligned} v_0 := t_*(u_0)^{-1/(q-2)}u_0. \end{aligned}$$

(1.9)

Here it is worth mentioning that such rescaled initial data form the set

$$\begin{aligned} {\mathcal {X}}&:= \{ t_*(u_0)^{-1/(q-2)}u_0 :u_0 \in H^1_0(\Omega ) \setminus \{0\}\} \nonumber \\&= \{w \in H^1_0(\Omega ) :t_*(w) = 1\} \end{aligned}$$

(1.10)

(see [5, Proposition 6] for the equality), and this plays a role of the phase set in stability analysis of asymptotic profiles (see Definition 1.2 below and [5] for more details). Now, the asymptotic profile $\phi (x)$ is reformulated as the limit of v(x, s) as $s \rightarrow \infty$; moreover, profiles are characterized as nontrivial solutions to the stationary problem

$$\begin{aligned}{} & {} \begin{aligned} - \Delta \phi&= \lambda _q|\phi |^{q-2}\phi{} & {} \text{ in } \Omega , \end{aligned} \end{aligned}$$

(1.11)

$$\begin{aligned}{} & {} \begin{aligned} \phi&= 0{} & {} \ \quad \quad \quad \quad \,\,\, \text{ on } \partial \Omega , \end{aligned} \end{aligned}$$

(1.12)

and vice versa. On the other hand, although quasi-convergence (i.e., convergence along a subsequence) of $v(\cdot ,s)$ follows from a standard argument (see, e.g., [5, 10, 26, 39, 44]), convergence (along the whole sequence) is more delicate. Actually, it is proved in [29] for non-negative bounded solutions with the aid of Łojasiewicz-Simon’s gradient inequality; however, it still seems open for possibly sign-changing solutions, unless asymptotic profiles are isolated in $H^1_0(\Omega )$ or q is even (i.e., analytic nonlinearity). Moreover, in [17], convergence of relative errors for non-negative solutions is also proved, that is,

$$\begin{aligned} \lim _{t \nearrow t_*} \left\| \frac{u(\cdot ,t)}{(t_*-t)^{1/(q-2)}\phi }-1\right\| _{C({\overline{\Omega }})} = \lim _{s \rightarrow \infty } \left\| \frac{v(\cdot ,s)}{\phi }-1\right\| _{C({\overline{\Omega }})} = 0. \end{aligned}$$

(1.13)

Furthermore, rates of convergence are discussed in [17], where an exponential convergence of the so-called relative entropy (see Corollary 1.5 below) was first proved; however, it still seems rather difficult to quantitatively estimate the rate of convergence. The sharp rate (see below) of convergence for non-degenerate (see below) positive asymptotic profiles was first discussed in [14] by developing the so-called nonlinear entropy method. We also refer the reader to recent developments [37, 38].

Throughout this paper, as in [14], we assume that $\phi$ is non-degenerate, i.e., the linearized problem

$$\begin{aligned} {\mathcal {L}}_\phi (u) := -\Delta u - \lambda _q (q-1) |\phi |^{q-2} u = 0, \quad u \in H_{0}^{1}(\Omega ) \end{aligned}$$

admits no non-trivial solution (or equivalently, ${\mathcal {L}}_\phi$ does not have zero eigenvalue), and hence, ${\mathcal {L}}_\phi :H_{0}^{1}(\Omega )\rightarrow H^{-1}(\Omega )$ is invertible. Then $\phi$ is also isolated in $H^1_0(\Omega )$ from the other solutions to (1.11), (1.12), that is, there exists a neighbourhood of $\phi$ in $H^1_0(\Omega )$ which does not involve any other solutions to (1.11), (1.12). We shall denote by $\{\mu _j\}_{j=1}^\infty$ the non-decreasing sequence consisting of all the eigenvalues for the eigenvalue problem

$$\begin{aligned} -\Delta e = \mu |\phi |^{q-2} e \ \text{ in } \Omega , \quad e = 0 \ \text{ on } \partial \Omega . \end{aligned}$$

(1.14)

Then, thanks to the spectral theory for compact self-adjoint operators (see, e.g., [23]), we find that $0< \mu _1 < \mu _2 \le \cdots \le \mu _j \rightarrow +\infty$ as $j \rightarrow +\infty$. Moreover, the eigenfunctions $\{e_j\}_{j=1}^\infty$ form a complete orthonormal system (CONS for short) in $H^1_0(\Omega )$ and also a CONS in a weighted $L^2$ space $L^2(\Omega ;|\phi |^{q-2}\text {d}x)$ with different normalization. As for positive profiles $\phi$, a slightly different form of the eigenvalue problem (1.14) has already been employed in [14] (see also Remark 1.8 below).

As in [14, Section 2], the sharp rate of convergence is defined for non-degenerate positive asymptotic profiles $\phi >0$ in view of a linearized analysis of (1.6)–(1.8). More precisely, we consider the (formally) linearized equation (i.e., linearization of (1.6)–(1.8) at $\phi$)

$$\begin{aligned} \begin{aligned} (q-1) \phi ^{q-2} \partial _s h&= \Delta h + \lambda _q (q-1) \phi ^{q-2} h \quad{} & {} \text{ in } \Omega \times (0,\infty ),\\ h&\,=\, 0 \ {}{} & {} \text{ on } \partial \Omega \times (0,\infty ),\\ h(\cdot ,0)&\,=\, h_0 := v_0-\phi \ {}{} & {} \text{ in } \Omega , \end{aligned} \end{aligned}$$

where the solution $h = h(x,s)$ may correspond to the difference between v(x, s) and $\phi (x)$. Then for a certain class of initial data $h_0$ the (linear) entropy

$$\begin{aligned} {\textsf{E}}[h(s)] = \int _\Omega h(x,s)^2 \phi (x)^{q-2} \, \text {d}x \end{aligned}$$

turns out to decay at the exponential rate $\textrm{e}^{-\lambda _0s}$ with the exponent

$$\begin{aligned} \lambda _0= \frac{2}{q-1} \left[ \mu _k - \lambda _q(q-1) \right] > 0, \end{aligned}$$

(1.15)

where $k \in {\mathbb {N}}$ is the least integer such that $\mu _k > \lambda _q(q-1)$ (that is, $\nu _k := \mu _k - \lambda _q (q-1)$ is the least positive eigenvalue of ${\mathcal {L}}_\phi$). Here and henceforth, the convergence rate mentioned above (or the exponent $\lambda _0$ as in (1.15)) is called a sharp rate. In contrast with the porous medium equation (i.e., the case for $1< q < 2$), which is studied in [7] by comparison arguments (see also [15, 20, 45] based on Global Harnack Principle or entropy methods and [17, Theorem 3.4], where an entropy method is developed for the PME), it is more difficult to directly prove the optimality of the convergence rate for (1.6)–(1.8) due to the nature of finite-time extinction phenomena of solutions for the fast diffusion equation. To be more precise, the major difficulty consists in comparing solutions with barriers near the extinction time; in particular, it is rather difficult to construct sub- and supersolutions that vanish at the same time as the solutions.

Define the energy functional $J : H^1_0(\Omega ) \rightarrow {\mathbb {R}}$ by

$$\begin{aligned} J(w) := \frac{1}{2} \int _\Omega |\nabla w(x)|^2 \, \text {d}x - \frac{\lambda _q}{q} \int _\Omega |w(x)|^q \, \text {d}x \end{aligned}$$

(1.16)

for $w \in H^1_0(\Omega )$. We are ready to state main results of the present paper.

Theorem 1.1

(Convergence with rates to sign-changing profiles) Let $v = v(x,s)$ be a (possibly sign-changing) weak solution to (1.6)–(1.8) and let $\phi = \phi (x)$ be a (possibly sign-changing) nontrivial solution to (1.11), (1.12) such that $v(\cdot ,s_n) \rightarrow \phi$ strongly in $H^1_0(\Omega )$ for some $s_n \rightarrow +\infty$. Suppose that $\phi$ is non-degenerate. Let $\lambda$ be a constant satisfying

$$\begin{aligned} 0< \lambda < \frac{2}{q-1} C_q^{-2} \Vert \phi \Vert _{L^q(\Omega )}^{-(q-2)} \frac{\mu _k - \lambda _q(q-1)}{\mu _k}, \end{aligned}$$

(1.17)

where $\mu _k$ is the least eigenvalue for (1.14) greater than $\lambda _q(q-1)$ and $C_q$ is the best constant of the Sobolev-Poincaré inequality,

$$\begin{aligned} \Vert w\Vert _{L^q(\Omega )} \le C_q \Vert \nabla w\Vert _{L^2(\Omega )} \quad \text{ for } \ w \in H^1_0(\Omega ). \end{aligned}$$

(1.18)

Then there exists a constant $C_\lambda > 0$ depending on the choice of $\lambda$ such that

$$\begin{aligned} 0 \le J(v(s)) - J(\phi ) \le C_\lambda \textrm{e}^{-\lambda s} \quad \text{ for } \ s \ge 0. \end{aligned}$$

(1.19)

Moreover, there exists a constant $M_\lambda > 0$ depending on the choice of $\lambda$ such that

$$\begin{aligned} \Vert v(s)-\phi \Vert _{H^1_0(\Omega )}^2 \le M_\lambda \textrm{e}^{-\lambda s} \quad \text{ for } \ s \ge 0. \end{aligned}$$

It is noteworthy that Theorem 1.1 is concerned with possibly sign-changing weak solutions to (1.6)–(1.8) and their limits, i.e., nontrivial solutions to (1.11), (1.12). It is well known that (1.11), (1.12) admits infinitely many sign-changing solutions in general (see, e.g., [42]). Moreover, in Section 9, we shall exhibit several examples of sign-changing initial data $u_0$ and domains $\Omega$ for which the (sign-changing) weak solutions $u = u(x,t)$ to (1.1)–(1.3) admit sign-definite and sign-changing asymptotic profiles, although sign-changing asymptotic profiles are often unstable (see [5]).

As a by-product of the theorem above, we can also prove exponential stability of non-degenerate asymptotic profiles which takes the least energy among all the profiles. Let us first recall the notion of stability and instability of asymptotic profiles for fast diffusion, which was introduced in [5] (see also [3, 4, 6]) and will also be used in §9. Here ${\mathcal {X}}$ is the phase set defined in (1.10).

Definition 1.2

(Stability and instability of asymptotic profiles (cf. [5])) Let $\phi$ be an asymptotic profile of a weak solution to (1.1)–(1.3) (equivalently, a nontrivial solution to (1.11), (1.12)).

• (i) $\phi$ is said to be stable, if for any $\varepsilon >0$ there exists $\delta > 0$ such that any solution v of (1.6), (1.7) satisfies

  $$\begin{aligned} \sup _{s \in [0, \infty )} \Vert v(s) -\phi \Vert _{H^1_0(\Omega )} < \varepsilon , \end{aligned}$$

  whenever $v(0) \in \mathcal X$ and $\Vert v(0) - \phi \Vert _{H^1_0(\Omega )}<\delta$.

• (ii) $\phi$ is said to be unstable, if $\phi$ is not stable.

• (iii) $\phi$ is said to be asymptotically stable, if $\phi$ is stable, and moreover, there exists $\delta _0 > 0$ such that any solution v of (1.6), (1.7) satisfies

  $$\begin{aligned} \lim _{s \nearrow \infty }\Vert v(s) - \phi \Vert _{H^1_0(\Omega )} = 0, \end{aligned}$$

  whenever $v(0) \in \mathcal X$ and $\Vert v(0) - \phi \Vert _{H^1_0(\Omega )}<\delta _0$.

• (iv) $\phi$ is said to be exponentially stable, if $\phi$ is stable, and moreover, there exist constants $C, \mu , \delta _1 > 0$ such that any solution v of (1.6), (1.7) satisfies

  $$\begin{aligned} \Vert v(s)-\phi \Vert _{H^1_0(\Omega )} \le C \textrm{e}^{-\mu s} \quad \text{ for } \text{ all } \ s \ge 0, \end{aligned}$$

  provided that $v(0) \in \mathcal X$ and $\Vert v(0)-\phi \Vert _{H^1_0(\Omega )} < \delta _1$.

In what follows, the least-energy solutions to (1.11), (1.12) (or least-energy asymptotic profiles) mean nontrivial solutions to (1.11), (1.12) minimizing the energy J among all the nontrivial solutions to (1.11), (1.12).

Corollary 1.3

(Exponential stability of non-degenerate least-energy profiles) Non-degenerate least-energy asymptotic profiles $\phi$ are exponentially stable in the sense of Definition 1.2. In particular, for any $\lambda$ satisfying (1.17), there exist constants $C, \delta _0 > 0$ such that any solution $v = v(x,s)$ of (1.6)–(1.8) satisfies

$$\begin{aligned} \Vert v(s)-\phi \Vert _{H^1_0(\Omega )} \le C \textrm{e}^{-\lambda s/2} \quad \text{ for } \text{ all } \ s \ge 0, \end{aligned}$$

provided that $v(0) \in \mathcal X$ and $\Vert v(0)-\phi \Vert _{H^1_0(\Omega )} < \delta _0$.

If we restrict ourselves to non-negative weak solutions, we can derive more precise results.

Theorem 1.4

(Sharp convergence rate of energy) Let $v = v(x,s)$ be a non-negative weak solution of (1.6)–(1.8) and let $\phi$ be a positive solution to (1.11), (1.12) such that $v(s_n) \rightarrow \phi$ strongly in $H^1_0(\Omega )$ for some $s_n \rightarrow +\infty$. Assume that $\phi$ is non-degenerate. Then there exists a constant $C > 0$ such that

$$\begin{aligned} 0 \le J(v(s))-J(\phi ) \le C \textrm{e}^{-\lambda _0s} \quad \text{ for } \ s \ge 0, \end{aligned}$$

(1.20)

where $\lambda _0>0$ is given as in (1.15).

The rate of convergence in (1.20) is faster than (1.19) obtained in Theorem 1.1 for (possibly) sign-changing solutions (see Remark 3.2 below). The preceding theorem yields the following corollary, which provides an alternative proof for [14, Theorem 1.2]:

Corollary 1.5

(Sharp convergence rate of relative entropy) Under the same assumptions as in Theorem 1.4, there exists a constant $C > 0$ such that

$$\begin{aligned} \int _\Omega |v(x,s)-\phi (x)|^2 \phi (x)^{q-2} \, \text {d}x \le C \textrm{e}^{-\lambda _0s} \quad \text{ for } \ s \ge 0, \end{aligned}$$

(1.21)

where $\lambda _0$ is given as in (1.15).

Thanks to the energy convergence (along with the entropic one), we can also derive the sharp convergence rate of the $H^1_0$-norm.

Corollary 1.6

(Sharp convergence rate of $H^1_0$-norm) Under the same assumptions as in Theorem 1.4, there exists a constant $C > 0$ such that

$$\begin{aligned} \int _\Omega |\nabla v(x,s)- \nabla \phi (x)|^2 \, \text {d}x \le C \textrm{e}^{-\lambda _0s} \quad \text{ for } \ s \ge 0, \end{aligned}$$

(1.22)

where $\lambda _0$ is given as in (1.15). Moreover, it also holds that

$$\begin{aligned} \left\| \partial _s \left( v^{q-1}\right) (s)\right\| _{H^{-1}(\Omega )} = \left\| J'(v(s)) \right\| _{H^{-1}(\Omega )} \le C \textrm{e}^{-\frac{\lambda _0}{2} s} \end{aligned}$$

(1.23)

for $s \ge 0$.

Scaling back to the original variable, we can readily rewrite Corollaries 1.5 and 1.6 as follows:

Corollary 1.7

Let $u = u(x,t)$ be a non-negative weak solution of (1.1)–(1.3) with a finite extinction time $t_* > 0$ and let $\phi$ be a positive solution to (1.11), (1.12) such that $(t_* - t)^{-1/(q-2)} u(t) \rightarrow \phi$ strongly in $H^1_0(\Omega )$ as $t \nearrow t_*$. Assume that $\phi$ is non-degenerate. Then there exists a constant $C > 0$ such that

$$\begin{aligned} \int _\Omega \left| \frac{u(x,t)}{(t_*-t)^{1/(q-2)}\phi (x)} - 1\right| ^2 \phi (x)^q \, \text {d}x&\le C \left( \frac{t_* - t}{t_*}\right) ^{\lambda _0}, \end{aligned}$$

(1.24)

$$\begin{aligned} \int _\Omega \left| (t_*-t)^{-1/(q-2)}\nabla u(x,t) - \nabla \phi (x) \right| ^2 \, \text {d}x&\le C \left( \frac{t_* - t}{t_*}\right) ^{\lambda _0}, \end{aligned}$$

(1.25)

where $\lambda _0$ is given as in (1.15), for $t \in [0,t_*)$.

The topology of convergence (with the sharp rate) in Corollary 1.6 seems slightly stronger than the main theorem of [14] (see Remark 1.8 below); however, with the aid of a recent boundary regularity result (for non-negative solutions on smooth domains) established by [37], convergences with the sharp rate in stronger topologies also follow from the relative error convergence in the weighted $L^2$ space obtained in [14] (see Corollary 1.5). On the other hand, the main results of the present paper will be proved in a different way, which relies on an energy method rather than the entropy method and which may be much simpler than the method used in [14]. In particular, we can avoid the argument to prove some improvement of the “almost orthogonality” along the nonlinear flow (see Sections 3.2–3.6 of [14]), which may be the most involved part of the paper [14]. Furthermore, it is also noteworthy that all the main results of the present paper can be proved for arbitrary bounded $C^{1,1}$ domains (see Remark 7.1 below for details).

Remark 1.8

(Comparison with [14]) Throughout this paper, we shall use the transformations (1.5), which are slightly different from those used in [14]. Moreover, [14] is concerned with an eigenvalue problem, which is also slightly different from (1.14) and whose eigenvalues $\lambda _{V,k}$, $k \ge 1$ coincide with $\mu _j/t_*$ of the present paper for $\sum _{\ell =1}^{k-1} N_\ell < j \le \sum _{\ell =1}^{k} N_\ell$ (here $N_\ell$ denotes the dimension of the $\ell$-th eigenspace), since the profile function V used in [14] corresponds to $t_*^{1/(q-2)}\phi$ of ours. On the other hand, the sharp rate $\lambda _0$ as in (1.15) coincides with $2T \lambda _m$ as in [14] with $T = t_*$; hence, (1.21) and (1.24) are completely same as the assertion of [14, (1.15) of Theorem 1.2 and (1.18) of Remark 1.3].

Plan of the paper. Sections 2–4 are devoted to a proof for Theorem 1.1. Sections 5–7 are concerned with a proof for Theorem 1.4. In Section 8, Corollaries 1.3, 1.5 and 1.6 will be proved. In Section 9, fast diffusion flows with changing signs are discussed; in particular, exponential stability of some sign-changing asymptotic profiles will be proved in dumbbell domains for initial data with certain symmetry. In Appendix, we shall recall Taylor’s theorem for operators in Banach spaces as well as some elementary inequalities.

Notation. We denote by C a generic non-negative constant which may vary from line to line. Moreover, $q' := q/(q-1)$ denotes the Hölder conjugate of $q \in (1,\infty )$. Furthermore, denote by $H^{-1}(\Omega )$ the dual space of the Sobolev space $H^1_0(\Omega )$ equipped with the inner product $(u,v)_{H^1_0(\Omega )} = \int _\Omega \nabla u \cdot \nabla v \, \text {d}x$ for $u,v \in H^1_0(\Omega )$. Moreover, an inner product of $H^{-1}(\Omega )$ is naturally defined as

$$\begin{aligned} (f,g)_{H^{-1}(\Omega )} = \langle f, (-\Delta )^{-1}g\rangle _{H^1_0(\Omega )} \quad \text{ for } \ f,g \in H^{-1}(\Omega ), \end{aligned}$$

(1.26)

which also gives $\Vert f\Vert _{H^{-1}(\Omega )}^2 = (f,f)_{H^{-1}(\Omega )}$ for $f \in H^{-1}(\Omega )$. Then $-\Delta$ is a duality mapping between $H^1_0(\Omega )$ and $H^{-1}(\Omega )$, that is,

$$\begin{aligned} \Vert u\Vert _{H^1_0(\Omega )}^2&= \Vert -\Delta u\Vert _{H^{-1}(\Omega )}^2 = \langle -\Delta u, u \rangle _{H^1_0(\Omega )},\\ \Vert f\Vert _{H^{-1}(\Omega )}^2&= \Vert (-\Delta )^{-1} f\Vert _{H^1_0(\Omega )}^2 = \langle f, (-\Delta )^{-1} f \rangle _{H^1_0(\Omega )} \end{aligned}$$

for $u \in H^1_0(\Omega )$ and $f \in H^{-1}(\Omega )$. Let X and Y be Banach spaces and denote by ${\mathscr {L}}^{(n)}(X,Y)$ the set of all bounded n-linear forms from X into Y for $n \in {\mathbb {N}}$ (in particular, ${\mathscr {L}}(X,Y) = {\mathscr {L}}^{(1)}(X,Y)$). In particular, we write ${\mathscr {L}}(X) = {\mathscr {L}}(X,X)$. Let $T : X \rightarrow Y$ be an operator. We denote by $\text {D}_G T$ the Gâteaux derivative of T. Moreover, the n-th Fréchet derivative of T is denoted by $T^{(n)}$ for $n \in {\mathbb {N}}$ (we shall write $T' = T^{(1)}$ and $T'' = T^{(2)}$ for short).

2 Convergence with Rates for Possibly Sign-Changing Asymptotic Profiles

Through the next three sections, we shall give a proof for Theorem 1.1. Let $v = v(x,s)$ be a (possibly sign-changing) weak solution to (1.6)–(1.8) and let $\phi = \phi (x)$ be a non-degenerate (possibly sign-changing) solution to (1.11), (1.12) such that $v(s_n) \rightarrow \phi$ strongly in $H^1_0(\Omega )$ for some $s_n \rightarrow +\infty$. Then we first claim that

$$\begin{aligned} v(s) \rightarrow \phi \quad \text{ strongly } \text{ in } H^1_0(\Omega ) \ \text{ as } \ s \rightarrow +\infty . \end{aligned}$$

(2.1)

Indeed, it is well known that every non-degenerate solution $\phi$ is isolated in $H^1_0(\Omega )$ (see, e.g., [5, Section 5.3]), that is, there exists $r > 0$ such that the ball $B_{H^1_0(\Omega )}(\phi ;r) = \{w \in H^1_0(\Omega ) :\Vert w - \phi \Vert _{H^1_0(\Omega )} < r\}$ does not involve any solutions to (1.11), (1.12) except for $\phi$. Now, suppose to the contrary that there exist a sequence $\sigma _n \rightarrow +\infty$ and a constant $r_0 > 0$ such that $\Vert v(\sigma _n) - \phi \Vert _{H^1_0(\Omega )} > r_0$ for any $n \in {\mathbb {N}}$. Then due to [5, Theorem 1], up to a (not relabeled) subsequence, $v(\sigma _n) \rightarrow \psi$ strongly in $H^1_0(\Omega )$ for another (nontrivial) solution $\psi$ to (1.11), (1.12). Then since $\Vert \phi - \psi \Vert _{H^1_0(\Omega )} \ge r$, one can take a sequence ${\tilde{s}}_n \rightarrow +\infty$ such that $\Vert v({\tilde{s}}_n) - \phi \Vert _{H^1_0(\Omega )} = r/2$ (cf. see [4, Proof of Theorem 3]). However, one can take a (not relabeled) subsequence of $({\tilde{s}}_n)$ such that $v({\tilde{s}}_n) \rightarrow {\tilde{\phi }}$ strongly in $H^1_0(\Omega )$ for some nontrivial solution ${{\tilde{\phi }}}$ to (1.11), (1.12) and $\Vert {{\tilde{\phi }}}-\phi \Vert _{H^1_0(\Omega )} = r/2$. It is a contradiction. Thus (2.1) follows. Moreover, we can assume $v(s) \ne \phi$ for any $s>0$; otherwise, $v(s) \equiv \phi$ for any $s > 0$ large enough.

Formally test (1.6) by $\partial _s v(s)$ to see that

$$\begin{aligned} \frac{4}{qq'} \left\| \partial _s (|v|^{(q-2)/2}v)(s) \right\| _{L^2(\Omega )}^2 \le - \dfrac{\text {d}}{\text {d}s} J(v(s)), \end{aligned}$$

(2.2)

where $J: H^1_0(\Omega ) \rightarrow {\mathbb {R}}$ is the functional given by (1.16) (this procedure can be justified via construction of weak solutions and their uniqueness; see, e.g., [2] and also [19] for the fractional case, cf. [22]). Noting that

$$\begin{aligned} \partial _s (|v|^{q-2}v)(s) = \frac{2(q-1)}{q} |v(s)|^{(q-2)/2} \partial _s (|v|^{(q-2)/2}v)(s), \end{aligned}$$

(2.3)

we also find from (2.1) along with the embedding $H^1_0(\Omega ) \hookrightarrow L^q(\Omega )$ that, for any $\varepsilon > 0$, there exists $s_\varepsilon > 0$ large enough such that

$$\begin{aligned}&\left\| \partial _s (|v|^{q-2}v) (s)\right\| _{H^{-1}(\Omega )}\\&\quad \le C_q \left\| \partial _s (|v|^{q-2}v) (s)\right\| _{L^{q'}(\Omega )}\\&\quad \le \frac{2(q-1)}{q} C_q \Vert v(s)\Vert _{L^q(\Omega )}^{(q-2)/2} \left\| \partial _s (|v|^{(q-2)/2}v)(s) \right\| _{L^2(\Omega )}\\&\quad \le \frac{2(q-1)}{q} C_q \left( \Vert \phi \Vert _{L^q(\Omega )}+\varepsilon \right) ^{(q-2)/2} \left\| \partial _s (|v|^{(q-2)/2}v)(s) \right\| _{L^2(\Omega )} \end{aligned}$$

for all $s \ge s_\varepsilon$. Here $C_q$ denotes the best constant of the Sobolev-Poincaré inequality (1.18). As above, we shall often use the dual inequality of (1.18),

$$\begin{aligned} \Vert f\Vert _{H^{-1}(\Omega )} \le C_q \Vert f\Vert _{L^{q'}(\Omega )} \quad \text{ for } \ f \in L^{q'}(\Omega ), \end{aligned}$$

(2.4)

which is equivalent to (1.18) by duality. Hence $C_q$ is also best for (2.4) (see also [20, Appendix 7.8] and [13]). Combining the above with (2.2), we infer that

$$\begin{aligned}&\frac{1}{q-1} C_q^{-2} \left( \Vert \phi \Vert _{L^q(\Omega )}+\varepsilon \right) ^{-(q-2)} \left\| \partial _s (|v|^{q-2}v) (s)\right\| _{H^{-1}(\Omega )}^2\quad \nonumber \\&\quad \le - \frac{\text {d}}{\text {d}s} J(v(s)) \quad \text{ for } \ s \ge s_\varepsilon . \end{aligned}$$

(2.5)

We shall next derive the following gradient inequality:

Lemma 2.1

(Gradient inequality) For any constant $\omega > Q_{\phi }^{1/2}/\sqrt{2}$, where

$$\begin{aligned} Q_\phi := \sup \left\{ \left\langle h, {\mathcal {L}}_\phi ^{-1}(h) \right\rangle _{H^1_0(\Omega )} :h \in H^{-1}(\Omega ), \ \Vert h\Vert _{H^{-1}(\Omega )}=1 \right\} > 0, \end{aligned}$$

there exists a constant $\delta > 0$ such that

$$\begin{aligned} \left( J(w)-J(\phi ) \right) _+^{1/2} \le \omega \Vert J'(w)\Vert _{H^{-1}(\Omega )} \quad \text{ for } \ w \in H^1_0(\Omega ), \end{aligned}$$

(2.6)

provided that $\Vert w-\phi \Vert _{H^1_0(\Omega )} < \delta$.

Proof

As J is of class $C^2$ in $H^1_0(\Omega )$, by Taylor’s theorem (see Theorem A.2 and Remark A.3 in Appendix), one finds that

$$\begin{aligned} J(\phi + h) = J(\phi ) + \frac{1}{2} \langle {\mathcal {L}}_\phi h, h \rangle _{H^1_0(\Omega )} + R(h) \quad \text{ for } \ h \in H^1_0(\Omega ), \end{aligned}$$

(2.7)

where we used the fact that $J'(\phi ) = 0$ and $R(\cdot )$ denotes a generic functional defined on $H^1_0(\Omega )$ satisfying

$$\begin{aligned} \lim _{\Vert h\Vert _{H^1_0(\Omega )} \rightarrow 0} \dfrac{|R(h)|}{\Vert h\Vert _{H^1_0(\Omega )}^2} = 0 \end{aligned}$$

(2.8)

and may vary from line to line. Moreover, one can take an operator $r : H^1_0(\Omega ) \rightarrow H^{-1}(\Omega )$ such that

$$\begin{aligned} J'(\phi +h) = {\mathcal {L}}_\phi h + r(h) \ \text{ in } H^{-1}(\Omega ) \quad \text{ for } \ h \in H^1_0(\Omega ) \end{aligned}$$

(2.9)

and

$$\begin{aligned} \lim _{\Vert h\Vert _{H^1_0(\Omega )}\rightarrow 0} \dfrac{\Vert r(h)\Vert _{H^{-1}(\Omega )}}{\Vert h\Vert _{H^1_0(\Omega )}} = 0. \end{aligned}$$

(2.10)

Hence it follows that

$$\begin{aligned}&J(w)-J(\phi )\nonumber \\&\quad {\mathop {=}\limits ^{(2.7)}} \frac{1}{2} \left\langle {\mathcal {L}}_\phi (w-\phi ), w-\phi \right\rangle _{H^1_0(\Omega )} + R(w-\phi )\nonumber \\&\quad {\mathop {=}\limits ^{(2.9)}} \frac{1}{2} \left\langle J'(w), {\mathcal {L}}_\phi ^{-1} \left( J'(w)\right) \right\rangle _{H^1_0(\Omega )} + R(w-\phi )\nonumber \\&\quad \le \frac{Q_\phi }{2} \Vert J'(w)\Vert _{H^{-1}(\Omega )}^2 + R(w-\phi ) \quad \text{ for } \ w \in H^1_0(\Omega ), \end{aligned}$$

(2.11)

where $Q_\phi$ is a positive constant given by

$$\begin{aligned} Q_\phi := \sup \left\{ \left\langle h, {\mathcal {L}}_\phi ^{-1}(h) \right\rangle _{H^1_0(\Omega )} :h \in H^{-1}(\Omega ), \ \Vert h\Vert _{H^{-1}(\Omega )} = 1 \right\} > 0. \end{aligned}$$

Indeed, ${\mathcal {L}}_\phi$ has positive eigenvalues. Moreover, by (2.8) and (2.10), for any $\nu > 0$ one can take $\delta _\nu > 0$ such that

$$\begin{aligned} |R(h)| \le \frac{\nu }{2}\Vert h\Vert _{H^1_0(\Omega )}^2 \quad \text{ and } \quad \Vert r(h)\Vert _{H^{-1}(\Omega )} \le \nu \Vert h\Vert _{H^1_0(\Omega )} \end{aligned}$$

(2.12)

for any $h \in H^1_0(\Omega )$ satisfying $\Vert h\Vert _{H^1_0(\Omega )} < \delta _\nu$. Now, we see that

$$\begin{aligned}&\Vert w-\phi \Vert _{H^1_0(\Omega )}\\&\quad = \left\| {\mathcal {L}}_\phi ^{-1} \circ {\mathcal {L}}_\phi (w-\phi ) \right\| _{H^1_0(\Omega )}\nonumber \\&\quad \le \Vert {\mathcal {L}}_\phi ^{-1}\Vert _{{\mathscr {L}}(H^{-1}(\Omega ),H^1_0(\Omega ))} \left\| {\mathcal {L}}_\phi (w-\phi )\right\| _{H^{-1}(\Omega )}\\&\quad {\mathop {\le }\limits ^{(2.9)}}\Vert {\mathcal {L}}_\phi ^{-1}\Vert _{{\mathscr {L}}(H^{-1}(\Omega ),H^1_0(\Omega ))} \left( \left\| J'(w) \right\| _{H^{-1}(\Omega )} + \left\| r(w-\phi ) \right\| _{H^{-1}(\Omega )} \right) , \end{aligned}$$

whence it follows from (2.12) that, for $0< \nu < \Vert {\mathcal {L}}_\phi ^{-1}\Vert _{{\mathscr {L}}(H^{-1}(\Omega ),H^1_0(\Omega ))}^{-1}$,

$$\begin{aligned} \Vert w-\phi \Vert _{H^1_0(\Omega )} \le \frac{ \Vert {\mathcal {L}}_\phi ^{-1}\Vert _{{\mathscr {L}}(H^{-1}(\Omega ),H^1_0(\Omega ))} }{ 1-\nu \Vert {\mathcal {L}}_\phi ^{-1}\Vert _{{\mathscr {L}}(H^{-1}(\Omega ),H^1_0(\Omega ))} } \left\| J'(w) \right\| _{H^{-1}(\Omega )} \end{aligned}$$

(2.13)

for any $w \in H^1_0(\Omega )$ satisfying $\Vert w-\phi \Vert _{H^1_0(\Omega )} < \delta _\nu$. Hence combining (2.11), (2.12) and (2.13), we conclude that (2.6) is satisfied for any $\omega > Q_\phi ^{1/2}/\sqrt{2}$ and some $\delta > 0$ small enough. This completes the proof. $\square$

Since $\partial _s (|v|^{q-2}v)(s) = -J'(v(s))$ (see (1.6)–(1.8)) and $J(v(s)) > J(\phi )$ for $s > 0$, we obtain

$$\begin{aligned}&\frac{1}{q-1} C_q^{-2} \left( \Vert \phi \Vert _{L^q(\Omega )}+\varepsilon \right) ^{-(q-2)} \omega ^{-2} \left[ J(v(s)) - J(\phi ) \right] \quad \\&\quad \le - \frac{\text {d}}{\text {d}s} \left[ J(v(s)) - J(\phi ) \right] \end{aligned}$$

for $s \ge s_\varepsilon$ with some $s_\varepsilon > 0$ large enough so that $\sup _{s \ge s_\varepsilon }\Vert v(s)-\phi \Vert _{H^1_0(\Omega )} < \delta$ (see (2.1)). Thus since $J(v(s_0)) \le J(v_0)$, we get

$$\begin{aligned} 0 < J(v(s)) - J(\phi )&\le \big [ J(v(s_0)) - J(\phi ) \big ] \textrm{e}^{- \lambda (s-s_0)}\nonumber \\&\le \big [ J(v_0) - J(\phi ) \big ] \textrm{e}^{\lambda s_0} \textrm{e}^{- \lambda s} \quad \text{ for } \ s \ge s_0, \end{aligned}$$

(2.14)

where $\lambda >0$ is any constant satisfying

$$\begin{aligned} \lambda < \frac{2}{q-1} C_q^{-2} \Vert \phi \Vert _{L^q(\Omega )}^{-(q-2)} Q_{\phi }^{-1} \end{aligned}$$

(2.15)

and $s_0 > 0$ is a constant depending on the choice of $\lambda$. Since $J(v(s)) \le J(v_0)$ for $s \ge 0$, setting $C_\lambda = [J(v_0) - J(\phi )] \textrm{e}^{\lambda s_0}$, we obtain

$$\begin{aligned} 0 < J(v(s)) - J(\phi ) \le C_\lambda \textrm{e}^{-\lambda s} \quad \text{ for } \ s \ge 0. \end{aligned}$$

3 Quantitative Estimates for the Rate of Convergence

In this section, we shall establish a quantitative estimate for the rate of convergence obtained in the last section. To this end, as in [14], let us introduce the weighted eigenvalue problem

$$\begin{aligned} -\Delta e = \mu |\phi |^{q-2} e \ \text{ in } \Omega , \quad e = 0 \ \text{ on } \partial \Omega , \end{aligned}$$

(3.1)

whose eigenpairs $\{(\mu _j,e_j)\}_{j=1}^\infty$ are such that

• $0< \mu _1 < \mu _2 \le \mu _3 \le \cdots \le \mu _k \rightarrow +\infty$ as $k \rightarrow +\infty$,

• The eigenfunctions $\{e_j\}_{j=1}^\infty$ forms a CONS in $H^1_0(\Omega )$; in particular, $(e_j,e_k)_{H^1_0(\Omega )} = \delta _{jk}$ for $j,k \in {\mathbb {N}}$

(see, e.g., [23]). Here we note that $|\phi | \ne 0$ a.e. in $\Omega$ (see [31] and [35]). Moreover, $\{-\Delta e_j\}_{j=1}^\infty$ forms a CONS in $H^{-1}(\Omega )$. In particular, if $\phi$ is a positive solution to (1.11), (1.12), then $\mu _1 = \lambda _q$ and $e_1 = \phi /\Vert \phi \Vert _{H^1_0(\Omega )}$.

For every $u \in H^1_0(\Omega )$, there exists a sequence $\{\alpha _j\}_{j=1}^\infty$ in $\ell ^2$ such that

$$\begin{aligned} u = \sum _{j=1}^\infty \alpha _j e_j \ \text{ in } H^1_0(\Omega ). \end{aligned}$$

Hence

$$\begin{aligned} {\mathcal {L}}_\phi (u)&= \sum _{j=1}^\infty \alpha _j {\mathcal {L}}_\phi (e_j)\\&= \sum _{j=1}^\infty \alpha _j \left[ - \Delta e_j - \lambda _q (q-1) |\phi |^{q-2} e_j \right] \\&= \sum _{j=1}^\infty \alpha _j \frac{\mu _j - \lambda _q (q-1)}{\mu _j} (-\Delta e_j) \quad \text{ in } H^{-1}(\Omega ). \end{aligned}$$

In what follows, we shall write $\nu _j := \mu _j - \lambda _q (q-1)$ for $j \in {\mathbb {N}}$. We particularly find that

$$\begin{aligned} {\mathcal {L}}_\phi (e_j) = \nu _j |\phi |^{q-2} e_j, \quad j \in {\mathbb {N}}. \end{aligned}$$

For any $f \in H^{-1}(\Omega )$, since $(-\Delta )^{-1}f$ lies on $H^1_0(\Omega )$, there exists a sequence $\{\beta _j\}_{j=1}^\infty$ in $\ell ^2$ such that

$$\begin{aligned} (-\Delta )^{-1}f = \sum _{j=1}^\infty \beta _j e_j \ \text{ in } H^1_0(\Omega ),\ \text{ i.e., } f = \sum _{j=1}^\infty \beta _j (-\Delta e_j) \ \text{ in } H^{-1}(\Omega ), \end{aligned}$$

and hence,

$$\begin{aligned} {\mathcal {L}}_\phi ^{-1}(f) = \sum _{j=1}^\infty \beta _j \frac{\mu _j}{\nu _j} e_j \ \text{ in } H^1_0(\Omega ). \end{aligned}$$

(3.2)

Therefore it follows that

$$\begin{aligned} \left\langle f, {\mathcal {L}}_\phi ^{-1}(f) \right\rangle _{H^1_0(\Omega )} = \sum _{j=1}^\infty \beta _j^2 \frac{\mu _j}{\nu _j}. \end{aligned}$$

Noting that

$$\begin{aligned} \Vert f\Vert _{H^{-1}(\Omega )}^2 = \sum _{j=1}^\infty \beta _j^2, \end{aligned}$$

we observe that

$$\begin{aligned} Q_\phi&= \sup \left\{ \left\langle f, {\mathcal {L}}_\phi ^{-1}(f) \right\rangle _{H^1_0(\Omega )} :f \in H^{-1}(\Omega ), \ \Vert f\Vert _{H^{-1}(\Omega )} = 1 \right\} \\&= \max _{j} \frac{\mu _j}{\nu _j} = \frac{\mu _k}{\nu _k} > 0, \end{aligned}$$

where $k \in {\mathbb {N}}$ is the number determining (1.15) (i.e., $\mu _k$ is the least eigenvalue for (1.14) greater than $\lambda _q(q-1)$).

Thus combining the observation above with (2.15), we conclude that

$$\begin{aligned} 0< \lambda < \frac{2}{q-1} C_q^{-2} \Vert \phi \Vert _{L^q(\Omega )}^{-(q-2)} \frac{\mu _k - \lambda _q(q-1)}{\mu _k}. \end{aligned}$$

Consequently, we obtain

Lemma 3.1

(Exponential convergence of energy) Let $v = v(x,s)$ be a (possibly sign-changing) weak solution to (1.6)–(1.8) and let $\phi = \phi (x)$ be a (possibly sign-changing) nontrivial solution to (1.11), (1.12) such that $v(s_n) \rightarrow \phi$ strongly in $H^1_0(\Omega )$ for some $s_n \rightarrow +\infty$. Suppose that $\phi$ is non-degenerate. Then for any constant $\lambda > 0$ satisfying

$$\begin{aligned} 0< \lambda < \frac{2}{q-1} C_q^{-2} \Vert \phi \Vert _{L^q(\Omega )}^{-(q-2)} \frac{\mu _k - \lambda _q(q-1)}{\mu _k}, \end{aligned}$$

(3.3)

where $\mu _k$ is the least eigenvalue for (1.14) greater than $\lambda _q(q-1)$ and $C_q$ is the best constant of the Sobolev-Poincaré inequality (1.18), there exists a constant $C_\lambda > 0$ depending on the choice of $\lambda$ such that

$$\begin{aligned} 0 \le J(v(s)) - J(\phi ) \le C_\lambda \textrm{e}^{-\lambda s} \quad \text{ for } \ s \ge 0. \end{aligned}$$

Remark 3.2

(Least-energy asymptotic profiles) In particular, if $\phi >0$ is a least-energy solution to (1.11), (1.12), it then holds that

$$\begin{aligned} C_q = \frac{\Vert \phi \Vert _{L^q(\Omega )}}{\Vert \nabla \phi \Vert _{L^2(\Omega )}} = \lambda _q^{-1/2} \Vert \phi \Vert _{L^q(\Omega )}^{(2-q)/2} \end{aligned}$$

(see [42, 48] and also [17, 18] for q close to 2), and hence, we can choose any $\lambda$ satisfying

$$\begin{aligned} 0< \lambda < \frac{2\lambda _q}{q-1} \frac{\mu _k - \lambda _q(q-1)}{\mu _k} = \lambda _0\frac{\lambda _q}{\mu _k} = \lambda _0\frac{\mu _1}{\mu _k}. \end{aligned}$$

Here we used the fact that $\mu _1 = \lambda _q$ because of $\phi > 0$. Moreover, noting that $\mu _1 < \mu _k$, we note that in Theorem 1.1 there still remains a gap from the sharp rate $\lambda _0$ even for least-energy asymptotic profiles (cf. Corollary 1.6).

4 Exponential Convergence of Rescaled Solutions

In this section, we shall derive exponential convergence of rescaled solutions $v = v(x,s)$ in $H^1_0(\Omega )$ as $s \rightarrow +\infty$. From (2.5) along with (2.6), we observe that

$$\begin{aligned}&\omega ^{-1} \left[ J(v(s)) - J(\phi ) \right] ^{1/2} \Vert \partial _s (|v|^{q-2}v)(s)\Vert _{H^{-1}(\Omega )}\\&\quad \le \Vert \partial _s (|v|^{q-2}v)(s)\Vert _{H^{-1}(\Omega )}^2 \le - C \frac{\text {d}}{\text {d}s} \left[ J(v(s))-J(\phi )\right] , \end{aligned}$$

whence it follows that

$$\begin{aligned} \Vert \partial _s (|v|^{q-2}v)(s)\Vert _{H^{-1}(\Omega )} \le - C \frac{\text {d}}{\text {d}s} \left[ J(v(s))-J(\phi )\right] ^{1/2}. \end{aligned}$$

Thus one can derive that

$$\begin{aligned}&\left\| |\phi |^{q-2}\phi - (|v|^{q-2}v)(s) \right\| _{H^{-1}(\Omega )}\\&\quad \le \int ^\infty _s \left\| \partial _s \left( |v|^{q-2}v\right) (\sigma ) \right\| _{H^{-1}(\Omega )} \, \text {d}\sigma \\&\quad \le C \left[ J(v(s)) - J(\phi ) \right] ^{1/2} \le M \textrm{e}^{- \frac{\lambda }{2} s} \quad \text{ for } \ s \ge 0 \end{aligned}$$

for some constant $M > 0$. Here we have used Lemma 3.1 with some $\lambda > 0$ satisfying (3.3). Then one has

$$\begin{aligned}&\frac{4}{qq'} \left\| (|v|^{(q-2)/2}v)(s) - |\phi |^{(q-2)/2}\phi \right\| _{L^2(\Omega )}^2 \nonumber \\&\quad \le \left\langle (|v|^{q-2}v)(s)-|\phi |^{q-2}\phi , v(s)-\phi \right\rangle _{H^1_0(\Omega )}\nonumber \\&\quad \le \left\| (|v|^{q-2}v)(s)-|\phi |^{q-2}\phi \right\| _{H^{-1}(\Omega )} \Vert v(s)-\phi \Vert _{H^1_0(\Omega )}\nonumber \\&\quad \le CM \textrm{e}^{- \frac{\lambda }{2} s} \quad \text{ for } \ s \ge 0. \end{aligned}$$

(4.1)

Here we used the inequality

$$\begin{aligned} \frac{4}{qq'} \left| |a|^{(q-2)/2}a - |b|^{(q-2)/2}b\right| ^2 \le \left( |a|^{q-2}a - |b|^{q-2}b\right) (a-b) \end{aligned}$$

(4.2)

for $a,b \in {\mathbb {R}}$ (see Appendix 10) as well as the fact that $\sup _{s \ge 0}\Vert v(s)\Vert _{H^1_0(\Omega )} < +\infty$. Moreover, using Taylor’s theorem (see Theorem A.2 and Remark A.3 in Appendix), we observe that

$$\begin{aligned}&J(v(s))- J(\phi )\nonumber \\&\quad = \frac{1}{2} \Vert \nabla (v(s)-\phi )\Vert _{L^2(\Omega )}^2 + \Big ( \nabla \phi , \nabla (v(s)-\phi ) \Big )_{L^2(\Omega )}\nonumber \\&\qquad - \frac{\lambda _q}{q} \Vert v(s)\Vert _{L^q(\Omega )}^q + \frac{\lambda _q}{q} \Vert \phi \Vert _{L^q(\Omega )}^q\nonumber \\&\quad = \frac{1}{2} \Vert \nabla (v(s)-\phi )\Vert _{L^2(\Omega )}^2 + \lambda _q \int _\Omega |\phi |^{q-2}\phi (v(s)-\phi ) \, \text {d}x\nonumber \\&\qquad - \frac{\lambda _q}{q} \Vert v(s)\Vert _{L^q(\Omega )}^q + \frac{\lambda _q}{q} \Vert \phi \Vert _{L^q(\Omega )}^q\nonumber \\&\quad = \frac{1}{2} \Vert \nabla (v(s)-\phi )\Vert _{L^2(\Omega )}^2 - \dfrac{\lambda _q}{2} (q-1) \int _\Omega |v(s)-\phi |^2 |\phi |^{q-2} \, \text {d}x\nonumber \\&\qquad + o \left( \Vert v(s)-\phi \Vert _{H^1_0(\Omega )}^2 \right) . \end{aligned}$$

(4.3)

One can verify that

$$\begin{aligned}&|v(x,t)-\phi (x)|^2\nonumber \\&\quad \le C |\phi (x)|^{2-q} \left| |v(x,t)|^{(q-2)/2}v(x,t) - |\phi (x)|^{(q-2)/2}\phi (x) \right| ^2, \end{aligned}$$

(4.4)

whenever $\phi (x) \ne 0$. Here we used the inequality,

$$\begin{aligned} 0 \le \frac{|a|^{p-1}a - |b|^{p-1}b}{a-b} \le 2^{1-p} |a|^{p-1} \quad \text{ for } \ a,b \in {\mathbb {R}}, \ a \ne 0, b \end{aligned}$$

(4.5)

for $p \in (0,1)$ (see Appendix 10), with the choice $p = 2/q \in (0,1)$, $a = |\phi |^{(q-2)/2}\phi$ and $b = |v|^{(q-2)/2}v$. Therefore it follows from (4.3) and (4.4) that

$$\begin{aligned}&J(v(s))-J(\phi )\\&\quad \ge \frac{1}{2} \Vert \nabla v(s)-\nabla \phi \Vert _{L^2(\Omega )}^2 - C \left\| (|v|^{(q-2)/2}v)(s) - |\phi |^{(q-2)/2}\phi \right\| _{L^2(\Omega )}^2 \\&\qquad + o \left( \Vert v(s)-\phi \Vert _{H^1_0(\Omega )}^2 \right) . \end{aligned}$$

Combining all these facts (see Lemma 3.1 and (4.1)), we deduce that

$$\begin{aligned} \Vert \nabla v(s)-\nabla \phi \Vert _{L^2(\Omega )}^2 \le C \left( \textrm{e}^{-\lambda s} + \textrm{e}^{-\frac{\lambda }{2}s}\right) \lesssim \textrm{e}^{-\frac{\lambda }{2}s} \end{aligned}$$

for $s \gg 1$. Now, turning back to (4.1) with the above, we can derive that

$$\begin{aligned} \left\| (|v|^{(q-2)/2}v)(s) - |\phi |^{(q-2)/2}\phi \right\| _{L^2(\Omega )}^2 \lesssim \textrm{e}^{- \frac{\lambda }{2} s} \textrm{e}^{-\frac{\lambda }{4} s} = \textrm{e}^{-\frac{\lambda }{2} (1 + \frac{1}{2})s}, \end{aligned}$$

which also leads us to obtain

$$\begin{aligned} \Vert \nabla v(s)-\nabla \phi \Vert _{L^2(\Omega )}^2 \lesssim \textrm{e}^{-\frac{\lambda }{2} (1 + \frac{1}{2})s}. \end{aligned}$$

Iterating these procedures, we can conclude that, for any $\mu < \lambda$, there exists a constant $C_\mu$ depending on the choice of $\mu$ such that

$$\begin{aligned} \Vert \nabla v(s)-\nabla \phi \Vert _{L^2(\Omega )}^2 \le C_\mu \textrm{e}^{-\mu s} \quad \text{ for } \ s \ge 0. \end{aligned}$$

(4.6)

Thus we obtain

Lemma 4.1

(Exponential convergence of rescaled solutions) Under the same assumptions as in Lemma 3.1, if $J(v(s))-J(\phi )$ converges to zero at an exponential rate $\textrm{e}^{-\lambda s}$ as $s \rightarrow +\infty$, then, for any $0< \mu < \lambda$, it holds that $v(s) \rightarrow \phi$ strongly in $H^1_0(\Omega )$ at the rate $\textrm{e}^{-\mu s/2}$ as $s \rightarrow +\infty$.

Proof of Theorem 1.1

Theorem 1.1 can be proved by combining Lemmata 3.1 and 4.1 . To be more precise, first fix $\lambda$ satisfying (1.17), and then, take another $\lambda '$ which is greater than $\lambda$ but still satisfies (1.17). Then apply Lemma 3.1 for the choice $\lambda '$ to get the decay of $J(v(s)) - J(\phi )$ at the rate $\textrm{e}^{-\lambda ' s}$. Finally, apply Lemma 4.1 by substituting $\lambda$ and $\lambda '$ to $\mu$ and $\lambda$ of the lemma, respectively, to get the conclusion. $\square$

5 Almost Sharp Rate of Convergence for Positive Asymptotic Profiles

In Theorem 1.1, the rate of convergence (1.19) is estimated by (1.17); however, it is still suboptimal (even for least-energy solutions, see Remark 3.2). In Sections 5–7, we shall more precisely estimate the rate of convergence for non-negative rescaled solutions to non-degenerate positive asymptotic profiles. We assume that $u_0 \ge 0$ a.e. in $\Omega$, and hence, $v = v(x,s)$ is always non-negative in $\Omega \times (0,+\infty )$. In what follows, we let $k \in {\mathbb {N}}$ be such that $\nu _k > 0$ and $\nu _\ell < 0$ for $\ell = 1,2,\ldots ,k-1$. Moreover, we denote by $L^2(\Omega ;\phi ^{q-2}\text {d}x)$ and $L^2(\Omega ;\phi ^{2-q}\text {d}x)$ the spaces of square-integrable functions with weights $\phi (x)^{q-2}$ and $\phi (x)^{2-q}$, respectively.

Moreover, we shall use the following fact:

$$\begin{aligned} \delta (s) := \left\| \frac{v(s)}{\phi }-1\right\| _{L^\infty (\Omega )} \rightarrow 0 \quad \text{ as } \ s \rightarrow +\infty . \end{aligned}$$

(5.1)

This fact was first proved by [17, Theorem 2.1] based on the Global Harnack Principle developed by [26, Proposition 6.2], where $\Omega$ is supposed to be of class $C^2$, and then, it was extended to a quantitative convergence by [14] with a proof independent of [17] and using only the $C^{1,1}$ regularity of $\Omega$ (see (6.2) and (6.3) in Lemma 6.1 below). Therefore (for bounded $C^{1,1}$ domains, using Theorem 1.1 and Lemma 6.1 below) we can take $s_1 > 0$ large enough so that

$$\begin{aligned} 0 < \frac{1}{2} \phi \le v(s) \le \frac{3}{2} \phi \ \text{ a.e. } \text{ in } \Omega \quad \text{ for } \ s \ge s_1 \end{aligned}$$

(5.2)

(cf. see [26, Proposition 6.2]). Hence since $v(s)/\phi$ is bounded a.e. in $\Omega$ for $s > s_1$, noting that $\partial _s (v^{q/2})(s) \in L^2(\Omega )$ by (2.2), we find from (2.3) that

$$\begin{aligned}&\int _\Omega \left| \partial _s (v^{q-1})(s)\right| ^2 \phi ^{2-q} \, \text {d}x\nonumber \\&\quad = \frac{4(q-1)^2}{q^2} \int _\Omega \left| \partial _s (v^{q/2})(s)\right| ^2 \left( \frac{v(s)}{\phi }\right) ^{q-2} \, \text {d}x < +\infty , \end{aligned}$$

which along with (1.6) implies $J'(v(s)) \in L^2(\Omega ;\phi ^{2-q}\text {d}x)$, for $s > s_1$. Therefore, due to (2.2) and (5.1), for any $\varepsilon > 0$, one can take $s_\varepsilon > s_1$ large enough that

$$\begin{aligned} \Vert J'(v(s))\Vert _{L^2(\Omega ;\phi ^{2-q}\text {d}x)}^2&\le \frac{4(q-1)^2}{q^2} (1+\varepsilon )^{q-2} \int _\Omega \left| \partial _s (v^{q/2})(s)\right| ^2 \, \text {d}x\nonumber \\&\le - (q-1)(1+\varepsilon )^{q-2} \dfrac{\text {d}}{\text {d}s} J(v(s)) \end{aligned}$$

(5.3)

for $s \ge s_\varepsilon$.

With the aid of Taylor’s theorem in Banach spaces, we can obtain the following:

Lemma 5.1

For each $s > s_1$, it holds that

$$\begin{aligned} J(v(s)) - J(\phi )&= \frac{1}{2} \left\langle {\mathcal {L}}_\phi (v(s)-\phi ), v(s)-\phi \right\rangle _{H^1_0(\Omega )} + E(s), \end{aligned}$$

(5.4)

$$\begin{aligned} J'(v(s))&= {\mathcal {L}}_\phi (v(s)-\phi ) + e(s). \end{aligned}$$

(5.5)

Here and henceforth, $E(s) \in {\mathbb {R}}$ and $e(s) \in H^{-1}(\Omega )$ denote generic functions satisfying

$$\begin{aligned} \lim _{s \rightarrow \infty } \frac{|E(s)|}{\Vert v(s)-\phi \Vert _{H^1_0(\Omega )}^{2+\gamma }}< +\infty ,\quad \lim _{s \rightarrow \infty } \frac{\Vert e(s)\Vert _{H^{-1}(\Omega )}}{\Vert v(s)-\phi \Vert _{H^1_0(\Omega )}^{1+\gamma }} < +\infty \end{aligned}$$

(5.6)

for some $\gamma \in (0,1]$ and may vary from line to line.

Proof

In case $q \ge 3$, J is of class $C^3$ in $H^1_0(\Omega )$ in the sense of Fréchet derivative (this fact may be standard, but it will be checked in Appendix A). Hence employing Taylor’s theorem (see Theorem A.2 in Appendix A ) and recalling that $J'(\phi ) = 0$, we can immediately verify (5.4) and (5.5) with $E(s) \in {\mathbb {R}}$ and $e(s) \in H^{-1}(\Omega )$ satisfying (5.6) with $\gamma = 1$. In case $2< q < 3$, $J''$ may fail to be Fréchet differentiable at $\phi$ in $H^1_0(\Omega )$; however, we can still prove the assertions for some $\gamma \in (0,1)$. A proof for this case will be detailed in Section 7. $\square$

Let $s > 0$ be fixed for a while. Since ${\mathcal {L}}_\phi$ is invertible, one can deduce from (5.4) and (5.5) along with (5.6) that

$$\begin{aligned} J(v(s)) - J(\phi ) = \frac{1}{2} \left\langle J'(v(s)), {\mathcal {L}}_\phi ^{-1}(J'(v(s))) \right\rangle _{H^1_0(\Omega )} + E(s). \end{aligned}$$

(5.7)

Since $J'(v(s))$ belongs to $H^{-1}(\Omega )$, there exists a sequence $\{\sigma _j(s)\}_{j=1}^\infty$ in $\ell ^2$ such that

$$\begin{aligned} J'(v(s)) = \sum _{j=1}^\infty \sigma _j(s) (-\Delta e_j) \ \text{ in } H^{-1}(\Omega ) \end{aligned}$$

(namely, we set $\sigma _j(s) = (J'(v(s)),-\Delta e_j)_{H^{-1}(\Omega )}$ for $j \in {\mathbb {N}}$). Hence, by virtue of (3.2),

$$\begin{aligned} {\mathcal {L}}_\phi ^{-1}(J'(v(s))) = \sum _{j=1}^\infty \sigma _j(s) \frac{\mu _j}{\nu _j} e_j \ \text{ in } H^1_0(\Omega ). \end{aligned}$$

Thus

$$\begin{aligned}&\frac{1}{2} \left\langle J'(v(s)), {\mathcal {L}}_\phi ^{-1}(J'(v(s))) \right\rangle _{H^1_0(\Omega )}\\&\quad = \frac{1}{2} \sum _{i=1}^\infty \sum _{j=1}^\infty \sigma _i(s) \sigma _j(s)\frac{\mu _j}{\nu _j} \langle -\Delta e_i , e_j \rangle _{H^1_0(\Omega )} = \frac{1}{2} \sum _{j=1}^\infty \sigma _j(s)^2 \frac{\mu _j}{\nu _j}. \end{aligned}$$

Consequently,

$$\begin{aligned} J(v(s)) - J(\phi ) - \frac{1}{2} \sum _{j=1}^{k-1} \sigma _j(s)^2 \frac{\mu _j}{\nu _j}&= \frac{1}{2} \sum _{j=k}^\infty \sigma _j(s)^2 \frac{\mu _j}{\nu _j} + E(s)\\&\le \frac{1}{2\nu _k} \sum _{j=k}^\infty \mu _j \sigma _j(s)^2 + E(s). \end{aligned}$$

On the other hand, we can check in a standard manner that $\{-\Delta e_j/\sqrt{\mu _j}\}_{j=1}^\infty$ forms a CONS in $L^2(\Omega ;\phi ^{2-q}\text {d}x)$ equipped with the inner product

$$\begin{aligned} (f,g)_{L^2(\Omega ;\phi ^{2-q}\text {d}x)} = \int _\Omega f g \phi ^{2-q} \, \text {d}x \quad \text{ for } \ f,g \in L^2(\Omega ;\phi ^{2-q}\text {d}x). \end{aligned}$$

Moreover, since $J'(v(s))$ belongs to $L^2(\Omega ;\phi ^{2-q}\text {d}x)$, recalling (1.14) and noting that $\langle f,u \rangle _{H^1_0(\Omega )} = (f, -\Delta u)_{H^{-1}(\Omega )}$ for $u \in H^1_0(\Omega )$ and $f \in H^{-1}(\Omega )$ (see (1.26)), we observe that

$$\begin{aligned}&\left( J'(v(s)), \frac{-\Delta e_j}{\sqrt{\mu _j}} \right) _{L^2(\Omega ; \phi ^{2-q} \text {d}x)}\\&\quad = \int _\Omega J'(v(s)) \frac{-\Delta e_j}{\sqrt{\mu _j}} \phi ^{2-q} \, \text {d}x \\&\quad {\mathop {=}\limits ^{(1.14)}} \sqrt{\mu _j} \int _\Omega J'(v(s)) e_j \, \text {d}x\\&\quad = \sqrt{\mu _j} \langle J'(v(s)), e_j \rangle _{H^1_0(\Omega )}\\&\quad {\mathop {=}\limits ^{(1.26)}} \sqrt{\mu _j} \left( J'(v(s)), -\Delta e_j \right) _{H^{-1}(\Omega )} = \sqrt{\mu _j} \sigma _j(s) \end{aligned}$$

for $j \in {\mathbb {N}}$. Therefore we have

$$\begin{aligned} J'(v(s)) = \sum _{j=1}^\infty \sqrt{\mu _j} \sigma _j(s) \frac{-\Delta e_j}{\sqrt{\mu _j}} \ \text{ in } L^2(\Omega ;\phi ^{2-q}\text {d}x), \end{aligned}$$

which implies

$$\begin{aligned} \sum _{j=k}^\infty \mu _j \sigma _j(s)^2 \le \sum _{j=1}^\infty \mu _j \sigma _j(s)^2 = \Vert J'(v(s))\Vert _{L^2(\Omega ;\phi ^{2-q}\text {d}x)}^2. \end{aligned}$$

Thus we obtain

$$\begin{aligned} J(v(s)) - J(\phi ) - \frac{1}{2} \sum _{j=1}^{k-1} \sigma _j(s)^2 \frac{\mu _j}{\nu _j} \le \frac{1}{2\nu _k} \Vert J'(v(s))\Vert _{L^2(\Omega ;\phi ^{2-q}\text {d}x)}^2 + E(s). \end{aligned}$$

Moreover, since $J'(v(s)) = {\mathcal {L}}_\phi (v(s)-\phi ) + e(s)$ and ${\mathcal {L}}_\phi$ is invertible, we observe that

$$\begin{aligned}&\Vert v(s)-\phi \Vert _{H^1_0(\Omega )}\\&\quad = \Vert {\mathcal {L}}_\phi ^{-1} (J'(v(s)) - e(s)) \Vert _{H^1_0(\Omega )} \\&\quad \le \Vert {\mathcal {L}}_\phi ^{-1}\Vert _{{\mathcal {L}}(H^{-1}(\Omega );H^1_0(\Omega ))} \left( \Vert J'(v(s))\Vert _{H^{-1}(\Omega )} + \Vert e(s)\Vert _{H^{-1}(\Omega )} \right) \\&\quad {\mathop {\le }\limits ^{(5.6)}}\Vert {\mathcal {L}}_\phi ^{-1}\Vert _{{\mathcal {L}}(H^{-1}(\Omega );H^1_0(\Omega ))} \Vert J'(v(s))\Vert _{H^{-1}(\Omega )} + \frac{1}{2} \Vert v(s)-\phi \Vert _{H^1_0(\Omega )} \end{aligned}$$

for s large enough (i.e., $\Vert v(s)-\phi \Vert _{H^1_0(\Omega )} \ll 1$ by (2.1)). Hence we find that

$$\begin{aligned} E(s)&\le C \Vert v(s)-\phi \Vert _{H^1_0(\Omega )}^{2+\gamma }\\&\le C \Vert v(s)-\phi \Vert _{H^1_0(\Omega )}^\gamma \Vert {\mathcal {L}}_\phi ^{-1}\Vert _{{\mathcal {L}}(H^{-1}(\Omega );H^1_0(\Omega ))}^2 \Vert J'(v(s))\Vert _{H^{-1}(\Omega )}^2\\&\le C \Vert v(s)-\phi \Vert _{H^1_0(\Omega )}^\gamma \Vert {\mathcal {L}}_\phi ^{-1}\Vert _{{\mathcal {L}}(H^{-1}(\Omega );H^1_0(\Omega ))}^2 \Vert J'(v(s))\Vert _{L^2(\Omega ;\phi ^{2-q}\text {d}x)}^2\\&=: \beta (s) \Vert J'(v(s))\Vert _{L^2(\Omega ;\phi ^{2-q}\text {d}x)}^2 \end{aligned}$$

for s large enough. Hence,

$$\begin{aligned}&J(v(s)) - J(\phi ) - \frac{1}{2} \sum _{j=1}^{k-1} \sigma _j(s)^2 \frac{\mu _j}{\nu _j}\nonumber \\&\quad \le \left( \frac{1}{2\nu _k} + \beta (s) \right) \Vert J'(v(s))\Vert _{L^2(\Omega ;\phi ^{2-q}\text {d}x)}^2. \end{aligned}$$

(5.8)

We also note that $\beta (s) \rightarrow 0$ as $s \rightarrow +\infty$, and in particular, we have $\beta (s) < \varepsilon$ for $s \ge s_\varepsilon$ large enough. Thus it follows from (5.3) that

$$\begin{aligned}&J(v(s)) - J(\phi ) - \frac{1}{2} \sum _{j=1}^{k-1} \sigma _j(s)^2 \frac{\mu _j}{\nu _j}\\&\quad \le - \left( \frac{1}{2\nu _k} + \varepsilon \right) (q-1)(1+\varepsilon )^{q-2} \frac{\text {d}}{\text {d}s} J(v(s)) \quad \text{ for } \ s \ge s_\varepsilon , \end{aligned}$$

whence it follows that, for any $0< \lambda < 2\nu _k/(q-1)$, one can take $s_1 > 0$ such that

$$\begin{aligned} J(v(s)) - J(\phi ) \le - \frac{1}{\lambda } \frac{\text {d}}{\text {d}s} \left[ J(v(s)) - J(\phi ) \right] \quad \text{ for } \ s \ge s_1. \end{aligned}$$

Eventually, we conclude that

$$\begin{aligned} 0 < J(v(s))-J(\phi )&\le \left[ J(v(s_1))-J(\phi )\right] \textrm{e}^{-\lambda (s-s_1)}\nonumber \\&\le \left[ J(v_0)-J(\phi )\right] \textrm{e}^{\lambda s_1} \textrm{e}^{-\lambda s} \end{aligned}$$

(5.9)

for all $s \ge s_1$. It is noteworthy that the exponent

$$\begin{aligned} \frac{2\nu _k}{q-1} = \frac{2}{q-1} \left[ \mu _k - \lambda _q(q-1)\right] = \lambda _0> 0 \end{aligned}$$

is the sharp rate of convergence for solutions to the linearized problem (see Section 1 and [14, Section 2] with Remark 1.8).

Remark 5.2

(Almost sharp rate) In order to verify (5.9), we do not need the differentiability of $J''$ at $\phi$ in $H^1_0(\Omega )$. Indeed, the argument so far runs as well even for $E(s) = o(\Vert v(s)-\phi \Vert _{H^1_0(\Omega )}^2)$ and $e(s) = o(\Vert v(s)-\phi \Vert _{H^1_0(\Omega )})$ as $s \rightarrow +\infty$. On the other hand, (5.6) will be needed for deriving the sharp rate of convergence (see next section).

6 Convergence with the Sharp Rate

Now, let us move on to a proof for the convergence with the sharp rate $\lambda _0$. We first recall that

$$\begin{aligned} 0&< J(v(s)) - J(\phi ) \\&\le - \left( \frac{q-1}{2\nu _k} + (q-1)\beta (s) \right) (1+\delta (s))^{q-2} \frac{\text {d}}{\text {d}s} J(v(s)) \end{aligned}$$

and $\beta (s) \le C \Vert v(s)-\phi \Vert _{H^1_0(\Omega )}^\gamma$ for some $\gamma \in (0,1]$ (see (5.3) with (5.1) for $\delta (s))$. Then we have

$$\begin{aligned}&\left[ \left( \frac{q-1}{2\nu _k} + (q-1)\beta (s) \right) (1+\delta (s))^{q-2}\right] ^{-1} \left[ J(v(s)) - J(\phi ) \right] \quad \\&\quad \le - \frac{\text {d}}{\text {d}s} \left[ J(v(s)) - J(\phi ) \right] . \end{aligned}$$

Furthermore, using Theorem 4.1 of [14] on a weighted smoothing effect that allows us to bound quantitatively the uniform relative error in terms of the weighted $L^2$ norm, we can derive an exponential convergence of the relative error from Theorem 1.1. More precisely, we have

Lemma 6.1

If $\Vert v(s)-\phi \Vert _{H^1_0(\Omega )}\lesssim \textrm{e}^{-\mu s}$ for some constant $\mu > 0$ and any $s > 0$ large enough, then there exist constants $C, b, s_* > 0$ such that

$$\begin{aligned} \delta (s) = \left\| \frac{v(s)}{\phi }- 1\right\| _{L^\infty (\Omega )} \le C \textrm{e}^{-bs} \end{aligned}$$

for all $s \ge s_*$.

Proof

Since $\Omega$ is a bounded $C^{1,1}$ domain, as in Theorem 4.1 of [14], we can verify that there exist positive constants $C, L, s_*$ such that

$$\begin{aligned}&\left\| \frac{v(s)}{\phi }-1\right\| _{L^\infty (\Omega )} \nonumber \\&\quad \le C \frac{\textrm{e}^{L (s-s_0)}}{s-s_0} (1 + s - s_0) \sup _{\sigma \in [s_0,+\infty )} \left( \int _\Omega |v(\sigma )^{q-1} - \phi ^{q-1}| \, \text {d}x \right) ^{\frac{1}{N}}\nonumber \\&\qquad + C (s-s_0) \textrm{e}^{L(s-s_0)} \end{aligned}$$

(6.1)

for any $s > s_0 \ge s_*$. Let $s > 0$ be large enough and set $s_0 = s - \textrm{e}^{-a s}$, where a is a positive number to be determined later. Then

$$\begin{aligned}&\left\| \frac{v(s)}{\phi }-1\right\| _{L^\infty (\Omega )} \nonumber \\&\quad \le C \frac{\textrm{e}^{L \textrm{e}^{-a s}}}{\textrm{e}^{-a s}} (1 + \textrm{e}^{-as}) \sup _{\sigma \in [s-\textrm{e}^{-a s},+\infty )} \left( \int _\Omega |v(\sigma )^{q-1}-\phi ^{q-1}| \, \text {d}x \right) ^{\frac{1}{N}}\nonumber \\&\qquad \ + C \textrm{e}^{-a s} \textrm{e}^{L \textrm{e}^{-a s}}. \end{aligned}$$

(6.2)

Moreover, we observe that

$$\begin{aligned} \int _\Omega |v(\sigma )^{q-1}-\phi ^{q-1}| \, \text {d}x&\le C \Vert v(\sigma )-\phi \Vert _{H^1_0(\Omega )}, \end{aligned}$$

(6.3)

where the constant C above depends on $\Vert \phi \Vert _{L^q(\Omega )}$ and $\sup _{\sigma \ge 0} \Vert v(\sigma )\Vert _{L^q(\Omega )}$.

Thus the assumption yields

$$\begin{aligned} \delta (s) = \left\| \frac{v(s)}{\phi }-1\right\| _{L^\infty (\Omega )} \le C \textrm{e}^L \textrm{e}^{as} (1 + \textrm{e}^{-as}) \textrm{e}^{-\frac{\mu }{N} (s-1)} + C \textrm{e}^{-a s} \textrm{e}^L. \end{aligned}$$

Hence it suffices to choose $0< a < \mu /N$. $\square$

Here we remark that the assumption of the lemma above can be verified with the aid of Lemma 4.1 along with (5.9) (or Theorem 1.1 directly). Hence it follows that

$$\begin{aligned} \beta (s) + \delta (s) \le C \textrm{e}^{-cs} \quad \text{ for } \text{ all } \ s \ge s_* \end{aligned}$$

for some $c,C,s_* > 0$. Therefore we observe that

$$\begin{aligned}&\left( \frac{q-1}{2\nu _k} + (q-1)\beta (s) \right) (1+\delta (s))^{q-2} \\&\quad \le \frac{q-1}{2\nu _k} \left( 1 + C \textrm{e}^{-ds}\right) \quad \text{ for } \text{ all } \ s \ge s_* \end{aligned}$$

for some $d,C > 0$. Hence

$$\begin{aligned}&\left[ \left( \frac{q-1}{2\nu _k} + (q-1)\beta (s) \right) (1+\delta (s))^{q-2} \right] ^{-1} \\&\quad \ge \frac{2\nu _k}{q-1} \left( 1 - \frac{C \textrm{e}^{-ds}}{1+C \textrm{e}^{-ds}} \right) \ge \frac{2\nu _k}{q-1} \left( 1 - C \textrm{e}^{-ds}\right) \end{aligned}$$

for $s \ge s_*$. Thus $H(s) := J(v(s))-J(\phi ) > 0$ satisfies

$$\begin{aligned} \frac{2\nu _k}{q-1} H(s)&\le - \frac{\text {d}}{\text {d}s} H(s) + C \textrm{e}^{-d s} H(s) \end{aligned}$$

for $s \ge s_*$. Solving the differential inequality above, one deduces that

$$\begin{aligned} H(s) \le H(s_*) \textrm{e}^{C/d} \exp \left( - \frac{2\nu _k}{q-1} (s-s_*)\right) \end{aligned}$$

for $s \ge s_*$. Thus we have proved the assertion of Theorem 1.4 for $q \ge 3$. It remains only to prove Lemma 5.1 for the case that $2< q < 3$, and it will be performed in the next section.

7 The Case Where $2< q < 3$

In this section, we shall prove Lemma 5.1 for $2< q < 3$ to complete the proof of Theorem 1.4. It is standard that J is of class $C^2$ in $H^1_0(\Omega )$ in the sense of Fréchet derivative and $J''(w) = - \Delta - \lambda _q (q-1)|w|^{q-2}$ for $w \in H^1_0(\Omega )$ (see, e.g., [48, Corollary 1.13]). On the other hand, $J'' :H^1_0(\Omega ) \rightarrow {\mathscr {L}}(H^1_0(\Omega ),H^{-1}(\Omega ))$ may not be even Gâteaux differentiable at $\phi$ anymore; however, it can be so in a stronger topology. We shall first claim that $J''$ is Gâteaux differentiable at $\phi _\theta := \phi + \theta (v(s)-\phi ) = (1-\theta ) \phi + \theta v(s) > 0$ a.e. in $\Omega$ for any $\theta \in [0,1]$ and $s > s_1$ (see (5.2)) in the strong topology of

$$\begin{aligned} X_1 := \left\{ w \in H^1_0(\Omega ) :w \phi ^{\frac{q-3}{2}} \in L^{2\cdot 2_*}(\Omega ) \right\} , \end{aligned}$$

where $2_* := (2^*)' = 2N/(N+2)$, equipped with the norm

$$\begin{aligned} \Vert w\Vert _{X_1}^2 := \Vert w\Vert _{H^1_0(\Omega )}^2 + \Vert w \phi ^{\frac{q-3}{2}}\Vert _{L^{2\cdot 2_*}(\Omega )}^2 \quad \text{ for } \ w \in X_1. \end{aligned}$$

Then $X_1 \hookrightarrow H^1_0(\Omega )$. Hence (the restriction) $J' : X_1 \rightarrow H^{-1}(\Omega )$ (onto $X_1$) turns out to be of class $C^1$ in $X_1$ in the sense of Fréchet derivative, and moreover, its derivative (still denoted by $J''$) can be regarded as a continuous map from $X_1$ into ${\mathscr {L}}(X_1,H^{-1}(\Omega ))$. Let $u,e \in X_1$ and $t \ne 0$. Since $\phi _\theta = (1-\theta )\phi + \theta v(s) > 0$ a.e. in $\Omega$ for $s > s_1$, it then follows that

$$\begin{aligned}&\left| \frac{ [J''(\phi _\theta +te)](u) - [J''(\phi _\theta )](u)}{t} + \lambda _q (q-1) (q-2) \phi _\theta ^{q-3} eu \right| \\&\quad = \lambda _q (q-1) \left| \frac{|\phi _\theta +te|^{q-2} - \phi _\theta ^{q-2}}{t} - (q-2) \phi _\theta ^{q-3}e \right| |u| \rightarrow 0 \end{aligned}$$

a.e. in $\Omega$ as $t \rightarrow 0$. Moreover,

$$\begin{aligned} \left| \frac{|\phi _\theta +te|^{q-2} - \phi _\theta ^{q-2}}{t} - (q-2) \phi _\theta ^{q-3}e \right| |u| \le (q-1)\phi _\theta ^{q-3} |e| |u|. \end{aligned}$$

(7.1)

Here we used the fact that $0< q-2 < 1$ and the inequality

$$\begin{aligned} |a^p - b^p| \le a^{p-1}|a-b| \quad \text{ for } \text{ any } \ a,b > 0 \ \text{ and } \ p \in (0,1). \end{aligned}$$

(7.2)

Then the right-hand side of (7.1) belongs to $L^{2_*}(\Omega ) \simeq (L^{2^*}(\Omega ))^* \hookrightarrow H^{-1}(\Omega )$ due to the following fact:

$$\begin{aligned} |\phi _\theta ^{\frac{q-3}{2}} u|&= \left| (1-\theta ) \phi + \theta v(s) \right| ^{\frac{q-3}{2}} |u|\\&= \left| 1 - \theta + \theta (v(s)/\phi ) \right| ^{\frac{q-3}{2}} \phi ^{\frac{q-3}{2}} |u| \le C \phi ^{\frac{q-3}{2}} |u| \in L^{2\cdot 2_*}(\Omega ). \end{aligned}$$

Indeed, $v(s)/\phi \ge 1/2$ a.e. in $\Omega$ for $s > s_1$ (see (5.2)). Using Lebesgue’s dominated convergence theorem, we can then deduce that $J'' : X_1 \rightarrow {\mathscr {L}}(X_1,H^{-1}(\Omega ))$ is Gâteaux differentiable at $\phi _\theta$. Moreover, we observe that the Gâteaux derivative $\text {D}_G J''(\phi _\theta ) = -\lambda _q (q-1) (q-2) \phi _\theta ^{q-3}$ of $J''$ at $\phi _\theta$ is bounded in ${\mathscr {L}}^{(2)}(X_1,H^{-1}(\Omega ))$ for $\theta \in [0,1]$. Hence employing Taylor’s theorem (see Theorem A.2 in Appendix) and recalling $J'(\phi )=0$ and $J''(\phi ) = {\mathcal {L}}_\phi$, we can still verify that

$$\begin{aligned} J'(v(s)) = {\mathcal {L}}_\phi (v(s)-\phi ) + \varepsilon _1(v(s)-\phi ), \end{aligned}$$

where $\varepsilon _1 : X_1 \rightarrow H^{-1}(\Omega )$ is a generic function fulfilling

$$\begin{aligned} \lim _{\Vert w\Vert _{X_1}\rightarrow 0} \frac{\Vert \varepsilon _1(w)\Vert _{H^{-1}(\Omega )}}{\Vert w\Vert _{X_1}^2} < +\infty . \end{aligned}$$

In particular, we put $w = v(s)-\phi$. Then noting that $\Vert w/\phi \Vert _{L^\infty (\Omega )} = \Vert (v(s)-\phi )/\phi \Vert _{L^\infty (\Omega )}$ is uniformly bounded for $s > s_1$ (see (5.2)), we infer that

$$\begin{aligned} \Vert w\phi ^{\frac{q-3}{2}}\Vert _{L^{2\cdot 2_*}(\Omega )}^2&= \Vert |w/\phi |^{3-q} |w|^{q-1}\Vert _{L^{2_*}(\Omega )}\\&\le \Vert w/\phi \Vert _{L^\infty (\Omega )}^{3-q} \Vert |w|^{q-1}\Vert _{L^{2_*}(\Omega )} \le C \Vert w/\phi \Vert _{L^\infty (\Omega )}^{3-q} \Vert w\Vert _{H^1_0(\Omega )}^{q-1}, \end{aligned}$$

and hence, we observe that

$$\begin{aligned} \Vert w\Vert _{X_1}^2&= \Vert w\Vert _{H^1_0(\Omega )}^2 + \Vert w\phi ^{\frac{q-3}{2}}\Vert _{L^{2\cdot 2_*}(\Omega )}^2\\&\le \Vert w\Vert _{H^1_0(\Omega )}^2 + C \Vert w/\phi \Vert _{L^\infty (\Omega )}^{3-q} \Vert w\Vert _{H^1_0(\Omega )}^{q-1}. \end{aligned}$$

Set

$$\begin{aligned} e(s) = \varepsilon _1(v(s)-\phi ), \end{aligned}$$

whence it follows that

$$\begin{aligned} \Vert e(s)\Vert _{H^{-1}(\Omega )} \le C\Vert v(s)-\phi \Vert _{H^1_0(\Omega )}^{1+ (q-2)} \quad \text{ for } \ s \gg 1. \end{aligned}$$

Similarly, setting

$$\begin{aligned} X_2 := \left\{ w \in H^1_0(\Omega ) :w \phi ^{\frac{q-3}{3}} \in L^3(\Omega ) \right\} \end{aligned}$$

equipped with

$$\begin{aligned} \Vert w\Vert _{X_2}^3 := \Vert w\Vert _{H^1_0(\Omega )}^3 + \Vert w \phi ^{\frac{q-3}{3}} \Vert _{L^3(\Omega )}^3 \quad \text{ for } \ w \in X_2, \end{aligned}$$

(then $X_2 \hookrightarrow H^1_0(\Omega )$) and repeating the same argument as above again, we can prove that (the restriction) $J'': X_2 \rightarrow {\mathscr {L}}^{(2)}(X_2,{\mathbb {R}})$ is Gâteaux differentiable at $\phi _\theta$ in $X_2$ for any $\theta \in [0,1]$, and moreover, the Gâteaux derivative $\text {D}_G J''(\phi _\theta )$ is bounded in ${\mathscr {L}}^{(3)}(X_2,{\mathbb {R}})$ for $\theta \in [0,1]$. Hence it follows that

$$\begin{aligned} J(v(s)) = J(\phi ) + \frac{1}{2} \left\langle {\mathcal {L}}_\phi (v(s)-\phi ), v(s)-\phi \right\rangle _{H^1_0(\Omega )} + \varepsilon _2(v(s)-\phi ), \end{aligned}$$

where $\varepsilon _2 : X_2 \rightarrow {\mathbb {R}}$ is a generic function satisfying

$$\begin{aligned} \lim _{\Vert w\Vert _{X_2} \rightarrow 0} \dfrac{|\varepsilon _2(w)|}{\Vert w\Vert _{X_2}^3} < +\infty \end{aligned}$$

(see Theorem A.2 in Appendix). Put $w = v(s)-\phi$ again. Then we find that

$$\begin{aligned} \Vert w \phi ^\frac{q-3}{3}\Vert _{L^3(\Omega )}^{3}&\le \Vert w/\phi \Vert _{L^\infty (\Omega )}^{3-q} \Vert w\Vert _{L^q(\Omega )}^q\\&\le C \Vert w/\phi \Vert _{L^\infty (\Omega )}^{3-q} \Vert w\Vert _{H^1_0(\Omega )}^q \end{aligned}$$

and that

$$\begin{aligned} \Vert w\Vert _{X_2}^3 \le \Vert w\Vert _{H^1_0(\Omega )}^3 + C \Vert w/\phi \Vert _{L^\infty (\Omega )}^{3-q} \Vert w\Vert _{H^1_0(\Omega )}^q. \end{aligned}$$

Set $E(s) = \varepsilon _2(v(s)-\phi )$. Then we obtain

$$\begin{aligned} |E(s)| \le C\Vert v(s)-\phi \Vert _{H^1_0(\Omega )}^{2+(q-2)} \quad \text{ for } \ s \gg 1. \end{aligned}$$

Thus we have checked (5.4) and (5.5) with $E(\cdot )$ and $e(\cdot )$ satisfying (5.6) with $\gamma = q-2 \in (0,1)$, and hence, we have completed the proof of Lemma 5.1 for $2< q < 3$ as well. $\square$

Thus the proof of Theorem 1.4 has been completed. We close this section with the following remark on assumptions for domains based on the arguments so far.

Remark 7.1

(Assumption for domains) All the results in § 1 can be proved for arbitrary bounded $C^{1,1}$ domains. The $C^{1,1}$ condition for domains is needed for: (i) the $C^2(\Omega ) \cap C^1(\overline{\Omega })$ regularity of solutions $\phi$ to (1.11), (1.12) (see, e.g., [33, Theorems 9.15 and 9.19]), (ii) Hopf’s lemma (see, e.g., [28, Section 6.4.2]; indeed, the interior sphere condition follows from the $C^{1,1}$ condition) and (iii) the proof for Lemma 6.1 in § 6. To be more precise for (iii), in the proof of Lemma 6.1, a quantitative estimate (see (6.1)) established as in Theorem 4.1 of [14] is employed and the estimate is proved with the use of Green function estimates under the $C^{1,1}$ condition (see [24, 34]) as well as (ii).

8 Proofs of Corollaries

This section is devoted to proving corollaries exhibited in §1. We first give a proof of Corollary 1.3.

Proof of Corollary 1.3

It is well known that every non-degenerate nontrivial solution to (1.11), (1.12) is isolated in $H^1_0(\Omega )$ from all the other solutions (see, e.g., [5, Section 5.3]). Moreover, we recall Theorem 2 of [5]: Let $\varphi$ be a least-energy solution of (1.11), (1.12). If $\varphi$ is isolated in $H^1_0(\Omega )$ from all the other (sign-definite) solutions of (1.11), (1.12), then $\varphi$ is an asymptotically stable profile in the sense of Definition 1.2. Therefore since $\phi$ is isolated from all the other solutions to (1.11), (1.12) and takes the least energy among all the nontrivial solutions of (1.11), (1.12), it turns out to be an asymptotically stable asymptotic profile in the sense of Definition 1.2. Hence, any (possibly sign-changing) weak solution $v = v(x,s)$ of (1.6)–(1.8) emanating from some small (in $H^1_0(\Omega )$) neighbourhood $B_{H^1_0(\Omega )}(\phi ;\delta )$ of $\phi$ on the phase set ${\mathcal {X}}$ (see (1.10)) converges to $\phi$ strongly in $H^1_0(\Omega )$ as $s \rightarrow +\infty$. Therefore Theorem 1.1 can guarantee the exponential convergence. Here we note that the constant $M_\mu$ in Theorem 1.1 can be chosen so as to be independent of $v_0$, whenever $\Vert v_0 - \phi \Vert _{H^1_0(\Omega )} < \delta$. Thus the exponential stability of $\phi$ has been proved. $\square$

We next prove Corollary 1.5.

Proof of Corollary 1.5

We first note from (5.2) that

$$\begin{aligned} \left\| v^{q-1}(s)\right\| _{L^2(\Omega ;\phi ^{2-q}\text {d}x)}^2 \le \Vert v(s)\Vert _{L^q(\Omega )}^q \left\| \frac{v(s)}{\phi }\right\| _{L^\infty (\Omega )}^{q-2} \le C \end{aligned}$$

for $s \ge s_1$. Hence it follows from (2.1) that

$$\begin{aligned} v^{q-1}(s) \rightarrow \phi ^{q-1} \quad \text{ weakly } \text{ in } L^2(\Omega ; \phi ^{2-q} \text {d}x) \ \text{ as } \ s \rightarrow +\infty . \end{aligned}$$

Recalling (5.3) and (5.8), we see that

$$\begin{aligned} \Vert J'(v(s))\Vert _{L^2(\Omega ;\phi ^{2-q}\text {d}x)} \le - C \dfrac{\text {d}}{\text {d}s} \left[ J(v(s))-J(\phi ) \right] ^{1/2}, \end{aligned}$$

whence it follows from Theorem 1.4 that

$$\begin{aligned}&\left\| \phi ^{q-1} - v^{q-1}(s) \right\| _{L^2(\Omega ;\phi ^{2-q}\text {d}x)} \\&\quad \le \liminf _{\sigma \rightarrow +\infty } \left\| v^{q-1}(\sigma ) - v^{q-1}(s) \right\| _{L^2(\Omega ;\phi ^{2-q}\text {d}x)}\\&\quad \le \int ^\infty _s \left\| \partial _s \left( v^{q-1} \right) (\sigma ) \right\| _{L^2(\Omega ;\phi ^{2-q}\text {d}x)} \, \text {d}\sigma \\&\quad \le C \left[ J(v(s)) - J(\phi ) \right] ^{1/2} \le C \textrm{e}^{- \frac{\lambda _0}{2} s}. \end{aligned}$$

On the other hand, we observe that

$$\begin{aligned}&\int _\Omega |v(x,s) - \phi (x)|^2 \phi (x)^{q-2} \, \text {d}x \\&\quad \le \int _\Omega \left| v(x,s)^{q-1} - \phi (x)^{q-1}\right| ^2 \phi (x)^{2-q} \, \text {d}x. \end{aligned}$$

Here we used (7.2). Thus (1.21) follows immediately. $\square$

Let us give a proof for Corollary 1.6.

Proof of Corollary 1.6

As in (4.3) and § 5 (see also Lemma 5.1), we observe that

$$\begin{aligned}&J(v(s))- J(\phi ) \\&\quad = \frac{1}{2} \Vert \nabla (v(s)-\phi )\Vert _{L^2(\Omega )}^2 - \dfrac{\lambda _q}{2} (q-1) \int _\Omega |v(s)-\phi |^2 \phi ^{q-2} \, \text {d}x\\&\qquad + O \left( \Vert v(s)-\phi \Vert _{H^1_0(\Omega )}^{2+\gamma }\right) \end{aligned}$$

for some $\gamma \in (0,1]$. Consequently, Theorem 1.4 and Corollary 1.5 yield

$$\begin{aligned} \Vert v(s)-\phi \Vert _{H^1_0(\Omega )}^2 \le C \textrm{e}^{-\lambda _0s} \quad \text{ for } \ s \ge 0. \end{aligned}$$

Finally, (1.23) follows immediately from (5.5). This completes the proof. $\square$

From the argument above, we can also observe the following:

Corollary 8.1

Under the same assumption as in Theorem 1.4, if (1.20) holds for some $\lambda > 0$, then (1.21) and (1.22) hold for the same $\lambda$.

With the aid of the regularity results [37, 38], one can also improve the topology of the relative error convergence (respectively, convergence of the difference) up to $C^q(\overline{\Omega })$ (respectively, $C^{q+1}(\overline{\Omega })$) for smooth domains (see [37, Corollary 1.4]).

9 Fast Diffusion Flows with Changing Signs

Although asymptotic behaviors of sign-definite solutions to the fast diffusion equation have been well studied, the dynamics of sign-changing ones has not yet been fully pursued. In particular, since sign-changing asymptotic profiles are often unstable (see [5]), existence of (non-stationary) weak solutions of (1.6)–(1.8) converging to sign-changing solutions of (1.11), (1.12) may still be rather nontrivial. In this section, we shall discuss such dynamics of fast diffusion flows with changing signs.

9.1 One-Dimensional Case

We first restrict ourselves to the one-dimensional case $\Omega = (0,1)$, where the set $\{\pm \phi _k :k \in {\mathbb {N}}\}$ of all non-trivial solutions to (1.11), (1.12) consists of the unique positive solution $\phi _1 > 0$ and sign-changing ones $\phi _k$ given by

$$\begin{aligned} \phi _k(x) = (-1)^j k^{2/(q-2)}\phi _1(kx - j), \quad x \in (j/k,(j+1)/k) \end{aligned}$$

for $j = 0,1,\ldots , k-1$. Hence $\pm \phi _k$ have $k-1$ zeros arranged at equal intervals in (0, 1) and $J(\pm \phi _1)< J(\pm \phi _2)< \cdots < J(\pm \phi _k) \rightarrow +\infty$ as $k \rightarrow +\infty$ (see [5, Section 5.4] for more details). Moreover, one can verify that $\phi _k$ is non-degenerate in a standard way. Note that, for any non-negative data $u_0 \in H^1_0(0,1) \setminus \{0\}$, the solution to (1.1)–(1.3) with $\Omega = (0,1)$ has the positive asymptotic profile $\phi _1$ in the sense of (1.4). Furthermore, for each $k \in {\mathbb {N}}$, we can construct a solution $u = u(x,t)$ (of (1.1)–(1.3)) whose asymptotic profile coincides with $\phi _k$. Indeed, for instance, set $u_0(x) = \sin (k\pi x)$ for $x \in (0,1)$. Then all the zeros of $u(\cdot ,t)$ do not move for $t \ge 0$. Hence the dynamics of $u(\cdot ,t)$ restricted on each subinterval $(j/k,(j+1)/k)$ is reduced to those of sign-definite solutions.

We can also construct sign-changing initial data $u_0 \in H^1_0(0,1) \setminus \{0\}$ such that the corresponding solutions of (1.1)–(1.3) have sign-definite asymptotic profiles and sign-changing ones having fewer zeros; hence, some zeros of such solutions move and eventually vanish. Let $u = u(x,t)$ be the solution for (1.1)–(1.3) in $\Omega = (0,1)$ with a smooth initial datum $u_0$ which is even with respect to $x = 1/2$, negative in $(0,a) \cup (1-a,1)$ and positive in $(a,1-a)$ for some $a \in (0,1/2)$ such that

$$\begin{aligned} \int ^1_0 (|u_0|^{q-2}u_0)(x) \, \text {d}x > 0 \end{aligned}$$

(hence $u_0$ has exactly two zeros in (0, 1)). Then $u(\cdot ,t)$ is also even with respect to $x=1/2$ for $t > 0$. Integrating both sides of (1.1) over $\Omega = (0,1)$ and utilizing the evenness of $u(\cdot ,t)$ with respect to $x = 1/2$, we observe that

$$\begin{aligned} \frac{\text {d}}{\text {d}t} \int ^1_0 (|u|^{q-2}u)(x,t) \, \text {d}x - 2 \partial _x u(1,t) = 0. \end{aligned}$$

Now, suppose to the contrary that $\partial _x u(1,t) \ge 0$ for all $t \ge 0$. Then one gets

$$\begin{aligned} \int ^1_0 (|u|^{q-2}u)(x,t) \, \text {d}x \ge \int ^1_0 (|u_0|^{q-2}u_0)(x) \, \text {d}x > 0 \quad \text{ for } \text{ all } \ t \ge 0, \end{aligned}$$

which is a contradiction to the finite-time extinction of $u = u(x,t)$. Hence $\partial _x u(1,t_0) < 0$ at some $t_0 \in (0,t_*)$. Since the number of zeros of $u(\cdot ,t)$ is non-increasing in t, $u(\cdot ,t_0)$ must be non-negative in $\Omega = (0,1)$ (see, e.g., [30]). Therefore the solution $u = u(x,t)$ has the positive asymptotic profile $\phi _1$. Furthermore, for each $k \in {\mathbb {N}}$, extending the function $u_0$ considered above to be an anti-periodic function in (0, k), i.e., $u_0(x+1) = - u_0(x)$ for $x \in (0,k-1)$, one can construct a sign-changing solution (for (1.1)–(1.3) with $\Omega = (0,k)$) which has a sign-changing asymptotic profile with fewer zeros (than its initial datum).

9.2 Multi-Dimensional Case

The multi-dimensional case is more complicated; indeed, the structure of nontrivial solutions to (1.11), (1.12) is not so simple as in the one-dimensional case. It is already difficult to check the non-degeneracy of sign-changing solutions (indeed, even in balls, although the positive solution is unique and non-degenerate, there exist non-radial sign-changing solutions, which are degenerate; see [1, Theorem 1.3]).

We shall consider dumbbell-shaped domains in ${\mathbb {R}}^N$. Set

$$\begin{aligned} B = B_+ \cup B_- \subset {\mathbb {R}}^N, \end{aligned}$$

where $B_\pm$ denotes the open unit ball in ${\mathbb {R}}^N$ centered at $x = \pm 2 e_1$, respectively, with a unit vector $e_1 \in {\mathbb {R}}^N$ (hence $\overline{B_+} \cap \overline{B_-} = \emptyset$) and let $C = \{t e_1 :t \in [-1,1]\}$. Moreover, let $(\Omega _n)$ be a sequence of smooth bounded domains of ${\mathbb {R}}^N$ involving ${\overline{B}} \cup C$ and symmetric with respect to the hyperplane

$$\begin{aligned} H := \{x \in {\mathbb {R}}^N :x \cdot e_1 = 0\} \end{aligned}$$

through the origin such that $\Omega _n \rightarrow B$ in a proper sense as $n \rightarrow +\infty$ (see [27, p.122] for more details). Furthermore, let ${\tilde{B}} \subset {\mathbb {R}}^N$ be a ball including $\Omega _n$ for n large enough.

In what follows, we let $\phi _{+-} \in H^1_0(B)$ coincide with the positive and negative radial solutions to (1.11), (1.12) in $B_+$ and $B_-$, respectively (thanks to [32], positive solutions in balls are radial and unique). Then $\phi _{+-}$ turns out to be a non-degenerate solution to (1.11), (1.12) with $\Omega = B$ (indeed, the restriction of $\phi _{+-}$ onto each of the disjoint balls is non-degenerate due to [41]). Thanks to [27, (i) of Theorem 1], for each $n \in {\mathbb {N}}$ large enough, there exists a non-degenerate solution $\phi _n \in H^1_0(\Omega _n)$ of (1.11), (1.12) with $\Omega = \Omega _n$ uniquely corresponding to $\phi _{+-}$ in the sense that $\phi _n \rightarrow \phi _{+-}$ strongly in $L^q({\tilde{B}})$ as $n \rightarrow +\infty$ and $\phi _n$ is the only solution in $H^1_0(\Omega _n)$ close to $\phi _{+-}$ in $L^q({\tilde{B}})$. Here and henceforth, we use the same notation for functions of class $H^1_0(B)$ (or $H^1_0(\Omega _n)$) and their zero extensions onto ${\tilde{B}}$, when no confusion can arise. Hence $\phi _n$ is sign-changing for $n \in {\mathbb {N}}$ large enough, since so is $\phi _{+-}$. Then $(\Omega _n,\phi _n)$ will turn out to be our desired domain and asymptotic profile for fast diffusion for $n \in {\mathbb {N}}$ large enough. This fact will be precisely stated in Theorem 9.2 below.

To this end, let us first recall several materials developed in [5]. The set of initial data for (1.6)–(1.8) via the scaling (1.5) is defined as

$$\begin{aligned} {\mathcal {X}}(\Omega ) :=&\left\{ t_*(u_0)^{-1/(q-2)} u_0 :u_0 \in H^1_0(\Omega ) \setminus \{0\} \right\} \\ =&\left\{ v_0 \in H^1_0(\Omega ) :t_*(v_0) = 1 \right\} \end{aligned}$$

(see [5, Proposition 6] for the equality). It is noteworthy that ${\mathcal {X}}(\Omega )$ is homeomorphic to the unit sphere in $H^1_0(\Omega )$ (see [5, Proposition 10]). We denote by ${\mathcal {S}}(\Omega )$ the set of all nontrivial solutions to (1.11), (1.12). We may simply write ${\mathcal {X}}$ and ${\mathcal {S}}$ instead of ${\mathcal {X}}(\Omega )$ and ${\mathcal {S}}(\Omega )$, respectively, when no confusion can arise. Then the following proposition holds true:

Proposition 9.1

(Properties of the set of initial data [5]) It holds that:

• (i) The set ${\mathcal {S}}$ is included in ${\mathcal {X}}$ (see [5, Proposition 10]).

• (ii) Moreover, the weak solution $v = v(x,s)$ emanating from $v_0 \in {\mathcal {X}}$ quasi-converges to a nontrivial solution for (1.11), (1.12) (see [5, Theorem 1] and § 1).

• (iii) Furthermore, ${\mathcal {X}}$ is an invariant set of the dynamical system generated by (1.6)–(1.8) (see [5, Proposition 5]).

• (iv) The set ${\mathcal {X}}$ is sequentially closed in the weak topology of $H^1_0(\Omega )$ (see [5, Proposition 7]).

Moreover, let ${\mathcal {S}}(B)$ be defined as above and let ${\mathcal {S}}_{H}(B)$ be its subset whose elements are odd with respect to the hyperplane H, that is, $\phi \in {\mathcal {S}}_H(B)$ means $\phi \in {\mathcal {S}}(B)$ and $\phi (x) = - \phi (\textrm{Ref}_H(x))$ for $x \in B$, where $\textrm{Ref}_H(x) := x - 2 (x \cdot e_1) e_1$ stands for the reflection of x with respect to the hyperplane H. In particular, $\phi _{+-} \in {\mathcal {S}}_H(B)$. Moreover, set

$$\begin{aligned} J_B(w) := \frac{1}{2} \int _B |\nabla w(x)|^2 \, \text {d}x - \frac{\lambda _q}{q} \int _B |w(x)|^q \, \text {d}x \quad \text{ for } \ w \in H^1_0(B). \end{aligned}$$

We define ${\mathcal {S}}(\Omega _n)$, ${\mathcal {S}}_H(\Omega _n)$ and $J_{\Omega _n}$ in an analogous way. Then we claim that

$$\begin{aligned} \phi _n \in {\mathcal {S}}_H(\Omega _n) \end{aligned}$$

for $n \in {\mathbb {N}}$ large enough. Indeed, since $\phi _n \rightarrow \phi _{+-}$ strongly in $L^q({\tilde{B}})$ as $n \rightarrow +\infty$, we find from the symmetry of $\Omega _n$ that $-\phi _n(\textrm{Ref}_H(\cdot )) \rightarrow -\phi _{+-}(\textrm{Ref}_H(\cdot )) = \phi _{+-}$ strongly in $L^q({\tilde{B}})$ as $n \rightarrow +\infty$. From the uniqueness of $(\phi _n)$ (see [27, (i) of Theorem 1]), we find that $\phi _n$ coincides with $-\phi _n(\textrm{Ref}_H(\cdot ))$, i.e., $\phi _n \in {\mathcal {S}}_H(\Omega _n)$, for $n \in {\mathbb {N}}$ large enough. Furthermore, we set

$$\begin{aligned} {\mathcal {X}}_H(\Omega _n) = \left\{ w \in {\mathcal {X}}(\Omega _n) :w \text{ is } \text{ odd } \text{ with } \text{ respect } \text{ to } \text{ the } \text{ hyperplane } H \right\} . \end{aligned}$$

Then all the assertions of Proposition 9.1 with ${\mathcal {X}}$ and ${\mathcal {S}}$ replaced by ${\mathcal {X}}_H$ and ${\mathcal {S}}_H$, respectively, hold true, since the oddness of initial data is inherited by the solutions to (1.6)–(1.8) (see [3, Theorem 2.5]). Moreover, we stress that for any $w \in H^1_0(\Omega _n) \setminus \{0\}$ which is odd with respect to the hyperplane H one can take a constant $x(w) > 0$ such that x(w)w lies on the set ${\mathcal {X}}_H(\Omega _n)$ (more precisely, we have $x(w) = t_*(w)^{-1/(q-2)}$).

The following theorem ensures exponential stability of the asymptotic profile $\phi _n$, which is sign-changing and non-degenerate, in ${\mathcal {X}}_H(\Omega _n)$:

Theorem 9.2

(Exponential stability of $\phi _n$ in ${\mathcal {X}}_H(\Omega _n)$) Let $(\Omega _n)$ and $(\phi _n)$ be defined as above. Then, for any $n \in {\mathbb {N}}$ large enough, $\phi _n$ is exponentially stable under the dynamical system generated by (1.6)–(1.8) in ${\mathcal {X}}_H(\Omega _n)$, that is, for any $\varepsilon > 0$ there exists $\delta _{n,\varepsilon } > 0$ such that any weak solution $v = v(x,s)$ to (1.6)–(1.8) with $\Omega = \Omega _n$ satisfies

$$\begin{aligned} \sup _{s \ge 0} \Vert v(s)-\phi _n\Vert _{H^1_0(\Omega _n)} < \varepsilon , \end{aligned}$$

provided that $v(0) \in {\mathcal {X}}_H(\Omega _n)$ and $\Vert v(0)-\phi _n\Vert _{H^1_0(\Omega _n)} < \delta _{n,\varepsilon }$; moreover, there exist constants $C_n, \lambda _n,\delta _{n,0} > 0$ such that any weak solution $v = v(x,s)$ to (1.6)–(1.8) with $\Omega = \Omega _n$ fulfills

$$\begin{aligned} \Vert v(s)-\phi _n\Vert _{H^1_0(\Omega )} \le C_n \textrm{e}^{-\lambda _n s/2} \quad \text{ for } \text{ all } \ s \ge 0, \end{aligned}$$

provided that $v(0) \in {\mathcal {X}}_H(\Omega _n)$ and $\Vert v(0)-\phi _n\Vert _{H^1_0(\Omega _n)} < \delta _{n,0}$. Here $\lambda _n$ can be chosen as in (1.17) for $\phi = \phi _n$ and $\Omega = \Omega _n$.

Before proving this theorem, we recall Theorem 3 of [5]: Let $\psi$ be a sign-changing profile of a solution of (1.1)–(1.3). If $\psi$ is isolated in $H^1_0(\Omega )$ from all the other solutions, then $\psi$ is unstable in the sense of Definition of 1.2. Therefore $\phi _n$ turns out to be unstable in ${\mathcal {X}}(\Omega _n)$, whose elements are not always odd, since $\phi _n$ is sign-changing and non-degenerate (hence isolated in $H^1_0(\Omega _n)$).

To prove Theorem 9.2, we need the following:

Lemma 9.3

There exists a constant $r_0 > 0$ such that

$$\begin{aligned} \left\{ \varphi \in {\mathcal {S}}_H(B) :J_B(\varphi ) \le J_B(\phi _{+-}) + r_0 \right\} = \{\pm \phi _{+-}\}. \end{aligned}$$

(9.1)

Proof

We first note that $\phi _{+-}$ attains the infimum of the energy $J_B$ over ${\mathcal {S}}_H(B)$, since the positive solution on each ball takes the least energy among all nontrivial solutions on the ball. We next let $\phi _{\pm \mp } \in {\mathcal {S}}_H(B)$ coincide with a least-energy nodal solution $\psi \in {\mathcal {S}}(B_+)$ in $B_+$, that is, $\psi \in {\mathcal {S}}(B_+)$ is sign-changing and attains the minimum value of $J_{B_+}$ among all sign-changing solutions in $B_+$ (see [1, 8]). Here we note that $\psi$ takes the second minimum value of $J_{B_+}$ among ${\mathcal {S}}(B_+)$, since the positive solution is unique in the ball $B_+$. Then from the oddness of $\phi _{\pm \mp }$ it follows that

$$\begin{aligned} \phi _{\pm \mp }(x) = - \phi _{\pm \mp }(\textrm{Ref}_H(x)) \quad \text{ for } x \in B_-. \end{aligned}$$

(9.2)

Hence $\phi _{+-}$ and $\phi _{\pm \mp }$ take the first and second minimum values of the energy $J_{B}$ among ${\mathcal {S}}_H(B)$, respectively. We take $0< r_0 < J_B(\phi _{\pm \mp }) - J_B(\phi _{+-})$. Then (9.1) follows immediately. $\square$

We further need the following:

Lemma 9.4

Let $n \in {\mathbb {N}}$ be large enough. The functions $\phi _n$ and $- \phi _n$ are minimizers of the functional $J_{\Omega _n}$ over the set ${\mathcal {X}}_H(\Omega _n)$. Moreover, it holds that $J_{\Omega _n}(w) > J_{\Omega _n}(\pm \phi _n)$ for any $w \in {\mathcal {X}}_H(\Omega _n) \setminus \{ \pm \phi _n\}$.

Proof

We first claim that

$$\begin{aligned} \left\{ \varphi \in {\mathcal {S}}_H(\Omega _n) :J_{\Omega _n}(\varphi ) \le J_B(\phi _{+-}) + r_0 \right\} = \{\pm \phi _n\} \end{aligned}$$

(9.3)

for any $n \in {\mathbb {N}}$ large enough. Here $r_0$ is given as in (9.1). Indeed, recalling that $\phi _n \in {\mathcal {S}}(\Omega _n)$, $\phi _n \rightarrow \phi _{+-}$ strongly in $L^q({\tilde{B}})$ as $n \rightarrow +\infty$ and $\phi _{+-} \in {\mathcal {S}}(B)$, we deduce that

$$\begin{aligned} J_{\Omega _n}(\phi _n)&= \frac{q-2}{2q} \lambda _q \Vert \phi _n\Vert _{L^q(\Omega _n)}^q = \frac{q-2}{2q} \lambda _q \Vert \phi _n\Vert _{L^q({\tilde{B}})}^q\\&\rightarrow \frac{q-2}{2q} \lambda _q \Vert \phi _{+-}\Vert _{L^q({\tilde{B}})}^q = J_B(\phi _{+-}) \end{aligned}$$

as $n \rightarrow +\infty$. Hence we find that the set given by the left-hand side of (9.3) includes $\pm \phi _n$ for $n \in N$ large enough. Therefore it suffices to prove the inverse inclusion. Suppose to the contrary that, up to a (not relabeled) subsequence, there exists a sequence $(\varphi _n)$ in ${\mathcal {S}}_H(\Omega _n) \setminus \{\pm \phi _n\}$ such that

$$\begin{aligned} J_{\Omega _n}(\varphi _n) \le J_B(\phi _{+-}) + r_0. \end{aligned}$$

Then by [27, (ii) of Theorem 1] we can take a (not relabeled) subsequence of (n) and $\varphi \in {\mathcal {S}}_H(B) \cup \{0\}$ such that, for any $\varepsilon > 0$, there exists $n_\varepsilon \in {\mathbb {N}}$ satisfying

$$\begin{aligned} J_B(\varphi ) \le J_B(\phi _{+-}) + r_0, \quad \Vert \varphi - \varphi _n\Vert _{H^1_0({\tilde{B}})} < \varepsilon \end{aligned}$$

for $n \in {\mathbb {N}}$ greater than $n_\varepsilon$. One may rule out $\varphi = 0$. Indeed, if $\varphi = 0$, then $\varphi _n \rightarrow 0$ strongly in $H^1_0({\tilde{B}})$ as $n \rightarrow +\infty$. On the other hand, we observe that

$$\begin{aligned} J_{\Omega _n}(\varphi _n) \ge \inf _{w \in {\mathcal {S}}(\Omega _n)} J_{\Omega _n}(w)&= \frac{q-2}{2q} \Big [ \lambda _q C_q(\Omega _n)^q \Big ]^{-2/(q-2)}\\&\ge \frac{q-2}{2q} \left[ \lambda _q C_q({\tilde{B}})^q \right] ^{-2/(q-2)} > 0, \end{aligned}$$

where $C_q(\Omega _n)$ denotes the best constant of the Sobolev-Poincaré inequality (1.18) with $\Omega = \Omega _n$ (see, e.g., [42] and also [5, p.571]). Here we also used the relation $C_q(\Omega _n) \le C_q({\tilde{B}})$. Hence it contradicts the fact that $J_{\Omega _n}(\varphi _n) = \frac{q-2}{2q} \Vert \nabla \varphi _n\Vert _{L^2(\Omega _n)}^2 \rightarrow 0$ as $n \rightarrow +\infty$. Thus we obtain $\varphi \ne 0$. Using (9.1), we can obtain either $\varphi = \phi _{+-}$ or $\varphi = - \phi _{+-}$. Hence $\varphi _n$ converges to either $\phi _{+-}$ or $-\phi _{+-}$ strongly in $H^1_0({\tilde{B}})$ as $n \rightarrow +\infty$. However, due to [27, (ii) of Theorem 1], we infer that $\varphi _n$ coincides with either $\phi _n$ or $-\phi _n$, and this fact yields a contradiction to the assumption $\varphi _n \ne \pm \phi _n$. Thus (9.3) follows. Moreover, we can deduce that $J_{\Omega _n}$ is minimized over ${\mathcal {S}}_H(\Omega _n)$ by $\phi _n$ and $-\phi _n$ only.

Finally, we shall prove that $\pm \phi _n$ also minimize $J_{\Omega _n}$ over ${\mathcal {X}}_H(\Omega _n)$. Let $v_{0,n} \in {\mathcal {X}}_H(\Omega _n)$ be such that $J_{\Omega _n}(v_{0,n}) \le J_B(\phi _{+-}) + r_0$. Then the solution $v_n = v_n(x,s)$ to (1.6)–(1.8) with $\Omega = \Omega _n$ and $v_0 = v_{0,n}$ quasi-converges to a limit $\psi _n \in {\mathcal {S}}_H(\Omega _n)$ strongly in $H^1_0(\Omega _n)$ as $s \rightarrow +\infty$. Since the energy $s \mapsto J_{\Omega _n}(v_n(s))$ is non-increasing, it follows that

$$\begin{aligned} J_{\Omega _n}(\psi _n) \le J_{\Omega _n}(v_{0,n}) \le J_B(\phi _{+-}) + r_0. \end{aligned}$$

By (9.3), we obtain either $\psi _n = \phi _n$ or $\psi _n = -\phi _n$. Combining these facts, we deduce that $J_{\Omega _n}(\phi _n) \le J_{\Omega _n}(v_{0,n})$. Hence $\pm \phi _n$ are minimizers of $J_{\Omega _n}$ over ${\mathcal {X}}_H(\Omega _n)$. Furthermore, if $v_{0,n} \in {\mathcal {X}}_H(\Omega _n)$ minimizes $J_{\Omega _n}$ over ${\mathcal {X}}_H(\Omega _n)$, that is, $J_{\Omega _n}(v_{0,n}) = J_{\Omega _n}(\phi _n)$, we obtain $v_{0,n} \in {\mathcal {S}}_H(\Omega _n)$. Indeed, we derive from (2.2) that

$$\begin{aligned} \dfrac{4}{qq'} \int ^s_0 \left\| \partial _s (|v_n|^{(q-2)/2}v_n)(s)\right\| _{L^2(\Omega _n)}^2 \, \text {d}s + J_{\Omega _n}(v_n(s)) \le J_{\Omega _n}(v_{0,n}), \end{aligned}$$

which along with the fact that $J_{\Omega _n}(\phi _n) = \inf _{w \in {\mathcal {X}}_H(\Omega _n)} J_{\Omega _n}(w)$ implies

$$\begin{aligned} J_{\Omega _n}(v_n(s)) \equiv J_{\Omega _n}(v_{0,n}) \quad \text{ and } \quad \partial _s (|v_n|^{(q-2)/2}v_n)(s) \equiv 0 \ \text{ a.e. } \text{ in } \Omega _n \end{aligned}$$

for $s \ge 0$. Hence $v_n(s) \equiv v_{0,n}$ and it solves (1.11), (1.12) with $\Omega = \Omega _n$. Thus $v_{0,n}$ turns out to be an element of ${\mathcal {S}}_H(\Omega _n)$, and therefore, by (9.3), $v_{0,n}$ coincides with either $\phi _n$ or $-\phi _n$. Consequently, we obtain $J_{\Omega _n}(w) > J_{\Omega _n}(\phi _n)$ if and only if $w \ne \pm \phi _n$. $\square$

Now, we are ready to prove Theorem 9.2, which can be proved along the same lines of Theorem 2 of [5] with the aid of lemmata proved so far. We provide here a proof for completeness.

Proof of Theorem 9.2

Since $\pm \phi _n$ are non-degenerate for $n \in {\mathbb {N}}$ large enough, they are isolated in $H^1_0(\Omega _n)$ from all the other non-trivial solutions for (1.11), (1.12). Hence let $r_n > 0$ be small enough that

$$\begin{aligned} B_{\Omega _n}(\phi _n;r_n) \cap {\mathcal {S}}(\Omega _n) = \{\phi _n\}, \end{aligned}$$

(9.4)

where $B_{\Omega _n}(\phi _n;r_n)$ denotes the ball in $H^1_0(\Omega _n)$ centered at $\phi _n$ with radius $r_n$. Let $\varepsilon \in (0,r_n)$ be fixed. Then we claim that

$$\begin{aligned} c_{n,\varepsilon }&:= \inf \left\{ J_{\Omega _n}(w) :w \in {\mathcal {X}}_H(\Omega _n), \ \Vert w - \phi _n\Vert _{H^1_0(\Omega _n)} = \varepsilon \right\} \nonumber \\&> J_{\Omega _n}(\phi _n) \end{aligned}$$

(9.5)

for $n \in {\mathbb {N}}$ large enough. Indeed, it has already been proved in Lemma 9.4 that $c_{n,\varepsilon } \ge J_{\Omega _n}(\phi _n)$. Hence it suffices to show that $c_{n,\varepsilon } \ne J_{\Omega _n}(\phi _n)$. Suppose to the contrary that $c_{n,\varepsilon } = J_{\Omega _n}(\phi _n)$. Then there exists a sequence $(w_m)$ in ${\mathcal {X}}_H(\Omega _n)$ such that $J_{\Omega _n}(w_m) \rightarrow J_{\Omega _n}(\phi _n)$ and $\Vert w_m - \phi _n\Vert _{H^1_0(\Omega _n)} = \varepsilon$. Hence we can extract a (not relabeled) subsequence of $(w_m)$ such that

$$\begin{aligned} w_m \rightarrow \psi _n \quad \text{ weakly } \text{ in } H^1_0(\Omega _n) \text{ and } \text{ strongly } \text{ in } L^q(\Omega _n) \end{aligned}$$

as $m \rightarrow +\infty$ for some $\psi _n \in H^1_0(\Omega _n)$. Since ${\mathcal {X}}_H(\Omega _n)$ is sequentially weakly closed in $H^1_0(\Omega _n)$, $\psi _n$ turns out to be an element of ${\mathcal {X}}_H(\Omega _n)$. It follows from Lemma 9.4 that $J_{\Omega _n}(\psi _n) \ge J_{\Omega _n}(\phi _n)$. Therefore we see that

$$\begin{aligned} \frac{1}{2} \Vert \nabla w_m\Vert _{L^2(\Omega _n)}^2&= J_{\Omega _n}(w_m) + \frac{\lambda _q}{q} \Vert w_m\Vert _{L^q(\Omega _n)}^q\\&\rightarrow J_{\Omega _n}(\phi _n) + \frac{\lambda _q}{q} \Vert \psi _n\Vert _{L^q(\Omega _n)}^q\\&\le J_{\Omega _n}(\psi _n) + \frac{\lambda _q}{q} \Vert \psi _n\Vert _{L^q(\Omega _n)}^q = \frac{1}{2} \Vert \nabla \psi _n\Vert _{L^2(\Omega _n)}^2. \end{aligned}$$

Thus we obtain

$$\begin{aligned} w_m \rightarrow \psi _n \quad \text{ strongly } \text{ in } H^1_0(\Omega _n) \end{aligned}$$

as $m \rightarrow +\infty$. Hence it follows that $J_{\Omega _n}(\psi _n) = J_{\Omega _n}(\phi _n)$ and $\Vert \psi _n - \phi _n\Vert _{H^1_0(\Omega _n)} = \varepsilon \in (0,r_n)$; however, by virtue of Lemma 9.4, they contradict each other. Thus we conclude that $c_{n,\varepsilon } > J_{\Omega _n}(\phi _n)$.

Since $J_{\Omega _n}(\cdot )$ is continuous in $H^1_0(\Omega _n)$, one can take $\delta _{n,\varepsilon } \in (0,\varepsilon )$ such that

$$\begin{aligned} J_{\Omega _n}(v_{0,n}) < c_{n,\varepsilon } \end{aligned}$$

for any $v_{0,n} \in {\mathcal {X}}_H(\Omega _n)$ satisfying $\Vert v_{0,n}-\phi _n\Vert _{H^1_0(\Omega _n)} < \delta _{n,\varepsilon }$. Hence let $v_{0,n} \in {\mathcal {X}}_H(\Omega _n)$ satisfy $\Vert v_{0,n}-\phi _n\Vert _{H^1_0(\Omega _n)} < \delta _{n,\varepsilon }$ and let $v_n = v_n(x,s)$ be the weak solution to (1.6)–(1.8) with $\Omega = \Omega _n$ and $v_0 = v_{0,n}$. Since $s \mapsto J_{\Omega _n}(v_n(s))$ is non-increasing, we have

$$\begin{aligned} J_{\Omega _n}(v_n(s)) \le J_{\Omega _n}(v_{0,n}) < c_{n,\varepsilon } \end{aligned}$$

for any $s \ge 0$. Therefore, by virtue of (9.5), $v_n(s)$ cannot go beyond the boundary of the ball $B_{\Omega _n}(\phi _n;\varepsilon )$ for any $s \ge 0$, that is, it holds that

$$\begin{aligned} \sup _{s \ge 0} \Vert v_n(s)-\phi _n\Vert _{H^1_0(\Omega _n)} \le \varepsilon \end{aligned}$$

(9.6)

(cf. [4, Proof of Theorem 3]). Thus $\phi _n$ turns out to be stable under the dynamical system in ${\mathcal {X}}_H(\Omega _n)$ generated by (1.6)–(1.8) with $\Omega = \Omega _n$.

Furthermore, since each solution $v_n(s)$ of (1.6)–(1.8) with $\Omega = \Omega _n$ and $v_0 = v_{0,n} \in {\mathcal {X}}_H(\Omega _n)$ quasi-converges to an element of ${\mathcal {S}}_H(\Omega _n)$ strongly in $H^1_0(\Omega _n)$ as $s \rightarrow +\infty$ and $\phi _n$ is isolated in $H^1_0(\Omega _n)$ from all the other elements of ${\mathcal {S}}_H(\Omega _n)$ (see (9.4)), we deduce from the stability of $\phi _n$ that $v_n(s) \rightarrow \phi _n$ strongly in $H^1_0(\Omega _n)$ as $s \rightarrow +\infty$, provided that $v_{0,n} \in {\mathcal {X}}_H(\Omega _n)$ and $\Vert v_{0,n} -\phi _n\Vert _{H^1_0(\Omega _n)}$ is small enough. Finally, the exponential stability follows from Theorem 1.1. This completes the proof. $\square$

Remark 9.5

(Positive and even asymptotic profiles in dumbbell domains) The above argument can also be applied to positive and even (with respect to the hyperplane H) solutions on dumbbell domains with thin channels by replacing odd functions with even ones.

Appendices

Appendix A: Taylor’s Theorem

In this section, we shall recall the well-known mean-value theorem as well as Taylor’s theorem for operators in Banach spaces for the convenience of the reader. We refer the reader to, e.g., [48, 49] for details on Fréchet and Gâteaux derivatives of operators defined on Banach spaces (see also Notation in Section 1). Let us start with the mean-value theorem.

Theorem A.1

(Mean-value theorem for operators) Let $x,y \in X$ and let $I = [x,y] = \{(1-\theta )x+\theta y :\theta \in [0,1]\}$. Let U be an open set in X such that $I \subset U$ and let $F : U \subset X \rightarrow Y$ be Gâteaux differentiable on I such that the Gâteaux derivative $\text {D}_G F : I \subset X \rightarrow {\mathscr {L}}(X,Y)$ of F is bounded in ${\mathscr {L}}(X,Y)$ on I. Then it holds that

$$\begin{aligned} \Vert F(y)-F(x)\Vert _Y \le \sup _{\theta \in [0,1]} \left\| \text {D}_G F((1-\theta )x+\theta y) \right\| _{{\mathscr {L}}(X,Y)} \Vert y-x\Vert _X. \end{aligned}$$

Proof

Let $\eta \in Y^*$ be such that $\Vert \eta \Vert _{Y^*} = 1$ and $\langle \eta , F(y) - F(x) \rangle _Y = \Vert F(y)-F(x)\Vert _Y$ (indeed, such an $\eta$ exists thanks to Hahn-Banach’s theorem; see, e.g., [23, Corollary 1.3]). Since F is Gâteaux differentiable on I, we see that $\theta \mapsto \varphi (\theta ) := \langle \eta , F((1-\theta )x + \theta y)\rangle _Y$ is differentiable on [0, 1]. Hence using the standard mean-value theorem, we can take $\theta _0 \in (0,1)$ such that $\varphi (1)-\varphi (0) = \varphi '(\theta _0) (1-0)$, that is,

$$\begin{aligned} \Vert F(y)-F(x)\Vert _Y&= \langle \eta , [\text {D}_G F((1-\theta _0)x + \theta _0 y)](y-x) \rangle _Y\\&\le \Vert [\text {D}_G F((1-\theta _0)x + \theta _0 y)](y-x)\Vert _Y\\&\le \Vert \text {D}_G F((1-\theta _0)x+\theta _0 y)\Vert _{{\mathscr {L}}(X,Y)} \Vert y-x\Vert _X\\&\le \sup _{\theta \in [0,1]}\Vert \text {D}_G F((1-\theta )x+\theta y)\Vert _{{\mathscr {L}}(X,Y)} \Vert y-x\Vert _X. \end{aligned}$$

This completes the proof. $\square$

   Here and henceforth, for each $j \in {\mathbb {N}}$, $T \in {\mathscr {L}}^{(j)}(X,Y)$ and $x \in X$, we shall simply write $T(x,x,\ldots ,x) = T x^j$.

Theorem A.2

(Taylor’s theorem for operators) Let $x,y \in X$ and let $I = [x,y] = \{(1-\theta )x+\theta y :\theta \in [0,1]\}$. Let U be an open set in X such that $I \subset U$ and let $F : U \subset X \rightarrow Y$ be $(n-1)$-times Fréchet differentiable in U such that the $(n-1)$-th Fréchet derivative $F^{(n-1)} : U \subset X \rightarrow {\mathscr {L}}^{(n-1)}(X,Y)$ of F is Gâteaux differentiable on I and the Gâteaux derivative $\text {D}_G F^{(n-1)}$ of $F^{(n-1)}$ is bounded in ${\mathscr {L}}^{(n)}(X,Y)$ on I. Then it holds that

$$\begin{aligned} F(y) = F(x) + \sum _{j=1}^{n-1} \dfrac{[F^{(j)}(x)]}{j!}(x-y)^j + e, \end{aligned}$$

(A.1)

where $e \in Y$ satisfies

$$\begin{aligned} \Vert e\Vert _Y \le \sup _{\theta \in [0,1]} \Vert \text {D}_G F^{(n-1)}((1-\theta )x+\theta y)\Vert _{{\mathscr {L}}^{(n)}(X,Y)} \Vert y-x\Vert _X^n. \end{aligned}$$

Proof

Set

$$\begin{aligned} P_{n-1}(w) = \sum _{j=0}^{n-1} \frac{[F^{(j)}(x)]}{j!}(w-x)^j \end{aligned}$$

and

$$\begin{aligned} G(w) = F(w) - P_{n-1}(w) \end{aligned}$$

for $w \in U$. Then G is $(n-1)$-times Fréchet differentiable on U such that

$$\begin{aligned} G^{(\ell )}(x) = 0 \quad \text{ for } \ \ell = 0,1,\ldots ,n-1. \end{aligned}$$

(A.2)

Moreover, by assumption, $G^{(n-1)}$ is Gâteaux differentiable on I and

$$\begin{aligned} \sup _{t \in [0,1]} \Vert \text {D}_G G^{(n-1)}([x,y]_t)\Vert _{{\mathscr {L}}^{(n)}(X,Y)} < +\infty . \end{aligned}$$

In what follows, we write $[x,y]_t = (1-t)x + ty$ and note that $[x,[x,y]_t]_s = [x,y]_{st}$ for $s,t \in [0,1]$. Moreover, using (A.2) and Theorem A.1 repeatedly, we see that

$$\begin{aligned} \Vert G(y)\Vert _Y&= \Vert G(y) - G(x)\Vert _Y\nonumber \\&\le \sup _{t_1 \in [0,1]} \Vert G'([x,y]_{t_1})\Vert _{{\mathscr {L}}(X,Y)} \Vert y-x\Vert _X\nonumber \\&= \sup _{t_1 \in [0,1]} \Vert G'([x,y]_{t_1}) - G'(x)\Vert _{{\mathscr {L}}(X,Y)} \Vert y-x\Vert _X\nonumber \\&\le \sup _{t_1 \in [0,1]} \sup _{t_2 \in [0,1]} \Vert G''([x,[x,y]_{t_1}]_{t_2}) \Vert _{{\mathscr {L}}^{(2)}(X,Y)} t_1 \Vert y-x\Vert _X^2\nonumber \\&\le \sup _{t \in [0,1]} \Vert G''([x,y]_t) \Vert _{{\mathscr {L}}^{(2)}(X,Y)} \Vert y-x\Vert _X^2\nonumber \\&\le \sup _{t \in [0,1]} \Vert G^{(n-1)}([x,y]_t) \Vert _{{\mathscr {L}}^{(n-1)}(X,Y)} \Vert y-x\Vert _X^{n-1}\\&\le \sup _{t \in [0,1]} \Vert \text {D}_G G^{(n-1)}([x,y]_t)\Vert _{{\mathscr {L}}^{(n)}(X,Y)} \Vert y-x\Vert _X^n,\nonumber \end{aligned}$$

(A.3)

which ensures the desired assertion for $e = G(y)$. This completes the proof. $\square$

Remark A.3

If $F : U \subset X \rightarrow Y$ is only of class $C^{n-1}$ in U in the sense of Fréchet derivative, then we can still obtain (A.1) along with $e \in Y$ satisfying only

$$\begin{aligned} \lim _{\Vert y-x\Vert _X \rightarrow 0} \frac{\Vert e\Vert _Y}{\Vert y-x\Vert _X^{n-1}} = 0. \end{aligned}$$

Indeed, as in (A.3), we can derive from the continuity of $G^{(n-1)}$ that

$$\begin{aligned} \frac{\Vert G(y)\Vert _Y}{\Vert y-x\Vert _X^{n-1}}&\le \sup _{t \in [0,1]} \Vert G^{(n-1)}([x,y]_t) - G^{(n-1)}(x)\Vert _{{\mathscr {L}}^{(n-1)}(X,Y)} \\&\rightarrow 0 \quad \text{ as } \ \Vert x-y\Vert _X \rightarrow 0. \end{aligned}$$

Setting $e = G(y)$, we obtain the desired conclusion.

   Finally, we shall give a proof for the fact that J is of class $C^3$ in $H^1_0(\Omega )$, provided that $q \ge 3$. Let $w \in H^1_0(\Omega )$ be arbitrarily fixed. It is well known that J is of class $C^2$ in $H^1_0(\Omega )$ and its second Fréchet derivative $J''(w) \in {\mathscr {L}}(H^1_0(\Omega ),H^{-1}(\Omega )) = {\mathscr {L}}^{(2)}(H^1_0(\Omega ),{\mathbb {R}})$ at w is represented by

$$\begin{aligned}{}[J''(w)](u, v) = \int _\Omega \nabla u \cdot \nabla v \, \text {d}x - \lambda _q (q-1) \int _\Omega |w|^{q-2} uv \, \text {d}x \end{aligned}$$

for $u,v \in H^1_0(\Omega )$ (see, e.g., [48, Corollary 1.13]). Therefore since $q \ge 3$, we can see that

$$\begin{aligned}&\left\| \frac{J''(w+ he) - J''(w)}{h} + \lambda _q (q-1)(q-2) |w|^{q-4}w e \right\| _{{\mathscr {L}}^{(2)}(H^1_0(\Omega ),{\mathbb {R}})} \\&\quad \le C \lambda _q (q-1) \left\| \frac{|w+he|^{q-2}-|w|^{q-2}}{h} - (q-2) |w|^{q-4}w e \right\| _{L^{q/(q-2)}(\Omega )}\\&\quad \rightarrow 0 \quad \text{ as } \ h \rightarrow 0 \end{aligned}$$

for $e \in H^1_0(\Omega )$. Thus $J'' : H^1_0(\Omega ) \rightarrow {\mathscr {L}}^{(2)}(H^1_0(\Omega );{\mathbb {R}})$ is Gâteaux differentiable at w and its derivative $\text {D}_G J''(w) \in {\mathscr {L}}^{(3)}(H^1_0(\Omega );{\mathbb {R}})$ at w is represented as

$$\begin{aligned}{}[\text {D}_G J''(w)](e,u,v) = \lambda _q (q-1)(q-2) \int _\Omega |w|^{q-4}w e u v \, \text {d}x \end{aligned}$$

for $e,u,v \in H^1_0(\Omega )$. Moreover, one can check from $q \ge 3$ that $w \mapsto \text {D}_G J''(w)$ is a continuous map from $H^1_0(\Omega )$ into ${\mathscr {L}}^{(3)}(H^1_0(\Omega ),{\mathbb {R}})$, and therefore, $J''$ also turns out to be Fréchet differentiable at w and its Fréchet derivative $J^{(3)}(w)$ at w coincides with $\text {D}_G J''(w)$.

Appendix B: Elementary Inequalities

We first prove (4.2). We can assume that $a > b$ and $a,b \ne 0$ without loss of generality. We see that

$$\begin{aligned} \left( |a|^{q-2}a - |b|^{q-2}b \right) (a - b)&= \left( \int ^a_b \frac{\text {d}}{\text {d}\xi } (|\xi |^{q-2} \xi ) \, \text {d}\xi \right) \left( \int ^a_b 1 \, \text {d}\xi \right) \\&\ge \left[ \int ^a_b \left( \frac{\text {d}}{\text {d}\xi } (|\xi |^{q-2} \xi ) \right) ^{1/2} 1^{1/2} \,\text {d}\xi \right] ^2\\&= (q-1)\left[ \int ^a_b |\xi |^{\frac{q-2}{2}} \text {d}\xi \right] ^2\\&= (q-1)\left[ \int ^a_b \frac{2}{q} \frac{\text {d}}{\text {d}\xi } (|\xi |^{\frac{q-2}{2}}\xi ) \, \text {d}\xi \right] ^2\\&= \frac{4(q-1)}{q^2} \left| |a|^{\frac{q-2}{2}}a - |b|^{\frac{q-2}{2}}b \right| ^2. \end{aligned}$$

Inequality (7.2) is standard. We next prove (4.5). In case a and b have the same sign, (7.2) is applicable. In case a and b have different signs, we may simply assume $a> 0 > b$. Set $c = -b > 0$. Since $p \in (0,1)$, we see that

$$\begin{aligned} \frac{a^p+c^p}{2} \le \left( \frac{a+c}{2}\right) ^p, \end{aligned}$$

which yields

$$\begin{aligned} a^p + c^p \le 2^{1-p} (a+c)^p. \end{aligned}$$

It follows that

$$\begin{aligned} \frac{a^p+c^p}{a+c} \le 2^{1-p} (a+c)^{p-1} \le 2^{1-p} a^{p-1}. \end{aligned}$$

Thus we obtain (4.5).