1 Introduction

We shall be concerned with time T-periodic solutions of the following parabolic problem

$$\begin{aligned} {\frac{\partial u}{\partial t}} (x,t) = \Delta u (x,t) + V(x)u(x,t)+f(t,x,u(x,t)),\quad x\in {\mathbb {R}}^N,\, t> 0, \end{aligned}$$
(1)

where \(\Delta \) is the Laplace operator (with respect to x), V is a Kato–Rellich type potential and \(f:[0,+\infty )\times {\mathbb {R}}^N\times {\mathbb {R}}\rightarrow {\mathbb {R}}\) is T-periodic in time:

$$\begin{aligned} f(t+T,x,u) = f(t,x,u), \end{aligned}$$
(2)

for all \(t\ge 0\), \(u\in {\mathbb {R}}\) and a.e. \(x\in {\mathbb {R}}^N\).

Periodic problems for parabolic equations were widely studied by many authors by use of various methods. Some early results are due to Brezis and Nirenberg [4], Amman and Zehnder [2], Nkashama and Willem [19], Hirano [15, 16], Prüss [25], Hess [14], Shioji [27] and many others; see also [29] and the references therein. These results treat the case where \(\Omega \) is bounded and are based either on topological degree and coincidence index techniques in the spaces of functions depending both on x and time t or on the translation along trajectories operator to which fixed point theory is applied. In this paper we shall study the case \(\Omega ={\mathbb {R}}^N\). In this case arguments based on the compactness of the heat semigroup are no longer valid, since \(H^1({\mathbb {R}}^N)\) is not compactly embedded into \(L^2({\mathbb {R}}^N)\). Here, we shall prove that the Poincaré translation along trajectories is eventually compact, which enables us to use fixed point index and averaging techniques (see e.g. [6, 7]).

We assume that V is of Kato–Rellich type, i.e. \(V = V_\infty + V_0\) where \(V_\infty \in L^\infty ({\mathbb {R}}^N)\) and \(V_0\in L^p({\mathbb {R}}^N)\) where

$$\begin{aligned} 2<p<\infty \quad \text{ for } N=1,2 \qquad \text{ and }\qquad N \le p <\infty \quad \text{ for } N\ge 3. \end{aligned}$$

The nonlinear perturbation \(f:[0,+\infty )\times {\mathbb {R}}^N\!\times \, {\mathbb {R}}\rightarrow {\mathbb {R}}\) is such that \(f(t,\cdot , u)\) is measurable, for all \(t\ge 0\) and \(u\in {\mathbb {R}}\),

$$\begin{aligned} |f(t,x,0)| \le m_0 (x),\quad \text{ for } \text{ all } t\ge 0 \text{ and } \text{ a.e. } x\in {\mathbb {R}}^N, \end{aligned}$$
(3)

for some \(m_0\in L^2({\mathbb {R}}^N)\), and there are \(\theta \in (0,1)\),

$$\begin{aligned}&|f(t,x,u)-f(s,x,u)| \le ({\tilde{k}}(x)+ k(x)|u|)|t-s|^{\theta }, \end{aligned}$$
(4)
$$\begin{aligned}&|f(t,x,u)-f(t,x,v)| \le l(t,x)|u-v|, \end{aligned}$$
(5)

for all \(t,s\in [0,+\infty )\), \(u, v\in {\mathbb {R}}\) and a.e. \(x\in {\mathbb {R}}^N\), where \({\tilde{k}}\in L^2({\mathbb {R}}^N)\), k is of Kato–Rellich type (i.e. satisfies the same assumption as V, possibly with different p), \(l=l_0+l_\infty \) with \(l_\infty \in L^\infty ([0,+\infty )\times {\mathbb {R}}^N)\) and \(\sup _{t\ge 0} \Vert l_0(t,\cdot )\Vert _{L^p} <\infty \). We shall also assume a sort of relaxed monotonicity:

$$\begin{aligned} (f(t,x,u) - f(t,x,v))\cdot (u-v) \le a(x) |u-v|^2, \end{aligned}$$
(6)

for any \(u, v \in {\mathbb {R}}^N\), \(t\ge 0\) and a.e. \(x\in {\mathbb {R}}^N\), where a is of Kato–Rellich type and

$$\begin{aligned} \lim _{r\rightarrow \infty } \mathop {\mathrm {esssup}}\limits _{|x|>r} ( V_\infty + a_\infty )(x)<0. \end{aligned}$$

Usually we consider separately two cases looking at the time averaged right hand-side \(\Delta +V+{\widehat{f}}\) where the time average function \({\widehat{f}}:{\mathbb {R}}^N\times {\mathbb {R}}\rightarrow {\mathbb {R}}\) of f is given by

$$\begin{aligned} {\widehat{f}}(x,u):=\frac{1}{T}\int _{0}^{T} f(t,x,u)\mathrm {\,d}t. \end{aligned}$$

The resonant case is when the linearization of the nonlinear operator \(\Delta + V+{\widehat{f}}\) at infinity and/or zero has nontrivial kernel and the non-resonant case when the mentioned linearizations have zero kernels. Both cases can be put in similar settings, however, are different geometrically. The resonant case was studied in [8]. Here we consider the non-resonant case. Our main results are the following criteria for the existence of T-periodic solutions.

Theorem 1.1

Suppose that f and V satisfy conditions (2), (3), (4), (5) and (6). If \({\mathrm {Ker}}\, (\Delta +V) = \{0\},\)

$$\begin{aligned} \lim _{|u|\rightarrow \infty } \frac{f(t,x,u)}{u} = 0 \quad \text{ for } \text{ any } t\ge 0 \text{ and } \text{ a.e. } x\in {\mathbb {R}}^N, \end{aligned}$$
(7)

and

$$\begin{aligned} \lim _{r\rightarrow \infty } \mathop {\mathrm {esssup}}\limits _{|x|>r} V_\infty (x) < 0, \end{aligned}$$
(8)

then the Eq. (1) admits a T-periodic solution

$$\begin{aligned} u\in C([0,+\infty ), H^2({\mathbb {R}}^N)) \cap C^1([0,+\infty ), L^2({\mathbb {R}}^N)). \end{aligned}$$

Our second result applies in the case where there exists a trivial periodic solution \(u\equiv 0\) and the previous theorem does not imply the existence of a nontrivial periodic one.

Theorem 1.2

Suppose that all the assumptions of Theorem 1.1 are satisfied and,  additionally,  assume that,  for any \(t\ge 0\) and a.e. \(x\in {\mathbb {R}}^N,\)

$$\begin{aligned} \lim _{u\rightarrow 0} \frac{f(t,x,u)}{u} = \alpha (x), \end{aligned}$$
(9)

where \(\alpha \) is of Kato–Rellich type. If \({\mathrm {Ker}}\, ( \Delta + V + \alpha ) = \{ 0 \},\)

$$\begin{aligned} \lim _{r\rightarrow \infty } \mathop {{\mathrm {esssup}}}\limits _{|x|>r} (V_\infty + \alpha _\infty )(x) < 0 \end{aligned}$$
(10)

and

$$\begin{aligned} m_+(\Delta +V)\not \equiv m_+(\Delta +V+\alpha ) \ \mathrm{mod}\ 2, \end{aligned}$$

where \(m_+ (\Delta +V)\) and \(m_+ (\Delta +V+\alpha )\) are the total multiplicities of the positive eigenvalues of \(\Delta +V\) and \(\Delta +V+\alpha ,\) respectively,  then the Eq. (1) admits a nontrivial T-periodic solution \(u\in C([0,+\infty ), H^2({\mathbb {R}}^N))\cap C^1([0,+\infty ),L^2({\mathbb {R}}^N)).\)

Remark 1.3

  1. (a)

    Due to the spectral theory assumption (8) implies that the essential spectrum \(\sigma _e(\Delta +V)\) is contained in \((-\infty ,0)\) and all the positive elements of the spectrum are actually eigenvalues. Therefore, \(0\not \in \sigma (\Delta +V)\) and the number \(m_+(\Delta +V)\) are well-defined. The same goes for the operator \(\Delta +V+\alpha \) (see also Remark 7.4 for more details).

  2. (b)

    Note that although it is not stated explicitly (9) implies that \(f(t,x,0)=0\) for all \(t\ge 0\) and a.e. \(x\in {\mathbb {R}}^N\).

  3. (c)

    Theorems 1.1 and 1.2 are straightforward consequences of more general results from Sect. 7, where V may depend on time as well. \(\square \)

The above results are obtained by use of the Poincaré (translation along trajectories) operator \({\varvec{\Phi }}_T: H^1({\mathbb {R}}^N) \rightarrow H^1({\mathbb {R}}^N)\) defined as

$$\begin{aligned} {\varvec{\Phi }}_T({\bar{u}}):=u(T),\quad {\bar{u}} \in H^1({\mathbb {R}}^N), \end{aligned}$$

where \(u: [0,T] \rightarrow H^1({\mathbb {R}}^N)\) is the solution of (1) satisfying the initial value condition \(u(0)={\bar{u}}\). We obtain T-periodic solutions as fixed points of \({\varvec{\Phi }}_T\). In case of parabolic equations on bounded domains, the Poincaré operator is compact, which is no longer the case on \({\mathbb {R}}^N\). Actually neither is \({\varvec{\Phi }}_T\) a set contraction. In consequence, we can not use standard Leray–Schauder degree/index. However, following the tail estimates techniques from Wang [31] (see also Prizzi [24] and [9]), we show that \({\varvec{\Phi }}_T\) is ultimately compact, i.e. belongs to a wider class of maps for which the fixed point index \(\mathrm {Ind}({\varvec{\Phi }}_T, U)\), with respect to open subsets of \(H^1({\mathbb {R}}^N)\), can be considered (see e.g. [1]). To determine the index \(\mathrm {Ind}({\varvec{\Phi }}_T, U)\) (for a properly chosen U), we use the averaging method, i.e. we consider the Eq. (1) as one of the family of problems

$$\begin{aligned} {\frac{\partial u}{\partial t}} = \Delta u + V(x)u+ f (t/\lambda ,x,u ),\quad x\in {\mathbb {R}}^N,\ t>0,\ \lambda >0. \end{aligned}$$
(11)

By use of averaging techniques we show that solutions of (11) converge as \(\lambda \rightarrow 0^+\) to a solution of the averaged equation

$$\begin{aligned} {\frac{\partial u}{\partial t}} = \Delta u + V(x)u+ {\widehat{f}}(x,u),\quad x\in {\mathbb {R}}^N, \ t>0. \end{aligned}$$
(12)

Our asymptotic assumptions on f imply a sort of a priori bounds conditions, i.e. that, for \(\lambda \in (0,1]\), the problem (11) has no \(\lambda T\)-periodic solution if only initial states are of sufficiently large \(H^1\) norm (in case of Theorem 1.1) and also of sufficiently small \(H^1\) norm (in case of Theorem 1.2). This will mean that initial states of \(\lambda T\)-periodic solutions are located inside some open bounded set \(U\subset H^1({\mathbb {R}}^N)\). Next we prove a sort of “averaged” Krasnosel’skii formula (see also [6]) stating that

$$\begin{aligned} \mathrm {Ind}({\varvec{\Phi }}_T, U) = \lim _{t\rightarrow 0^+} \mathrm {Ind}(\widehat{\varvec{\Phi }}_t, U), \end{aligned}$$
(13)

where \(\widehat{\varvec{\Phi }}_t\) is the translation along trajectories operator for (12). To determine \(\mathrm {Ind}(\widehat{\varvec{\Phi }}_t, U)\), for small \(t>0\), we use linearization techniques and strongly rely on spectral properties of \(\Delta +V\) and \(\Delta +V+\alpha \).

Remark 1.4

  1. (a)

    By the applied approach the (topological) fixed point index of the Poincaré operator \({\varvec{\Phi }}_T\) is determined. The fact that we consider operators in the phase space and use topological methods has some stability and continuity implications. Namely, if the nonlinearity f is perturbed by a small term so that the time T-periodicity and regularity conditions are satisfied, then the conclusions of Theorems 1.2 and 1.1 hold. Actually, one can even perturb the nonlinearity, by a term that does not push it out of the class determined by assumptions (2)–(7) as well as (9) and (10). Moreover, it comes from fixed point index theory (see [1, 12]) that T-periodic solutions of the problem perturbed with a small term are localized near the T-periodic solutions of (1) with the original f. Another advantage is that the knowledge of the fixed point index of the translation operator carries additional information on the dynamics of the evolution system generated by (1). In fact we can compute the indices of all the iterations of \({\varvec{\Phi }}_T\), which are related to the stability of periodic solutions (see e.g. [20, 21]).

  2. (b)

    There are several other approaches to forced oscillations for nonlinear evolution problems where the solutions are found in the space of time-periodic functions, see e.g. [30] or [17] for problems on bounded domains. Another method uses a separation of the so-called steady state part and the oscillatory one like in recent paper [11, 18], in which maximal regularity for semi-linear problems on \({\mathbb {R}}^N\) were obtained (it is not clear yet if this setting adapts to nonlinear state depending f).

  3. (c)

    We may assume that the time T-periodic f is defined on \({\mathbb {R}}\times {\mathbb {R}}^N\) and consider T-periodic solutions on \({\mathbb {R}}\). Then we are able to consider the problem where \(\partial u/\partial t\) is replaced by \(- \partial u/\partial t\) provided \(-V\) and \(-f(-t,x,u)\) satisfy the above assumptions for V and f.

The paper is organized as follows. In Sect. 2 we recall the concept of ultimately compact maps and fixed point index theory. In Sect. 3 we strengthen in a general setting of sectorial operators the initial condition continuity property and Henry’s averaging principle, that we apply to the parabolic equation in Sect. 4. Section 5 is devoted to the ultimate compactness property of the translation operator. In Sect. 6 we adapt the ideas of [6] to the case \(\Omega ={\mathbb {R}}^N\), proving the averaging index formula (13) as well as verify a priori bounds conditions for \(\lambda T\)-periodic solutions of (11) with \(\lambda \in (0,1]\). Finally, in Sect. 7 the main results are proved.

2 Preliminaries

Notation. If X is a normed space with the norm \(\Vert \cdot \Vert \), then, for \(x_0\in X\) and \(r>0\), we put \(B_X(x_0,r):=\{x\in X \mid \Vert x-x_0\Vert <r\}\). By \(\partial U\) and \(\overline{U}\) we denote the boundary and the closure of \(U\subset X\). \(\mathrm {conv}\, V\) and \(\overline{\mathrm {conv}}^{X}\, V\) stand for the convex hull and the closed (in X) convex hull of \(V\subset X\), respectively. By \((\cdot ,\cdot )_0\) is denoted the inner product in X.

Measure of noncompactness. If X is a Banach space and \(V\subset X\) is bounded, then by \(\beta _X (V)\) we denote the infimum over all \(r>0\) such that V can be covered with a finite number of open balls of radius r. Clearly \(\beta _X (V)\) is finite and it is called the Hausdorff measure of noncompactness of the set V in the space X. It is not hard to show that \(\beta _X (V)=0\) implies that V is relatively compact in X. More properties of the measure of noncompactness can be found in [10] or [1].

Fixed point index. Below we recall basic definitions and facts from the fixed point index theory for ultimately compact maps. For details we refer to [1].

We say that a map \(\Phi : D \rightarrow X\), defined on a subset D of a Banach space X is ultimately compact if \(W\subset X\) is such that \(\overline{\mathrm {conv}} \,\Phi (W\cap D)=W\), then W is compact. We shall say that an ultimately compact map \(\Phi :\overline{U} \rightarrow X\), defined on the closure of an open bounded set \(U\subset X\), is called admissible if \(\Phi (u)\ne u\) for all \(u\in \partial U\). By an admissible homotopy between two admissible maps \(\Phi _0, \Phi _1:\overline{U}\rightarrow X\) we mean a continuous map \({{\varvec{\Psi }}}:\overline{U} \times [0,1]\rightarrow X\) such that \({\varvec{\Psi }}(\cdot , 0) = \Phi _0, \, \, {\varvec{\Psi }}(\cdot , 1)=\Phi _1\), \({\varvec{\Psi }} (u,\mu )\ne u\) for all \(u\in \partial U\) and \(\mu \in [0,1]\), and, for any \(W\subset X\), if \(\overline{\mathrm {conv}} \,{\varvec{\Psi }} ( (W\cap \overline{U}) \times [0,1]) = W\), then W is relatively compact. \(\Phi _0, \Phi _1\) are called homotopic then. A fixed point index for ultimately compact maps was constructed in [1, 1.6.3 and 3.5.6]. Basic properties of the fixed point index are collected in the following.

Proposition 2.1

  1. (i)

    (Existence) If \(\mathrm {Ind}(\Phi , U)\ne 0,\) then there exists \(u\in U\) such that \(\Phi (u)=u\).

  2. (ii)

    (Additivity) If \(U_1, U_2 \subset U\) are open and \(\Phi (u)\ne u\) for all \(u\in \overline{U{\setminus } (U_1\cup U_2)},\) then

    $$\begin{aligned} \mathrm {Ind}(\Phi , U)=\mathrm {Ind}(\Phi , U_1) + \mathrm {Ind}(\Phi , U_2). \end{aligned}$$
  3. (iii)

    (Homotopy invariance) If \(\Phi _0, \Phi _1:\overline{U}\rightarrow X\) are homotopic,  then

    $$\begin{aligned} \mathrm {Ind}(\Phi _0, U)=\mathrm {Ind}(\Phi _1, U). \end{aligned}$$
  4. (iv)

    (Normalization) Let \(u_0\in X\) and \(\Phi _{u_0}:\overline{U}\rightarrow X\) be defined by \(\Phi _{u_0}(u)=u_0\) for all \(u\in \overline{U}\). Then \(\mathrm {Ind}(\Phi _{u_0},U)\) is equal 0 if \(u_0\not \in U\) and 1 if \(u_0\in U\).

Remark 2.2

If \(\Phi : \overline{U} \rightarrow X\) is a compact map then \(\mathrm {Ind}(\Phi , U)\) is equal to the Leray–Schauder index \(\mathrm {Ind}_{LS} (\Phi , U)\) (see e.g. [12]).

3 Remarks on abstract continuity and averaging principle

Let \(A:D(A)\rightarrow X\) be a sectorial operator such that for some \(a>0\), \(A+a I\) has its spectrum in the half-plane \(\{z\in \mathbb {C}\mid \mathrm {Re}\, z >0 \}\). Let \(X^\alpha \), \(0\le \alpha <1\), be the fractional power space determined by \(A+aI\). It is well-known that there exist \(C_0, C_\alpha >0\) such that for all \(t>0\)

$$\begin{aligned}&\Vert e^{-t A}u\Vert _\alpha \le C_0 e^{at} \Vert u\Vert _\alpha \quad \text{ for } \text{ all } u\in X^\alpha ,\\&\Vert e^{-tA} u\Vert _{\alpha } \le C_\alpha t^{-\alpha }e^{at} \Vert u\Vert _{0} \quad \text{ for } \text{ all } u\in X, \end{aligned}$$

where \(\{e^{-tA}\}_{t\ge 0}\) is the semigroup generated by \(-A\). Consider the equation

$$\begin{aligned} \left\{ \begin{array}{l} \dot{u}(t) = - A u(t) + F(t,u(t)), \quad t>0,\\ u(0) = {\bar{u}}, \end{array} \right. \end{aligned}$$
(14)

where \({\bar{u}} \in X^\alpha \) and \(F:[0,+\infty ) \times X^\alpha \rightarrow X\) is such that there exists \(C\ge 0\) with

$$\begin{aligned} \Vert F(t,u)\Vert \le C(1+\Vert u\Vert _\alpha )\quad \text{ for } \text{ all } u\in X, \, t>0, \end{aligned}$$
(15)

and, for any bounded \(Z\subset X^\alpha \times [0,+\infty )\) there exist \(D, L\ge 0\) and \(\theta \in (0,1)\) with the property

$$\begin{aligned} \Vert F(t,u)-F(s,v)\Vert \le D |t-s|^\theta + L \Vert u-v\Vert _\alpha \quad \text{ for } \text{ all } (u,t),(v,s) \in Z. \end{aligned}$$
(16)

We shall say that \(u: [0,+\infty ) \rightarrow X^\alpha \) is a solution of above initial value problem if

$$\begin{aligned} u\in C([0,+\infty ),X^\alpha )\cap C((0,+\infty ), D(A))\cap C^1((0,+\infty ), X) \end{aligned}$$

and satisfies (14). By classical results (see [5] or [13]), the problem (14) admits a unique global solution \(u\in C([0,+\infty ),X^\alpha )\cap C((0,+\infty ), D(A))\cap C^1((0,+\infty ), X)\). Moreover, it is known that u being solution of (14) satisfies the following Duhamel formula:

$$\begin{aligned} u(t)= e^{-tA} u(0) + \int _0^t e^{-(t-s)A}F(s,u(s)) \mathrm {\,d}s,\quad t>0. \end{aligned}$$
(17)

Remark 3.1

Assume that \(u:[0,T]\rightarrow X^\alpha \) is a solution of (14) with \(T>0\). Then clearly, by (15) and (17), there is a constant \({\tilde{C}}={\tilde{C}}(C,C_0, C_\alpha , a,T)>0\) such that for all \(t\in (0,T]\)

$$\begin{aligned} \Vert u(t)\Vert _\alpha\le & {} C_0 e^{at}\Vert {\bar{u}}\Vert _\alpha +\int _{0}^{t} C_\alpha (t-s)^{-\alpha } e^{a(t-s)}\Vert F(s,u(s))\Vert \mathrm {\,d}s,\\\le & {} {\tilde{C}}(1+\Vert {\bar{u}}\Vert _\alpha ) + {\tilde{C}} \int _{0}^{t}(t-s)^{-\alpha } \Vert u(s)\Vert _\alpha \mathrm {\,d}s. \end{aligned}$$

This in view of [5, Lemma 1.2.9] implies that there exists \({\bar{C}}={\bar{C}}(C,C_0, C_\alpha , a,T,\alpha )>0\) such that

$$\begin{aligned} \Vert u(t)\Vert _\alpha \le {\bar{C}}(1+\Vert {\bar{u}}\Vert _\alpha ) \quad \text{ for } \text{ all }\ t\in [0,T]. \end{aligned}$$
(18)

Theorem 3.2

Assume that \((\alpha _n)\) is a sequence of positive numbers such that \(\alpha _n\rightarrow \alpha _0\) as \(n\rightarrow +\infty \) for some \(\alpha _0>0\) and that \(A_n:= \alpha _n A\) for \(n\ge 0\). Let \(F_n : [0,T]\times X^\alpha \rightarrow X,\) \(T>0,\) \(n\ge 0,\) satisfy (15) and (16) with common constants CL (independent of n) and let,  for each \(u\in X^\alpha ,\)

$$\begin{aligned} \int _{0}^{t} F_n (s,u) \mathrm {\,d}s\rightarrow \int _{0}^{t} F_0(s,u) \mathrm {\,d}s\quad \text{ in } \ X \text{ as } n\rightarrow +\infty \end{aligned}$$

uniformly with respect to \(t\in [0,T]\). If \(u_n:[0,T]\rightarrow X^\alpha ,\) \(n\ge 0,\) are solutions of

$$\begin{aligned} \dot{u}(t)=-A_n u(t)+F_n (t,u(t)),\quad t\in [0,T], \end{aligned}$$

and \(u_n(0) \rightarrow u_0(0)\) in X,  then \(u_n (t)\rightarrow u_0 (t)\) in \(X^\alpha \) uniformly with respect to t from compact subsets of (0, T].

Remark 3.3

Recall that Henry’s result from [13] states that, under the above assumptions with \(\alpha _n \equiv 1\), if \(u_n (0) \rightarrow u_0 (0)\) in \(X^\alpha \), as \(n\rightarrow +\infty \), then \(u_n(t)\rightarrow u_0(t)\) in \(X^\alpha \) uniformly on compact subsets of [0, T). Here, inspired by the proof of Proposition 2.3 of [24], we modify Henry’s proof.

In the proof we shall use the following lemma.

Lemma 3.4

Under the assumptions of Theorem 3.2, for any continuous \(u:[0,T]\rightarrow X^\alpha ,\)

$$\begin{aligned} \int _{0}^{t} e^{-(t-s)A_n} F_n (s, u(s)) \mathrm {\,d}s \rightarrow \int _{0}^{t} e^{-(t-s)A_0} F_0 (s, u(s)) \mathrm {\,d}s\quad \text{ in } X^\alpha \text{ as } \ n \rightarrow +\infty , \end{aligned}$$

uniformly with respect to \(t\in [0,T]\).

Proof

We shall adjust arguments from the proof of [13, Lemma 3.4.7]. First observe that due to the assumptions concerning the constant L for \(F_n\)’s it is sufficient to show the assertion for \(u\equiv {\bar{u}}\) where \({\bar{u}}\in X^{\alpha }\). Take any \(\varepsilon >0\). There exist \(\delta >0\), \({\tilde{C}}>0\) and \(\tilde{a}>0\) such that, for any \(n\ge 0\) and \(t\in [0,\delta ]\)

$$\begin{aligned}&\left\| \int _{0}^{t} e^{ - (t-s) A_n } F_n (s,{\bar{u}}) \mathrm {\,d}s \right\| _\alpha \le \int _{0}^{t} C_\alpha \alpha _{n}^{-\alpha } (t-s)^{-\alpha } e^{\alpha _n a (t-s)}C (1+\Vert {\bar{u}}\Vert _\alpha ) \mathrm {\,d}s\nonumber \\&\quad \le {\tilde{C}} \int _{0}^{t} \tau ^{-\alpha } e^{{\tilde{a}} \tau } \mathrm {\,d}\tau \le {\tilde{C}} e^{{\tilde{a}} T} (1-\alpha )^{-1}\delta ^{1-\alpha }\le {\tilde{C}} e^{{\tilde{a}} T} \delta ^{1-\alpha }< \varepsilon /4 \end{aligned}$$
(19)

and, for any \(n\ge 0\) and \(t \in [\delta ,T]\),

$$\begin{aligned} \left\| \int _{t-\delta }^{t} e^{- (t-s) A_n} F_n (s,{\bar{u}}) \mathrm {\,d}s \right\| _\alpha \le {\tilde{C}} \int _{0}^{\delta } \tau ^{-\alpha } e^{{\tilde{a}} \tau } \mathrm {\,d}\tau \le {\tilde{C}} e^{{\tilde{a}} T} (1-\alpha )^{-1}\delta ^{1-\alpha } < \varepsilon /4.\nonumber \\ \end{aligned}$$
(20)

Observe that, for any \(n\ge 0\) and \(t\in [\delta ,T]\),

$$\begin{aligned} \int _{0}^{t-\delta } e^{-(t-s)A_n} F_n (s, {\bar{u}}) \mathrm {\,d}s= & {} e^{-tA_n}\int _{0}^{t} F_n (\tau ,{\bar{u}})\mathrm {\,d}\tau -e^{-\delta A_n}\int _{t-\delta }^{t} F_n (\tau , {\bar{u}})\mathrm {\,d}\tau \\&+ \int _{0}^{t-\delta } A_n e^{-(t-s)A_n} \int _{s}^{t} F_n (\tau ,\bar{u}) \mathrm {\,d}\tau \, \mathrm {\,d}s. \end{aligned}$$

Clearly,

$$\begin{aligned} e^{-tA_n} \int _{0}^{t} F_n (\tau ,{\bar{u}})\mathrm {\,d}\tau \rightarrow e^{-t A_0 }\int _{0}^{t} F_0(\tau ,{\bar{u}})\mathrm {\,d}\tau , \quad \text{ in } X^{\alpha }, \end{aligned}$$

uniformly with respect to \(t\in [\delta ,T]\). Note also that, for all \(t\in [\delta , T]\) and all \(n\ge 1\),

$$\begin{aligned} \left\| e^{-\delta A_n}\int _{t-\delta }^{t} F_n (\tau , {\bar{u}})\mathrm {\,d}\tau \right\| _\alpha \le {\tilde{C}} e^{{\tilde{a}} \delta } \delta ^{1-\alpha }\le \varepsilon /4. \end{aligned}$$

Finally, for large n and all \(t\in [\delta , T]\) and \(s\in [0,t-\delta ]\), one has

$$\begin{aligned}&\left\| A_n e^{-(t-s)A_n} \!\! \int _{s}^{t} F_n (\tau ,{\bar{u}}) \mathrm {\,d}\tau \!-\! A_0 e^{-(t-s)A_0}\!\! \int _{s}^{t} F_0 (\tau ,{\bar{u}}) \mathrm {\,d}\tau \right\| _{\alpha }\\&\quad \le |\alpha _n-\alpha _0| \left\| A e^{-(t-s)A_n} \!\! \int _{s}^{t} F_n (\tau ,{\bar{u}}) \mathrm {\,d}\tau \right\| _{\alpha } \\&\qquad + \alpha _0 \left\| A e^{-(t-s)A_n} \!\! \int _{s}^{t} F_n (\tau ,{\bar{u}}) \mathrm {\,d}\tau \!-\! Ae^{-(t-s)A_0}\!\! \int _{s}^{t} F_0 (\tau ,{\bar{u}}) \mathrm {\,d}\tau \right\| _{\alpha } \\&\quad \le {\bar{C}} |\alpha _n-\alpha _0| \left\| e^{-(t-s)A_n} \!\! \int _{s}^{t} F_n (\tau ,{\bar{u}}) \mathrm {\,d}\tau \right\| _{1+\alpha } \\&\qquad + {\bar{C}} \alpha _0 \left\| e^{-(t-s)A_n}\left( \int _{s}^{t} F_n (\tau ,{\bar{u}}) \mathrm {\,d}\tau -\int _{s}^{t} F_0 (\tau ,{\bar{u}}) \mathrm {\,d}\tau \right) \right\| _{1+\alpha }\\&\qquad + {\bar{C}}\alpha _0 \left\| \left( e^{-(t-s)A_n}-e^{-(t-s)A_0}\right) \int _{s}^{t} F_0 (\tau ,{\bar{u}}) \mathrm {\,d}\tau \right\| _{1+\alpha } \\&\quad \le |\alpha _n-\alpha _0|\frac{{\bar{C}} C_{1+\alpha } e^{{\tilde{a}} T}}{\delta ^{1+\alpha }}\left\| \int _{s}^{t} F_n (\tau ,{\bar{u}}) \mathrm {\,d}\tau \right\| \\&\qquad + \alpha _0 \frac{{\bar{C}} C_{1+\alpha }e^{{\tilde{a}} T}}{\delta ^{1+\alpha }} \left\| \int _{s}^{t} F_n (\tau ,{\bar{u}}) \mathrm {\,d}\tau -\int _{s}^{t} F_0 (\tau ,{\bar{u}}) \mathrm {\,d}\tau \right\| \\&\qquad + \alpha _0 \frac{{\bar{C}} C_{1+\alpha } e^{{\tilde{a}} T}}{(\alpha _0\delta /2)^{1+\alpha }}\left\| \left( e^{-((t-s)\alpha _n/\alpha _0 - \delta /2)A_0}-e^{-(t-s-\delta /2 )A_0}\right) \int _{s}^{t} F_0 (\tau ,{\bar{u}}) \mathrm {\,d}\tau \right\| , \end{aligned}$$

where \({\bar{C}}>0\) is such that \(\Vert Aw\Vert _\alpha \le {\bar{C}} \Vert w \Vert _{1+\alpha }\) for all \(w\in X^{1+\alpha }\). Therefore, for large n and all \(t\in [\delta ,T]\)

$$\begin{aligned} \left\| \int _{0}^{t-\delta } e^{-(t-s)A_n} F_n (s, {\bar{u}}) \mathrm {\,d}s - \int _{0}^{t-\delta } e^{-(t-s)A_0} F_0 (s, {\bar{u}}) \mathrm {\,d}s \right\| _\alpha \le \varepsilon /4 + \varepsilon /4 + \varepsilon /4 = 3\varepsilon /4, \end{aligned}$$

which together with (19) and (20) ends the proof. \(\square \)

Proof of Theorem 3.2

By the Duhamel formula, for \(t\in (0,T]\) and \(n\ge 1\),

$$\begin{aligned} u_n(t)\!-\!u_0(t)= & {} e^{-tA_n}u_n(0)-e^{-t A_0}u_0(0) +\\&+ \int _{0}^{t} e^{-(t-s)A_n}F_n(s, u_0(s))-e^{-(t-s)A_0}F_0(s,u_0(s)))\mathrm {\,d}s\\&+\int _{0}^{t} e^{-(t-s)A_n}(F_n(s, u_n(s))-F_n(s,u_0(s)))\mathrm {\,d}s. \end{aligned}$$

This gives, for all \(t\in (0,T]\) and \(n\ge 1\),

$$\begin{aligned} \Vert u_n(t)-u_0(t)\Vert _{\alpha }\le \gamma _n(t) + C_\alpha L\int _{0}^{t} e^{a\alpha _n (t-s)}(\alpha _n(t-s))^{-\alpha } \Vert u_n(s)-u_0(s)\Vert _\alpha \mathrm {\,d}s \end{aligned}$$

with

$$\begin{aligned} \gamma _n(t)&:= \!\frac{C_\alpha e^{a \alpha _n t}}{(\alpha _n t)^{\alpha }} \Vert u_n(0)-u_0(0)\Vert _{0}+ \left\| (e^{-t A_n}-e^{-t A_0})u_0(0)\right\| _{\alpha } \\&\quad + \left\| \int _{0}^{t}\!\! \left( e^{-(t-s)A_n}\!F_n(s, u_0(s))\!-\!e^{-(t-s)A_0}\! F_0(s,u_0(s))\right) \!\!\mathrm {\,d}s\right\| _{\alpha }. \end{aligned}$$

This means that there are \({\tilde{a}}>0\) and \({\tilde{C}}>0\) such that, for all \(t\in (0,T]\) and \(n\ge 1\),

$$\begin{aligned} \Vert u_n(t)-u_0(t)\Vert _{\alpha }\le \gamma _n(t) + {\tilde{C}}\int _{0}^{t} e^{{\tilde{a}}(t-s)}(t-s)^{-\alpha } \Vert u_n(s)-u_0(s)\Vert _\alpha \mathrm {\,d}s. \end{aligned}$$

By use of Lemma 7.1.1 of [13], we get

$$\begin{aligned} \Vert u_n(t)-u_0(t)\Vert _\alpha \le \gamma _n(t) +K \int _{0}^{t} (t-s)^{-\alpha }\gamma _n(s)\mathrm {\,d}s \end{aligned}$$

for some constant \(K>0\). Now let us fix \(t \in [0,T]\) and take an arbitrary \(\delta \in (0, t)\). Observe also that

$$\begin{aligned} \int _{0}^{t} (t-s)^{-\alpha }\gamma _n(s)\mathrm {\,d}s\le & {} \frac{2^\alpha }{\delta ^\alpha } \int _{0}^{t-\delta /2} \gamma _n(s)\mathrm {\,d}s + \int _{t-\delta /2}^{t} (t-s)^{-\alpha }\gamma _n(s)\mathrm {\,d}s\\\le & {} \frac{2^\alpha }{\delta ^\alpha } \int _{0}^{T} \gamma _n(s)\mathrm {\,d}s + \frac{(\delta /2)^{1-\alpha }}{1-\alpha }\cdot \sup _{s\in [\delta /2,T]} \gamma _n(s). \end{aligned}$$

Since, in view of Lemma 3.4, \(\gamma _n(t)\rightarrow 0\) uniformly with respect to t from compact subsets of (0, T] and the functions \(\gamma _n\), \(n\ge 1\), are estimated from above by an integrable function we infer, by the dominated convergence theorem, that \(\Vert u_n(t)-u_0(t)\Vert _{\alpha }\rightarrow 0\) as \(n \rightarrow +\infty \) uniformly with respect to \(t\in [\delta ,T]\).

\(\square \)

The above theorem allows us to strengthen Henry’s averaging principle. We assume that mappings \(F_n:[0,+\infty )\times X^\alpha \rightarrow X\), \(n\ge 1\), satisfy (15) and (16) with common constants CL (independent of n) and that there exists \({\widehat{F}}: X^\alpha \rightarrow X\) such that, for all \({\bar{u}}\in X^\alpha \),

$$\begin{aligned} \lim _{\tau \rightarrow +\infty ,\, n\rightarrow +\infty } \frac{1}{\tau } \int _{0}^{\tau } F_n (t,{\bar{u}}) \mathrm {\,d}t = {\widehat{F}} ({\bar{u}}) \quad \text{ in }\ X. \end{aligned}$$
(21)

Theorem 3.5

Suppose \(F_n\) and \({\widehat{F}}\) are as above,  \({\bar{u}}_n \rightarrow {\bar{u}}\) in X\(\lambda _n \rightarrow 0^+\) as \(n\rightarrow +\infty ,\) and \(u_n: [0,+\infty ) \rightarrow X^\alpha ,\) \(n\ge 1,\) are solutions of

$$\begin{aligned} \left\{ \begin{array}{l} \dot{u}(t) = - A u(t) + F_n (t/\lambda _n,u(t)), \quad t>0,\\ u(0)={\bar{u}}_n. \end{array}\right. \end{aligned}$$

Then \(u_n(t)\rightarrow {\widehat{u}}(t)\) in \(X^\alpha \) uniformly with respect to t from compact subsets of \((0,+\infty )\) where \(\widehat{u}:[0,+\infty )\rightarrow X^\alpha \) is the solution of

$$\begin{aligned} \left\{ \begin{array}{l} \dot{u}(t)=-Au(t)+{\widehat{F}}(u(t)), \quad t>0,\\ u(0)={\bar{u}}. \end{array}\right. \end{aligned}$$

Proof

Let \({\tilde{F}}_n:=F_n(\cdot /\lambda _n,\cdot )\) and \({\tilde{F}}_0:={\widehat{F}}\). Observe that, using (21), we get, for any \({\bar{u}} \in X^\alpha \) and \(t>0\),

$$\begin{aligned} \int _{0}^{t} {\tilde{F}}_n (s,{\bar{u}})\mathrm {\,d}s= \lambda _n \int _{0}^{t/\lambda _n} F_n(\rho ,{\bar{u}}) \mathrm {\,d}\rho \rightarrow t{\widehat{F}}({\bar{u}}) = \int _{0}^{t} {\tilde{F}}_0 ({\bar{u}}) \mathrm {\,d}s \quad \text{ in } X, \text{ as } n\rightarrow +\infty . \end{aligned}$$

Clearly, \({\tilde{F}}_n\), \(n\ge 1\), and \({\tilde{F}}_0\) satisfy (15) and (16) with the common constants CL. It can be easily verified that the convergence above is uniform with respect to t from bounded subintervals of \([0,+\infty )\). Now, an application of Theorem 3.2 yields the assertion. \(\square \)

Remark 3.6

(a) The above result is an improvement of the continuation theorem and the Henry averaging principle [13, Th. 3.4.9] to the case when initial values from \(X^\alpha \) converge in the topology of X (not \(X^\alpha \)). This will appear crucial when establishing the ultimate compactness property of the Poincaré operator and verifying a priori estimates in the proofs of main results. We shall need to consider solutions in the phase space \(X^{1/2}=H^1({\mathbb {R}}^N)\) while the compactness of sequences of initial values is possible with respect to the \(L^2({\mathbb {R}}^N)\) topology only.

(b) An averaging principle for parabolic equations on \({\mathbb {R}}^N\) was also proved in [3] in a specific case, where the elliptic operator with time dependent coefficients was considered.

4 Continuity and averaging for parabolic equations

We transform (1) into an abstract evolution equation. To this end define an operator \(\mathbf{A}:D(\mathbf{A})\rightarrow X\) in the space \(X:=L^2({\mathbb {R}}^N)\) by

$$\begin{aligned} \mathbf{A} u:=-\sum _{i,j=1}^{N} a_{ij} \frac{\partial ^2 u}{\partial x_j \partial x_i} , \quad \text{ for } u\in D(\mathbf{A}):=H^2({\mathbb {R}}^N), \end{aligned}$$

where \(a_{ij}\in {\mathbb {R}}\), \(i,j=1,\ldots , N\), are such that

$$\begin{aligned} \sum _{i,j=1}^{N} a_{ij}\xi _i \xi _j > 0,\quad \text{ for } \text{ any } \xi \in {\mathbb {R}}^N, \end{aligned}$$

and \(a_{ij}=a_{ji}\) for \(i,j=1,\ldots ,N\). It is well-known that \(\mathbf{A}\) is a self-adjoint, positive and sectorial operator in \(L^2({\mathbb {R}}^N)\).

Suppose that f satisfies (3)–(5). Define \(\mathbf{F}:[0,+\infty ) \times H^1({\mathbb {R}}^N)\rightarrow L^2({\mathbb {R}}^2)\) by \([\mathbf{F}(t,u)](x):=f(t,x,u(x))\) for a.e. \(x\in {\mathbb {R}}^N\).

Lemma 4.1

Under the above assumptions there are constants \(D >0,\) depending only on k\({\tilde{k}},\) N and p\(L>0,\) depending only lN and p,  and \(C>0,\) depending only on \(m_0, l, N\) and p,  such that,  for all \(t_1, t_2 \ge 0\) and \(u_1, u_2\in H^1({\mathbb {R}}^N),\)

$$\begin{aligned}&\Vert \mathbf{F}(t_1,u_1)-\mathbf{F}(t_2, u_2)\Vert _{L^2} \le D(1+\Vert u_1\Vert _{H^1}) |t_1-t_2|^{\theta } + L \Vert u_1-u_2\Vert _{H^1}\quad \text{ and }\\&\Vert \mathbf{F}(t,u)\Vert _{L^2} \le C(1+\Vert u\Vert _{H^1}) \quad \text{ for } \text{ any } t\ge 0 \text{ and } u\in H^1({\mathbb {R}}^N). \end{aligned}$$

In the proof of Lemma 4.1 we shall use the following technical lemma (see, e.g. [8, Lemma 4.2]).

Lemma 4.2

There exist constants \(C_1=C_1(N,p)>0\) and \(C_2=C_2(N,p)>0\) such that for any \(u \in H^1({\mathbb {R}}^N)\)

$$\begin{aligned} \Vert u\Vert _{L^{2p/(p-1)}}\le C_1\Vert u\Vert _{H^1}, \end{aligned}$$
(22)

and

$$\begin{aligned} \Vert u\Vert _{L^{2p/(p-2)}}\le C_2 \Vert u\Vert _{H^1}. \end{aligned}$$
(23)

Proof of Lemma 4.1

By use of (4) and (5), the Hölder inequality and Lemma 4.2, one finds constants \(D=D(k,{\tilde{k}}, N, p)>0\) and \(L=L(l,N,p)>0\) such that, for any \(t_1,t_2\ge 0\) and \(u_1,u_2\in H^1({\mathbb {R}}^N)\),

$$\begin{aligned} \Vert \mathbf{F}(t_1,u_1)-\mathbf{F}(t_2, u_2)\Vert _{L^2}\le & {} (\Vert \tilde{k}\Vert _{L^2} + C_2\Vert k_{0}\Vert _{L^p}\Vert u_1\Vert _{H^1}+ \Vert k_{\infty }\Vert _{L^\infty }\Vert u_1\Vert _{L^2} )|t_1-t_2|^{\theta }\\&+ C_2\Vert l_0 (t_2,\cdot )\Vert _{L^p}\Vert u_1-u_2\Vert _{H^1} + \Vert l_\infty (t_2,\cdot )\Vert _{L^\infty }\Vert u_1-u_2\Vert _{L^2}\\\le & {} D(1+\Vert u_1\Vert _{H^1})|t_1-t_2|^{\theta } + L\Vert u_1-u_2\Vert _{H^1}. \end{aligned}$$

Furthermore, by (5), one also has \(|f(t,x,u)|\le |f(t,x,0)| + l(t,x)|u|\) for \(t\ge 0\)\(x\in {\mathbb {R}}^N, \, u\in {\mathbb {R}}.\) This, together with (3), gives the existence of \(C=C(m_0, l, N, p)>0\) such that

$$\begin{aligned} \Vert \mathbf{F}(t,u)\Vert _{L^2} \le \Vert m_0\Vert _{L^2} + C_2 \Vert l_0(t,\cdot )\Vert _{L^p} \Vert u\Vert _{H^1} + \Vert l_\infty (t,\cdot )\Vert _{L^\infty }\Vert u\Vert _{L^2} \le C (1+\Vert u\Vert _{H^1}) \end{aligned}$$

for any \(t\ge 0\) and \(u\in H^1({\mathbb {R}}^N)\). \(\square \)

Consider now the evolutionary problem

$$\begin{aligned} \dot{u}(t) = - \mathbf{A}u(t) +\mathbf{F} (t,u(t)), \, t\ge 0, \quad u(0) = {\bar{u}}\in H^1({\mathbb {R}}^N). \end{aligned}$$
(24)

Due to Lemma 4.1 and standard results in theory of abstract evolution equations (see [13] or [5]), the problem (24) admits a unique global solution \(u\in C([0,+\infty ),H^1({\mathbb {R}}^N))\) \(\cap \ \ C((0,+\infty ), H^2({\mathbb {R}}^N))\cap C^1((0,+\infty ), L^2({\mathbb {R}}^N))\). We shall say that \(u:[0,T_0) \rightarrow H^1({\mathbb {R}}^N)\), \(T_0>0\), is a solution ( \(H^1\) -solution) of

$$\begin{aligned} \left\{ \begin{array}{ll} {\frac{\partial u}{\partial t}} (x,t) = {{\mathcal {A}}} u(x,t)+f(t,x,u(x,t)), &{}\quad x\in {\mathbb {R}}^N, \, t\in (0, T_0),\\ u(x,0)={\bar{u}} (x), &{}\quad x\in {\mathbb {R}}^N, \end{array}\right. \end{aligned}$$

for some \({\bar{u}} \in H^1({\mathbb {R}}^N)\), where \({{\mathcal {A}}} = \sum _{i,j=1}^{N} a_{ij} \frac{\partial ^2 }{\partial x_j \partial x_i}\), if

$$\begin{aligned} u\in C([0,+\infty ),H^1({\mathbb {R}}^N))\cap C((0,+\infty ), H^2({\mathbb {R}}^N))\cap C^1((0,+\infty ), L^2({\mathbb {R}}^N)) \end{aligned}$$

and (24) holds. In this sense we have global in time existence and uniqueness of solutions for the parabolic partial differential equation.

The continuity of solutions properties are collected below.

Proposition 4.3

(Compare [24, Prop. 2.3]). Assume that functions \(f_n: [0,+\infty ) \times {\mathbb {R}}^N\times {\mathbb {R}}\rightarrow {\mathbb {R}},\) \(n\ge 0,\) satisfy the assumptions (3) with common \(m_0\) and (4) with common l and that \(f_n(t,x,u) \rightarrow f_0(t,x,u),\) for all \(t\ge 0,\) \(u\in {\mathbb {R}},\) a.e. \(x\in {\mathbb {R}}^N,\) and \(f_n (t,\cdot ,0)\rightarrow f_0(t,\cdot ,0)\) in \(L^2({\mathbb {R}}^N)\) for all \(t\ge 0\). Suppose that \((\alpha _n)\) is a sequence of positive numbers such that \(\alpha _n \rightarrow \alpha _0,\) as \(n\rightarrow +\infty ,\) for some \(\alpha _0>0.\) Let \(u_n:[0,T] \rightarrow H^1({\mathbb {R}}^N),\) \(n\ge 0,\) be a solution of

$$\begin{aligned} {\frac{\partial u}{\partial t}} (x,t) = \alpha _n {{\mathcal {A}}} u(x,t)+f_n (t,x,u(x,t)), \quad x\in {\mathbb {R}}^N, \ t\in (0, T], \end{aligned}$$

such that, for some \(R >0\), \(\Vert u_n (t)\Vert _{H^1} \le R\), for all \(t\in [0,T]\) and \(n\ge 0\). Then \(f_n(t,\cdot , u(\cdot ))\rightarrow f_0(t,\cdot , u(\cdot ))\) in \(L^2({\mathbb {R}}^N)\) for any \(u\in H^1({\mathbb {R}}^N)\) and \(t\ge 0\) and

  1. (i)

    if \(u_n(0)\rightarrow u_0(0)\) in \(L^2({\mathbb {R}}^N)\) as \(n\rightarrow \infty ,\) then \(u_n(t)\rightarrow u(t)\) in \(H^1({\mathbb {R}}^N)\) for t from compact subsets of (0, T].

  2. (ii)

    if \(u_n(0)\rightarrow u_0(0)\) in \(H^1({\mathbb {R}}^N)\) as \(n\rightarrow \infty ,\) then \(u_n(t)\rightarrow u_0(t)\) in \(H^1({\mathbb {R}}^N)\) uniformly for \(t\in [0,T]\).

Proof

Define \(\mathbf{F}_n:[0,+\infty )\times H^1({\mathbb {R}}^N)\rightarrow L^2({\mathbb {R}}^N)\), \(n\ge 0\), by \([\mathbf{F}_n(t,u)](x):=f_n(t,x,u(x))\). Note that, in view of (4), for any \(t\ge 0\), \(u\in H^1({\mathbb {R}}^N)\) and a.e. \(x\in {\mathbb {R}}^N\),

$$\begin{aligned} |f_n(t,x,u(x))\!-\!f_0(t,x,u(x))|^2 \!\le \! 2|f_n(t,x,0)-f_0(t,x,0)|^2\! \!+\! 4 |l(x,t)u|^2. \end{aligned}$$

Since, for any \(t\ge 0\), \(f_n(t,\cdot ,0) \rightarrow f_0(t,\cdot ,0)\) in \(L^2({\mathbb {R}}^N)\) as \(n\rightarrow +\infty \), the right hand side can be estimated by an integrated function, which due to the Lebesgue dominated convergence theorem implies \(\mathbf{F}_n(t,u)\rightarrow \mathbf{F}_0 (t,u)\) in \(L^2({\mathbb {R}}^N)\) as \( n \rightarrow +\infty \). Moreover, by use of Lemma 4.1, we may pass to the limit under the integral to get \(\int _{0}^{t} \mathbf{F}_n (s, u)\mathrm {\,d}s \rightarrow \int _{0}^{t} \mathbf{F}_0 (s,u)\mathrm {\,d}s \ \text{ in } \ L^2({\mathbb {R}}^N)\) for any \(u\in H^1({\mathbb {R}}^N)\) and \(t\ge 0\). This in view of Theorem 3.2 implies the assertion (i). The assertion (ii) comes from the standard continuity theorem from [13]. \(\square \)

Let us also state an averaging principle.

Proposition 4.4

Assume that functions \(f_n: [0,+\infty ) \times {\mathbb {R}}^N\times {\mathbb {R}}\rightarrow {\mathbb {R}},\) \(n\ge 0,\) satisfy the assumptions of Proposition 4.3 and additionally (2). Suppose that \({\bar{u}}_n \rightarrow {\bar{u}}_0\) in \(L^2({\mathbb {R}}^N),\) \(\lambda _n\rightarrow 0^+\) as \(n\rightarrow +\infty \) and that \(u_n: [0,+\infty ) \rightarrow H^1({\mathbb {R}}^N),\) \(n\ge 1,\) are solutions of

$$\begin{aligned} \left\{ \begin{array}{ll} {\frac{\partial u}{\partial t}} = {{\mathcal {A}}} u+f_n(t/\lambda _n,x,u), &{}\quad x\in {\mathbb {R}}^N, \, t>0,\\ u(x,0)={\bar{u}}_n (x), &{}\quad x\in {\mathbb {R}}^N. \end{array}\right. \end{aligned}$$

Then \(u_n(t)\rightarrow {\widehat{u}}(t)\) in \(H^1({\mathbb {R}}^N)\) uniformly on compact subsets of \((0,+\infty ),\) where \({\widehat{u}}:[0,+\infty )\rightarrow H^1({\mathbb {R}}^N)\) is the solution of

$$\begin{aligned} \left\{ \begin{array}{ll} {\frac{\partial u}{\partial t}} = {{\mathcal {A}}} u+{\widehat{f}}_0 (x,u), &{}\quad x\in {\mathbb {R}}^N, \, t>0,\\ u(x,0)={\bar{u}}_0 (x), &{}\quad x\in {\mathbb {R}}^N, \end{array}\right. \end{aligned}$$

with \({\widehat{f}}_0:{\mathbb {R}}^N \times {\mathbb {R}}\rightarrow {\mathbb {R}}\) given by \({\widehat{f}}_0 (x,u):=\frac{1}{T}\int _{0}^{T} f_0(t,x,u)\mathrm {\,d}t\) for all \(u\in {\mathbb {R}}\) and a.e. \(x\in {\mathbb {R}}^N\).

Proof

Define \({\mathbf{F}_n}:[0,+\infty )\times H^1({\mathbb {R}}^N) \rightarrow L^2({\mathbb {R}}^N)\), \(n\ge 0\), by \([\mathbf{F}_n(t,u)](x):=f_n(t,x,u(x))\) and \(\widehat{\mathbf{F}}_0: H^1({\mathbb {R}}^N)\rightarrow L^2({\mathbb {R}}^N)\) by \(\widehat{\mathbf{F}}_0 (u): = \frac{1}{T}\int _{0}^{T} \mathbf{F}_0(t,u)\mathrm {\,d}t\). Clearly, for all \(u\in H^1({\mathbb {R}}^N)\), \(\widehat{\mathbf{F}}(u)(x) = {\widehat{f}}(x,u(x))\) for a.e. \(x\in {\mathbb {R}}^N\). Fix any \({\bar{u}}\in H^1({\mathbb {R}}^N)\) and \((\tau _n)\) in \((0,+\infty )\) such that \(\tau _n\rightarrow +\infty \). Clearly, \(\mathbf{F}_n\), \(n\ge 1\), are T-periodic in time. Consequently, one has

$$\begin{aligned} I_n:=\frac{1}{\tau _n} \int _{0}^{\tau _n} \mathbf{F}_n (t,{\bar{u}}) \mathrm {\,d}t = \frac{[\tau _n/T]}{\tau _n/T} \cdot \frac{1}{T} \int _{0}^{T} \mathbf{F}_n (t,{\bar{u}})\mathrm {\,d}t+ \frac{1}{\tau _n}\int _{0}^{\tau _n-[\tau _n/T]T} \mathbf{F}_n (t,{\bar{u}})\mathrm {\,d}t. \end{aligned}$$

Hence, to see that \(I_n\rightarrow \widehat{\mathbf{F}}_0({\bar{u}})\) it is sufficient to prove that

$$\begin{aligned} I_n^{(T)}:=\frac{1}{T}\int _{0}^{T} \mathbf{F}_n (t,{\bar{u}})\mathrm {\,d}t \rightarrow \widehat{\mathbf{F}}_0 ({\bar{u}}) \ \text{ in } L^2({\mathbb {R}}^N), \ \text{ as } n\rightarrow +\infty . \end{aligned}$$

To this end observe that, for a.e. \(x\in {\mathbb {R}}^N\),

$$\begin{aligned} I_{n}^{(T)} (x) = \frac{1}{T} \int _{0}^{T} \!\! f_n(t,x,{\bar{u}}(x)) \mathrm {\,d}t\rightarrow \frac{1}{T} \int _{0}^{T}\!\! f_0 (t,x,{\bar{u}}(x)) \mathrm {\,d}t = {\widehat{f}}_0(x,u(x))= [\widehat{\mathbf{F}}_0 ({\bar{u}})](x). \end{aligned}$$

Moreover, by use of the assumptions on \(f_n\)’s, one has

$$\begin{aligned} |I_{n}^{(T)} (x)|= & {} \left| \frac{1}{T} \int _{0}^{T} f_n (t,x,\bar{u}(x)) \mathrm {\,d}t\right| \le m_0 (x)+ g(x), \end{aligned}$$

where \(g(x):= \frac{1}{T} \int _{0}^{T} |l (t,x)||{\bar{u}}(x)| \mathrm {\,d}t\). and, by use of Jensen’s inequality,

$$\begin{aligned} \int _{{\mathbb {R}}^N} |g(x)|^2 \mathrm {\,d}x \le \frac{1}{T} \int _{{\mathbb {R}}^N}\int _{0}^{T} |l(t,x)|^2 |{\bar{u}}(x)|^2 \mathrm {\,d}t \mathrm {\,d}x <+\infty \end{aligned}$$

(see the proof of Lemma 4.1). Hence, by the dominated convergence theorem we infer that \(I_n^{(T)}\rightarrow \widehat{\mathbf{F}}_0({\bar{u}})\) in \(L^2({\mathbb {R}}^N)\). Since \((\tau _n)\) was arbitrary it follows that

$$\begin{aligned} \lim _{\tau \rightarrow +\infty ,\ n\rightarrow +\infty } \frac{1}{\tau }\int _{0}^{\tau } \mathbf{F}_n (t,{\bar{u}})\mathrm {\,d}t \rightarrow \widehat{\mathbf{F}}_0 ({\bar{u}}). \end{aligned}$$

Finally, we get the assertion by use of Theorem 3.5. \(\square \)

5 Ultimate compactness of Poincaré operator

We start the section with the following version of standard tail estimates (see [24]).

Lemma 5.1

Assume that \(f:[0,+\infty )\times {\mathbb {R}}^N\times {\mathbb {R}}\rightarrow {\mathbb {R}}\) satisfies (3)–(5) and

$$\begin{aligned} (f(t,x,u)-f(t,x,v))\cdot (u-v) \le a(x)|u-v|^2. \end{aligned}$$
(25)

where a is of Rellich–Kato type and

$$\begin{aligned} {\bar{a}}: = \lim _{r \rightarrow +\infty } \mathop {\mathrm {essup}}\limits _{|x|>r} a_\infty (x) < 0. \end{aligned}$$
(26)

Suppose that \(u:[0,T]\rightarrow H^{1}({\mathbb {R}}^N)\) is a solution of

$$\begin{aligned} {\frac{\partial u}{\partial t}} = \Delta u + f(t,x,u), \quad x\in {\mathbb {R}}^N, t\ge 0, \end{aligned}$$
(27)

such that \(\Vert u(t)\Vert _{H^1}\le R\) for all \(t\in [0,T]\). Then,  for any \(\gamma \in (0,|{\bar{a}}|)\) there exists a sequence \((\alpha _n)\) with \(\alpha _n\rightarrow 0^+\) as \(n\rightarrow \infty \) such that

$$\begin{aligned} \int _{{\mathbb {R}}^N {\setminus } B(0,n)} |u(t)|^2 \mathrm {\,d}x \le R^2 e^{- 2 \gamma t} + \alpha _n \quad \text{ for } \text{ all } t\in [0,T], \text{ and } n\ge 1, \end{aligned}$$

where \(\alpha _n's\) depend only on \(N, p, R, m_0, a, \gamma \) and \(a_{ij}'s\).

Proof

It goes along the lines of [24, Prop. 2.2]. The only difference is that here we have the modified dissipativity condition (25) that implies

$$\begin{aligned} f(t,x,u)u\le a(x)|u|^2 + f(t,x,0)u \end{aligned}$$

for all \(t\ge 0\), \(u\in {\mathbb {R}}\) and a.e. \(x\in {\mathbb {R}}^N\). Hence, there exists \(r=r_\gamma > 0\) such that

$$\begin{aligned} f(t,x,u)u \le -\gamma |u|^2 + a_0 (x) |u|^2 + m_0(x)u \end{aligned}$$

for all \(t\ge 0\), \(u\in {\mathbb {R}}\) and a.e. \(x\in {\mathbb {R}}^N {\setminus } B(0,r)\). Therefore, one only needs to modify the proof of [24, Prop. 2.2] (compare also the proof of Lemma 5.3).

\(\square \)

Remark 5.2

Lemma 5.1 is sufficient to show the asymptotic compactness or Conley index admissibility of the parabolic semi-flow (when f is independent of time). However, to prove the ultimate compactness of the Poincareé operator we shall need tail estimates involving any two solutions of the parabolic equations as well as parameter dependence.

Suppose that \(a_{ij}\in C([0,1],{\mathbb {R}})\), \(i,j=1,\ldots , N\), are such that

$$\begin{aligned} \sum _{i,j=1}^{N} a_{ij}(\mu )\xi _i \xi _j >0 \quad \text{ for } \text{ any } \xi \in {\mathbb {R}}^N \text{ and } \mu \in [0,1] \end{aligned}$$

and consider \(h:[0,+\infty )\!\times \! {\mathbb {R}}^N\! \times \! {\mathbb {R}}\! \times \! [0,1]\rightarrow {\mathbb {R}}\) such that, for all \(t,s \ge 0\), \(u,v\in {\mathbb {R}}\), \(\mu ,\nu \in [0,1]\) and a.e. \(x\in {\mathbb {R}}^N\),

$$\begin{aligned}&h(t,\cdot ,u,\mu ) \text{ is } \text{ measurable } \text{ and } |h(t,x,0,\mu )| \le m_0(x), \end{aligned}$$
(28)
$$\begin{aligned}&|h(t,x,u,\mu )-h(s,x,v,\mu )| \le ({\tilde{k}} (x) + k(x)|u|)|t-s|^{\theta } + l(s,x)|u-v|, \end{aligned}$$
(29)
$$\begin{aligned}&|h(t,x,u,\mu ) - h(t,x,u,\nu )| \le l(t,x)\, |u|\, |\rho (\mu )-\rho (\nu )|, \end{aligned}$$
(30)
$$\begin{aligned}&(h(t,x,u,\mu )-h(t,x,v,\mu )) \cdot (u-v) \le a(x) |u-v|^2, \end{aligned}$$
(31)

where \(m_0\in L^2({\mathbb {R}}^N)\), \(\theta \in (0,1)\), \({\tilde{k}} \in L^2({\mathbb {R}}^N)\), k is of Rellich–Kato type, \(l=l_0+l_\infty \) with \(l_\infty \in L^\infty ([0,+\infty ) \times {\mathbb {R}}^N)\), and \(\sup _{t\ge 0} \, \Vert l_0 (t,\cdot ) \Vert _{L^p} <+\infty \), \(\rho \in C([0,1],{\mathbb {R}})\) and a is of Rellich–Kato type satisfying (26).

Let \(\mathbf{A}^{(\mu )}: D(\mathbf{A}^{(\mu )}) \rightarrow L^2({\mathbb {R}}^N)\), \(\mu \in [0,1]\), be given by

$$\begin{aligned} \mathbf{A}^{(\mu )} u:= - \sum _{i,j=1}^{N} a_{ij}(\mu ) \frac{\partial ^2 u}{\partial x_j\partial x_i}, \ u\in D(\mathbf{A}^{(\mu )}):=H^2({\mathbb {R}}^N), \end{aligned}$$

and \(\mathbf{H}:[0,+\infty ) \times H^1 ({\mathbb {R}}^N) \times [0,1] \rightarrow L^2 ({\mathbb {R}}^N)\) be defined by

$$\begin{aligned}{}[\mathbf{H}(t,u,\mu )](x):= h(t,x, u(x),\mu ) \text{ for } t\ge 0,\, u\in H^{1}({\mathbb {R}}^N),\, \mu \in [0,1], \text{ a.e. } x\in {\mathbb {R}}^N. \end{aligned}$$

We shall consider

$$\begin{aligned} \dot{u} (t) = - \mathbf{A}^{(\mu )} u(t) + \mathbf{H}(t,u(t),\mu ), \ t>0. \end{aligned}$$
(32)

Clearly, due to Lemma 4.1, we get the existence and uniqueness of solutions on \([0,+\infty )\). Denote by \(u(\cdot ; \bar{u}, \mu )\) the solution of (32) satisfying the initial value condition \(u(0)={\bar{u}}\). The following tail estimates will be crucial in studying the compactness properties of the translation along trajectories operator of (32).

Lemma 5.3

Take any \({\bar{u}}_1, {\bar{u}}_2\in H^{1}({\mathbb {R}}^N)\) and \(\mu _1, \mu _2\in [0,1]\) and suppose that there are solutions \(u(\cdot ;\bar{u}_i,\mu _i):[0,T]\rightarrow H^1({\mathbb {R}}^N),\) \(i=1,2,\) of (32), for some fixed \(T>0\). If \(\Vert u(t;{\bar{u}}_1,\mu _1)\Vert _{H^1}\le R\) and \(\Vert u(t;{\bar{u}}_2,\mu _2)\Vert _{H^1} \le R\) for all \(t\in [0,T]\) and some fixed \(R>0,\) then,  for any \(\gamma \in (0,|{\bar{a}}|)\) there exists a sequence \((\alpha _n)\) with \(\alpha _n\rightarrow 0^+\) as \(n\rightarrow \infty \) such that

$$\begin{aligned} \int _{{\mathbb {R}}^N{\setminus } B(0,n)} |u (t;{\bar{u}}_1,\mu _1)-u (t;\bar{u}_2,\mu _2)|^2 \mathrm {\,d}x \le e^{-2 \gamma t} \Vert {\bar{u}}_1 - {\bar{u}}_2 \Vert _{L^2}^{2} + Q \eta (\mu _1,\mu _2) + \alpha _n, \end{aligned}$$

for all \(t\in [0,T]\) and \(n\ge 1,\) where \(\alpha _n\ge 0\) and \(Q>0\) depend only on \(N, p, R, l, a, \gamma \) and \(a_{ij}'s,\)

$$\begin{aligned} \eta (\mu _1,\mu _2):= \max \left\{ |\rho (\mu _1)-\rho (\mu _2)|, \max _{i,j=1,\ldots , N} |a_{ij} (\mu _1) - a_{ij} (\mu _2)| \right\} . \end{aligned}$$

Proof

Let \(\phi :[0,+\infty )\rightarrow {\mathbb {R}}\) be a smooth function such that \(\phi (s)\in [0,1]\) for \(s\in [0,+\infty )\), \(\phi _{|[0,\frac{1}{2}]}\equiv 0\) and \(\phi _{|[1,+\infty )}\equiv 1\) and let \(\phi _n:{\mathbb {R}}^N\rightarrow {\mathbb {R}}\) be defined by \(\phi _n (x):=\phi (|x|^2/n^2)\), \(x\in {\mathbb {R}}^N\). Put \(u_1:=u(\cdot ; \bar{u}_1,\mu _1)\), \(u_2:=u(\cdot ; {\bar{u}}_2,\mu _2)\) and \(v:=u_1-u_2\). Observe that

$$\begin{aligned} \frac{1}{2}\frac{\mathrm {\,d}}{\mathrm {\,d}t} (v(t), \phi _n v(t))_{0}= & {} \frac{1}{2}\left( (v(t),\phi _n \dot{v}(t))_{0} + (\dot{v}(t), \phi _n v(t))_{0} \right) = (\phi _n v(t),\dot{v}(t)))_{0} \\= & {} I_1(t)+I_2(t) + I_3(t) \end{aligned}$$

where

$$\begin{aligned}&I_1 (t)&:= ( \phi _n v(t), - \mathbf{A}^{(\mu _1)}u_1(t) + \mathbf{A}^{(\mu _1)}u_2(t) )_0,\\&I_2 (t)&:= ( \phi _n v(t), -\mathbf{A}^{(\mu _1)}u_2(t) + \mathbf{A}^{(\mu _2)}u_2(t) )_0,\\&I_3 (t)&:= ( \phi _n v(t), \mathbf{H} (t, u_1(t), \mu _1) - \mathbf{H} (t, u_2(t), \mu _2) )_0. \end{aligned}$$

As for the first term we notice that

$$\begin{aligned} I_1(t)= & {} ( \phi _n v(t), - \mathbf{A}^{(\mu _1)} v(t) )_0 \\= & {} -\int _{{\mathbb {R}}^N}\sum _{i,j=1}^{N} a_{ij}(\mu _1) \frac{\partial }{\partial x_j}(\phi _n (x) v(t)) \frac{\partial }{\partial x_i}(v(t)) \mathrm {\,d}x\\= & {} - \int _{{\mathbb {R}}^N} \phi _n (x) \sum _{i,j=1}^{N} a_{ij}(\mu _1) \frac{\partial }{\partial x_j}( v(t)) \frac{\partial }{\partial x_i}(v(t)) \mathrm {\,d}x \\&-\frac{2}{n^2} \int _{{\mathbb {R}}^N} \sum _{i,j=1}^{N} \phi '(|x|^2/n^2)v(t) x_j a_{ij}(\mu _1) \frac{\partial }{\partial x_i} (v(t)) \mathrm {\,d}x\\\le & {} \frac{2 L_\phi }{n^2} \int _{\big \{\frac{\sqrt{2}}{2}n\le |x|\le n\big \}} \sum _{i,j=1}^{N} a_{ij}(\mu _1) |x| |v(t)||\nabla _x v(t)|\mathrm {\,d}x\\\le & {} \frac{2 L_\phi M N^2}{n} \Vert v(t)\Vert _{L^2}\Vert v(t)\Vert _{H^1} \end{aligned}$$

where \(L_\phi := \sup _{s\in [0,+\infty )} |\phi '(s)|< \infty \) (as \(\phi '\) is smooth and nonzero on a bounded interval) and \(M:=\max _{1\le i,j \le N, \, \mu \in [0,1]} |a_{ij}(\mu )|\). Further, in a similar manner

$$\begin{aligned} I_2(t)= & {} - \int _{{\mathbb {R}}^N}\sum _{i,j=1}^{N} \frac{\partial }{ \partial x_j} (\phi _n v(t)) (a_{ij}(\mu _1)-a_{ij}(\mu _2)) \frac{\partial }{\partial x_i }(u_2(t)) \mathrm {\,d}x \\= & {} - \int _{{\mathbb {R}}^N} \phi _n (x) \sum _{i,j=1}^{N} (a_{ij}(\mu _1)- a_{ij}(\mu _2)) \frac{\partial }{\partial x_j}( v (t) ) \frac{\partial }{\partial x_i}( u_2(t) ) \mathrm {\,d}x\\&-\frac{2}{n^2} \int _{{\mathbb {R}}^N} \sum _{i,j=1}^{N} \phi '(|x|^2/n^2) v(t) x_j (a_{ij}(\mu _1)- a_{ij}(\mu _2)) \frac{\partial }{\partial x_i} (u_2 (t)) \mathrm {\,d}x\\\le & {} \eta (\mu _1,\mu _2) \Vert v(t)\Vert _{H^1} \Vert u_2(t)\Vert _{H^1} + \frac{4 L_\phi \eta (\mu _1,\mu _2) N^2}{n} \Vert v(t)\Vert _{L^2} \Vert u_2(t)\Vert _{H^1}. \end{aligned}$$

To estimate \(I_3 (t)\), take \(r = r_\gamma >0\) such that

$$\begin{aligned} ( h(t,x,u,\mu ) - h(t,x,v,\mu ) ) \cdot (u-v) \le -\gamma |u-v|^2 + a_0 (x) |u-v|^2 \end{aligned}$$

for all \(t\ge 0\), \(u,v\in {\mathbb {R}}\), \(\mu \in [0,1]\), and a.e. \(x\in {\mathbb {R}}^N{\setminus } B(0,r)\). Therefore, for sufficiently large n, we have

$$\begin{aligned} I_3(t)= & {} \! \int _{{\mathbb {R}}^N}\!\! \phi _n(x) \left( \mathbf{H} (t,u_1(t),\mu _1) \! - \! \mathbf{H} (t, u_2(t), \mu _2 ) \right) v(t) \mathrm {\,d}x \\\le & {} \int _{{\mathbb {R}}^N} \phi _n(x) \left( \mathbf{H}(t,u_1(t),\mu _1)\!-\! \mathbf{H}(t, u_2(t),\mu _1)\right) v(t) \mathrm {\,d}x \\&+ \int _{{\mathbb {R}}^N} \phi _n(x) \left( \mathbf{H}(t,u_2(t),\mu _1)\!-\! \mathbf{H}(t, u_2(t),\mu _2) \right) v(t) \mathrm {\,d}x \\\le & {} - \gamma \int _{{\mathbb {R}}^N}\phi _n(x)|v(t)|^2 \mathrm {\,d}x + \int _{{\mathbb {R}}^N}\!\!\!\phi _n(x) a_0 (x)|v(t)|^2 \mathrm {\,d}x \\&+\, \int _{{\mathbb {R}}^N} l(t,x) |\rho (\mu _1)-\rho (\mu _2)| |u_2(t)| |v(t)| \mathrm {\,d}x. \end{aligned}$$

By use of the Hölder inequality together with Lemma 4.2 one can get, for sufficiently large n,

$$\begin{aligned} I_3(t)\le & {} - \gamma \int _{{\mathbb {R}}^N}\phi _n(x)|v(t)|^2 \mathrm {\,d}x + C_1\Vert v(t)\Vert _{H^1}^2 \bigg (\int _{\big \{|x|\ge \frac{\sqrt{2}}{2}a_0\big \}}a_0(x)^p\mathrm {\,d}x\bigg )^{1/p} \nonumber \\&+ \eta (\mu _1,\mu _2)(C_2\Vert l_0(t,\cdot )\Vert _{L^p}\Vert u_2(t)\Vert _{H^1} +\Vert l_{\infty }(t,\cdot )\Vert _{L^\infty }\Vert u_2(t)\Vert _{L^2})\Vert v(t)\Vert _{L^2}. \nonumber \\ \end{aligned}$$
(33)

Hence we get, for sufficiently large \(n\ge 1\),

$$\begin{aligned} \frac{\mathrm {\,d}}{\mathrm {\,d}t} (v(t), \phi _n v(t))_{0} \le - 2 \gamma (v(t), \phi _n v(t))_{0} + {\tilde{C}}\eta (\mu _1,\mu _2) + \alpha _n \end{aligned}$$

for some constant \({\tilde{C}}={\tilde{C}}(l, p, N, R)>0\), where \((\alpha _n)_{n \in \mathbb {N}}\) is a sequence such that \(\alpha _n \rightarrow 0\) as \(n \rightarrow +\infty \). Multiplying by \(e^{2\gamma t}\) and integrating over \([0,\tau ]\) one obtains

$$\begin{aligned} e^{2\gamma \tau }(v(\tau ), \phi _n v(\tau ))_{0}- (v(0), \phi _n v(0))_{0} \le (2\gamma )^{-1}(e^{2\gamma \tau }-1)\, ({\tilde{C}} \eta (\mu _1,\mu _2) +\alpha _n), \end{aligned}$$

which gives

$$\begin{aligned} (v(\tau ), \phi _n v(\tau ))_{0} \le e^{-2\gamma \tau }\Vert v(0)\Vert ^2_{L^2} + (2\gamma )^{-1} \left( {\tilde{C}} \eta (\mu _1,\mu _2)+\alpha _n\right) . \end{aligned}$$

And this finally implies the assertion as \(\Vert \phi _n v(\tau )\Vert _{L^2}^{2}\le (v(\tau ), \phi _n v(\tau ))_{0}\). \(\square \)

Let \({\varvec{\Psi }}_t: H^1({\mathbb {R}}^N)\times [0,1]\rightarrow H^1({\mathbb {R}}^N)\), \(t>0\), be the translation operator for (32), i.e. \({\varvec{\Psi }}_t ({\bar{u}},\mu ) = u (t;{\bar{u}},\mu )\) for \({\bar{u}}\in H^1({\mathbb {R}}^N)\) and \(\mu \in [0,1]\).

Proposition 5.4

Suppose that (28)–(31) are satisfied.

  1. (i)

    For any bounded \(W\subset H^1({\mathbb {R}}^N)\) and \(t>0,\) \(\beta _{L^2} ({\varvec{\Psi }}_t(W\times [0,1])) \le e^{-|{\bar{a}}|t} \beta _{L^2}(W);\)

  2. (ii)

    If a bounded \(W\subset H^1({\mathbb {R}}^N)\) is relatively compact as a subset of \(L^2({\mathbb {R}}^N),\) then \({\varvec{\Psi }}_t (W\times [0,1])\) is relatively compact in \(H^1({\mathbb {R}}^N);\)

  3. (iii)

    If \(W\subset \overline{\mathrm {conv}}^{H^1} {\varvec{\Psi }}_t(W\times [0,1])\) for some bounded \(W\subset H^{1}({\mathbb {R}}^N)\) and \(t>0,\) then W is relatively compact in \(H^1({\mathbb {R}}^N)\).

Proof

(i) Observe that, for each \(n\ge 1\),

$$\begin{aligned} {\varvec{\Psi }}_t(W\times [0,1])\subset \{u(t;{\bar{u}}, \mu )\mid \bar{u}\in W, \, \mu \in [0,1]\} \subset W_n +R_n \end{aligned}$$

where \(W_n:=\{\chi _n u(t;{\bar{u}},\mu ) \mid {\bar{u}}\in W,\,\mu \in [0,1]\}\) and \(R_n:=\{ (1-\chi _n) u(t;{\bar{u}},\mu ) \mid {\bar{u}}\in W,\,\mu \in [0,1]\}\) where \(\chi _n\) is the characteristic function of the ball B(0, n). Note that \(W_n\) may be viewed as a subset of \(H^1(B(0,n))\). Therefore, due to the Rellich–Kondrachov theorem, \(W_n\) is relatively compact in \(L^2({\mathbb {R}}^N)\). Hence

$$\begin{aligned} \beta _{L^2}({\varvec{\Psi }}_t (W\times [0,1])) \le \beta _{L^2}(R_n), \quad \text{ for } \text{ all }\ n\ge 1. \end{aligned}$$
(34)

Now we need to estimate the measure of noncompactness of \(R_n\) in \(L^2({\mathbb {R}}^N)\). To this end fix an arbitrary \(\varepsilon >0\). Choose a finite covering of W consisting of balls \(B_{L^2} ({\bar{u}}_k, r_\varepsilon )\), \(k=1,\ldots , m_\varepsilon \), with \(r_\varepsilon :=\beta _{L^2} (W)+\varepsilon \) and such that \({\bar{u}}_k \in W\) for each \(k=1,\dots , m_{\varepsilon }\) and cover [0, 1] with intervals \((\mu _l-\delta , \mu _l+\delta )\), \(l=1,\ldots , n_\delta \) where \(\delta >0\) is such that \(\eta (\mu _1, \mu _2)<\varepsilon \) whenever \(|\mu _1-\mu _2|<\delta \). Put \({\bar{u}}_{k,l}:=(1-\chi _n) u(t;{\bar{u}}_k,\mu _l)\), \(k=1,\ldots ,m_\varepsilon \), \(l=1,\ldots , n_\delta \).

Now take any \({\bar{v}}\in R_n\). There are \({\bar{u}}\in W\) and \(\mu \in [0,1]\) such that \({\bar{v}} = (1-\chi _n) u(t;{\bar{u}},\mu )\). Clearly there exist \(k_0\in \{ 1,\ldots , m_\varepsilon \}\) and \(l_0\in \{1,\ldots , n_\delta \}\) such that \(\Vert {\bar{u}} - \bar{u}_{k_0}\Vert <r_\varepsilon \) and \(|\mu -\mu _{l_0}|<\delta .\) In view of Lemma 5.3, for fixed \(\gamma \in (0,|{\bar{a}}|)\) and all n,

$$\begin{aligned} \Vert {\bar{v}} - {\bar{u}}_{k_0, l_0} \Vert _{L^2}^{2}= & {} \int _{{\mathbb {R}}^N{\setminus } B(0,n)} |u(t;{\bar{u}},\mu )- u(t;\bar{u}_{k_0},\mu _{l_0} )|^2 \mathrm {\,d}x \\\le & {} e^{-2 \gamma t} \Vert {\bar{u}}-{\bar{u}}_{k_0} \Vert _{L^2}^{2} + Q\, \eta (\mu ,\mu _{l_0}) + \alpha _n \\\le & {} r_{\varepsilon ,n}:= e^{-2\gamma t}r_\varepsilon ^2 +Q\, \varepsilon + \alpha _n, \end{aligned}$$

which means that \(R_n\) is covered by the balls \(B_{L^2}(\bar{u}_{k,l},\sqrt{r_{\varepsilon ,n}})\), \(k=1,\ldots ,m_\varepsilon \), \(l=1,\ldots , n_\delta \). This means that \(\beta _{L^2}(R_n) \le \sqrt{r_{\varepsilon ,n}}\) for any \(\varepsilon >0\), and, in consequence, \(\beta _{L^2}(R_n)\le (e^{-2\gamma t}(\beta _{L^2}(W))^2 +\alpha _n)^{1/2}\). Using (34) we get

$$\begin{aligned} \beta _{L^2} ( {\varvec{\Psi }}_t(W\times [0,1]) ) \le (e^{-2\gamma t}(\beta _{L^2}(W))^2 +\alpha _n)^{1/2},\quad \text{ for } n\ge 1. \end{aligned}$$

Finally, by a passage to the limit with \(n\rightarrow \infty \) and then another one with \(\gamma \rightarrow {\bar{a}}\), we obtain the required inequality.

(ii) Take any \(({\bar{u}}_n)\) in W and \((\mu _n)\) in [0, 1]. We may assume that \(\mu _n \rightarrow \mu _0\) for some \(\mu _0 \in [0,1]\), as \(n \rightarrow +\infty \). Since \(({\bar{u}}_n)\) is bounded, by the Banach–Alaoglu theorem, we may suppose that \(({\bar{u}}_n)\) converges weakly in \(H^{1}({\mathbb {R}}^N)\) to some \({\bar{u}}\in H^{1}({\mathbb {R}}^N)\). By the relative compactness of W in \(L^2({\mathbb {R}}^N)\) we may assume that \(\bar{u}_n\rightarrow {\bar{u}}\) in \(L^2({\mathbb {R}}^N)\). Therefore, by use of Proposition 4.3, one has \({\varvec{\Psi }}_t ({\bar{u}}_n, \mu _n) \rightarrow {\varvec{\Psi }}_t({\bar{u}}, \mu _0) \) in \(H^1({\mathbb {R}}^N)\), which ends the proof.

(iii) Observe that here, by use of (i), one gets

$$\begin{aligned} \beta _{L^2}(W) \le \beta _{L^2} ({\varvec{\Psi }}_t(W\times [0,1])) \le e^{-|{\bar{a}}| t} \beta _{L^2}(W). \end{aligned}$$

This implies \(\beta _{L^2}(W)=0\), i.e. that W is relatively compact in \(L^2({\mathbb {R}}^N)\). To see that W is relatively compact in \(H^1({\mathbb {R}}^N)\) observe that, by (ii), \({\varvec{\Psi }}_t(W\times [0,1])\) is relatively compact in \(H^{1}({\mathbb {R}}^N)\). \(\square \)

6 Averaging index formula

Consider the following parameterized equation

$$\begin{aligned} {\frac{\partial u}{\partial t}} (x,t) = \Delta u (x,t) + h(t/\lambda ,x,u(x,t),\mu ), \quad t>0, \, x\in {\mathbb {R}}^N, \end{aligned}$$
(35)

where h is as in the previous section and \(\lambda >0\). Combining the compactness result with the averaging principle we get the following result.

Lemma 6.1

Suppose h satisfies conditions (28)–(31) and is T-periodic in the time variable \((T>0).\) If \(({\bar{u}}_n)\) is a bounded sequence in \(H^1({\mathbb {R}}^N),\) \((\mu _n)\) in [0, 1],  \((\lambda _n)\) in \((0,+\infty )\) with \(\lambda _n \rightarrow 0^+\) as \(n\rightarrow +\infty \) and \(u_n: [0,+\infty ) \rightarrow H^1({\mathbb {R}}^N),\) \(n\ge 1,\) are solutions of (35) with \(\lambda =\lambda _n,\) \(\mu =\mu _n\) such that \(u_n(0)=u_n(\lambda _n T)= {\bar{u}}_n,\) then there are a subsequence \(({\bar{u}}_{n_k})\) of \(({\bar{u}}_n)\) converging in \(H^1 ({\mathbb {R}}^N)\) to some \({\bar{u}}_0 \in H^2({\mathbb {R}}^N)\) and a subsequence \((\mu _{n_k})\) of \((\mu _n)\) converging to some \(\mu _0\in [0,1],\) as \(k\rightarrow +\infty ,\) such that \({\bar{u}}_0\) is a solution of

$$\begin{aligned} \Delta u(x)+{\widehat{h}}(x,u(x),\mu _0)=0,\quad x\in {\mathbb {R}}^N, \end{aligned}$$

where \({\widehat{h}}:\!{\mathbb {R}}^N\! \times \! {\mathbb {R}}\times [0,1]\!\rightarrow \!{\mathbb {R}},\) \({\widehat{h}}(x,u,\mu )\! := {\frac{1}{T}}{\int _{0}^{T}}\!\! h(t,x,u,\mu ) \mathrm {\,d}t,\) \((x,u,\mu )\!\in \! {\mathbb {R}}^N\!\times {\mathbb {R}}\times [0,1]\). Moreover,  \(u_{n_k} (t)\rightarrow {\bar{u}}_0\) in \(H^1({\mathbb {R}}^N),\) as \(k\rightarrow +\infty ,\) uniformly with respect to t from compact subsets of \((0,+\infty )\).

Proof

Recall that \(u_n\) are solutions of \(\dot{u} = -\mathbf{A}u+{\mathbf{H}(t/\lambda _n, u,\mu _n)}\) with \(u_n(0) = u_n(\lambda _n T)={\bar{u}}_n\), \(n\ge 1\), where \(\mathbf{A}\) and \(\mathbf{H}\) are as in the previous section (with \(a_{ij}=0\) if \(i \ne j\) and \(a_{ij}=1\) if \(i=j\)). Clearly, by the sublinear growth, there exists \(R>0\) such that \(\Vert u_n(t)\Vert _{H^1}\le R\) for all \(t>0\) and \(n\ge 1\). For an arbitrary \(M>0\) and \(n\ge 1\) take \(k_n\in \mathbb {N}\) such that \(k_n \lambda _n T>M\). In view of Lemma 5.1, for a fixed \(\gamma \in (0,|{\bar{a}}|)\) and all sufficiently large \(m\ge 1\) and \(n\ge 1\),

$$\begin{aligned} \Vert (1-\chi _m){\bar{u}}_n \Vert _{L^2}^2 = \Vert (1-\chi _m) u_n (k_n\lambda _n T)\Vert _{L^2}^2 \le R^2 e^{-2\gamma k_n \lambda _n T} +\alpha _m \le R^2e^{-2\gamma M}+\alpha _m, \end{aligned}$$

where \(\chi _m\) is the characteristic function of B(0, m) . Since \(M>0\) is arbitrary we see that \(\Vert (1-\chi _m) {\bar{u}}_n\Vert _{L^2}\le \sqrt{\alpha _m}\). Since, due to the Rellich–Kondrachov for all \(m \ge 1\), the set \(\left\{ \chi _m {\bar{u}}_n\right\} _{n\ge 1}\) is relatively compact in \(L^2({\mathbb {R}}^N)\), we infer that \(\{ {\bar{u}}_n \}_{n\ge 1}\) is relatively compact in \(L^2({\mathbb {R}}^N)\). And since it is bounded in \(H^1({\mathbb {R}}^N)\) we get a subsequence \(({\bar{u}}_{n_k})\), denoted in the sequel again by \(({\bar{u}}_{n})\), such that \(\bar{u}_{n_k} \rightarrow {\bar{u}}_0\) in \(L^2({\mathbb {R}}^N)\) for some \({\bar{u}}_0\in H^1({\mathbb {R}}^N)\). We may also assume that \(\mu _{n_k} \rightarrow \mu _0\) for some \(\mu _0\in [0,1]\). Hence, in view of Theorem 3.5, \(u_n(t)\rightarrow {\widehat{u}}(t)\) uniformly for t from compact subsets of \((0,+\infty )\) where \({\widehat{u}}:[0,+\infty )\rightarrow H^1({\mathbb {R}}^N)\) is a solution to

$$\begin{aligned} \dot{u} = - \mathbf{A}u + \widehat{\mathbf{H}} (u,\mu _0), \quad t>0, \end{aligned}$$

with \(\widehat{\mathbf{H}}(u,\mu ):= \frac{1}{T}\int _{0}^{T} \mathbf{H}(t,u,\mu )\mathrm {\,d}t\) for \(u\in H^1({\mathbb {R}}^N)\), \(\mu \in [0,1]\). Here note that, for each \(u\in H^1({\mathbb {R}}^N)\) and \(\mu \in [0,1]\),

$$\begin{aligned}{}[\widehat{\mathbf{H}}(u,\mu )](x) = {\widehat{h}}(x,u(x),\mu ) \quad \text{ for } \text{ all } \text{ a.a. } x\in {\mathbb {R}}^N. \end{aligned}$$

Finally, for any \(t>0\), we put \(k_n:=[t/\lambda _n T]\), \(n\ge 1\), and see that

$$\begin{aligned} {\bar{u}}_n = u_n(0)=u_n(k_n\lambda _n T) \rightarrow {\widehat{u}} (t) \quad \text{ in } H^1({\mathbb {R}}^N), \text{ as } n\rightarrow +\infty . \end{aligned}$$

Hence \({\widehat{u}}(t)= {\widehat{u}}(0)={\bar{u}}_0\) and \({\bar{u}}_n\rightarrow \bar{u}_0\) in \(H^1({\mathbb {R}}^N)\). \(\square \)

Remark 6.2

Clearly that it follows from the proof of Lemma 6.1 that if \(f_n:[0,+\infty ) \times {\mathbb {R}}^N\times {\mathbb {R}}\rightarrow {\mathbb {R}}\) are as in Proposition 4.3 and satisfy (6) with common a and b, then for any bounded sequence \(({\bar{u}}_n)\) in \(H^1({\mathbb {R}}^N)\), \((\lambda _n)\) in \((0,+\infty )\) with \(\lambda _n \rightarrow 0^+\) as \(n\rightarrow +\infty \) and \(u_n: [0,+\infty ) \rightarrow H^1({\mathbb {R}}^N)\) being \(\lambda _nT\)-periodic solutions of

$$\begin{aligned} \left\{ \begin{array}{ll} {\frac{\partial u}{\partial t}} = \Delta u + f_n(t/\lambda _n,x, u),&{} \quad x\in {\mathbb {R}}^N, \, t>0,\\ u(x,0)= u(x,\lambda _n T)={\bar{u}}_n(x), &{}\quad x\in {\mathbb {R}}^N, \end{array}\right. \end{aligned}$$

there is a subsequence \(({\bar{u}}_{n_k})\) of \(({\bar{u}}_n)\) converging in \(H^1 ({\mathbb {R}}^N)\) to some \({\bar{u}}_0 \in H^2({\mathbb {R}}^N)\) being a solution of

$$\begin{aligned} \Delta u(x)+{\widehat{f}}_0 (x,u(x))=0 \quad \text{ on } {\mathbb {R}}^N. \end{aligned}$$

Moreover, \(u_{n_k} (t)\rightarrow {\bar{u}}_0\) in \(H^1({\mathbb {R}}^N)\), as \(k\rightarrow +\infty \), uniformly with respect to t from compact subsets of \((0,+\infty )\). \(\square \)

Now consider the following problem

$$\begin{aligned} {\frac{\partial u}{\partial t}} (x,t) = \Delta u (x,t) + f ( t/\lambda ,x,u(x,t)), \quad t>0, \ x\in {\mathbb {R}}^N, \end{aligned}$$
(36)

where f satisfies conditions (2)–(4) and (25). We intend to prove an averaging index formula that allows us to express the fixed point index of translation along trajectories operator for (36) in terms of the averaged equation

$$\begin{aligned} {\frac{\partial u}{\partial t}} (x,t) = \Delta u (x,t) + {\widehat{f}}(x,u(x,t)), \quad t>0, \ x\in {\mathbb {R}}^N, \end{aligned}$$
(37)

where \({\widehat{f}}:{\mathbb {R}}^N\times {\mathbb {R}}\rightarrow {\mathbb {R}}\) is defined by

$$\begin{aligned} {\widehat{f}}(x,u):= \frac{1}{T}\int _{0}^{T} f(t,x,u)\mathrm {\,d}t, \quad x\in {\mathbb {R}}^N, \ u\in {\mathbb {R}}. \end{aligned}$$

Theorem 6.3

Let \(U\subset H^1({\mathbb {R}}^N)\) be an open bounded set and by \({\varvec{\Phi }}_{t}^{(\lambda )}\) and \(\widehat{\varvec{\Phi }}_t,\) \(t>0,\) denote the translation along trajectories operators (by time t) for the Eqs. (36) and (37), respectively. If the problem

$$\begin{aligned} \left\{ \begin{array}{l} - \Delta u (x) ={\widehat{f}}(x,u(x)), \quad x\in {\mathbb {R}}^N,\\ u \in H^1 ({\mathbb {R}}^N), \end{array} \right. \end{aligned}$$
(38)

has no solution in \(\partial U,\) then there exists \(\lambda _0>0\) such that,  for all \(\lambda \in (0,\lambda _0],\) \({\varvec{\Phi }}_{\lambda T}^{(\lambda )}({\bar{u}})\ne {\bar{u}},\) \(\widehat{\varvec{\Phi }}_{\lambda T}({\bar{u}}) \ne {\bar{u}}\) for all \({\bar{u}} \in \partial U,\) and

$$\begin{aligned} \mathrm {Ind}({\varvec{\Phi }}_{\lambda T}^{(\lambda )}, U) = \mathrm {Ind}(\widehat{\varvec{\Phi }}_{\lambda T}, U). \end{aligned}$$

Proof

Define \(\mathbf{H}:[0,+\infty ) \times H^1({\mathbb {R}}^N)\times [0,1]\rightarrow L^2({\mathbb {R}}^N)\) by

$$\begin{aligned} {[}{} \mathbf{H} (t,u,\mu )](x):= (1-\mu ) f(t,x,u(x)) + \mu {\widehat{f}} (x,u(x)), \quad \text{ for } \text{ a.e. } x\in {\mathbb {R}}^N, \end{aligned}$$

and all \(t>0, \, u\in H^1({\mathbb {R}}^N)\). For a parameter \(\lambda >0\) consider

$$\begin{aligned} \dot{u}(t) = - \mathbf{A} u(t) + \mathbf{H} (t/\lambda ,u(t),\mu ),\quad t\in [0,T], \end{aligned}$$
(39)

and the parameterized translation operator \({\varvec{\Psi }}_{t}^{(\lambda )}:H^{1}({\mathbb {R}}^N)\times [0,1]\rightarrow H^{1}({\mathbb {R}}^N)\) defined by

$$\begin{aligned} {\varvec{\Psi }}_{t}^{(\lambda )} ({\bar{u}}, \mu ):= u(t), \end{aligned}$$

where \(u:[0,T]\rightarrow H^{1}({\mathbb {R}}^N)\) is the solution of (39) with \(u(0)={\bar{u}}\). Observe that for \(\mu =0\), (39) becomes

$$\begin{aligned} \dot{u} (t) = -\mathbf{A} u(t)+ \mathbf{F}(t/\lambda ,u(t)), \quad t\in [0,T], \end{aligned}$$

and we have \({\varvec{\Phi }}_t^{(\lambda )} = {\varvec{\Psi }}_{t}^{(\lambda )}(\cdot ,0)\). In the same way for \(\mu =1\) the equation (39) becomes

$$\begin{aligned} \dot{u} (t)= -\mathbf{A} u(t) +\widehat{\mathbf{F}}(u(t)), \quad t\in [0,T], \end{aligned}$$

and one has \(\widehat{\varvec{\Phi }}_t = {\varvec{\Psi }}_t^{(\lambda )}(\cdot , 1)\) (it does not depend on \(\lambda \)).

We claim that there exists \(\lambda _0>0\) such that, for all \(\lambda \in (0,\lambda _0]\),

$$\begin{aligned} {\varvec{\Psi }}_{\lambda T}^{(\lambda )}({\bar{u}}, \mu ) \ne {\bar{u}} \quad \text{ for } \text{ all } {\bar{u}}\in \partial U, \ \mu \in [0,1]. \end{aligned}$$
(40)

Suppose the claim does not hold. Then there exist \(({\bar{u}}_n)\) in \(\partial U\), \((\mu _n)\) in [0, 1] and \((\lambda _n)\) with \(\lambda _n\rightarrow 0^+\) as \(n\rightarrow \infty \) such that

$$\begin{aligned} {\varvec{\Psi }}_{\lambda _n T}^{(\lambda _n)}({\bar{u}}_n, \mu _n) ={\bar{u}}_n \quad \text{ for } \text{ all } \ n\ge 1. \end{aligned}$$

This means that for each \(n\ge 1\) there is a \(\lambda _n T\)-periodic solution \(u_n:[0,+\infty ) \rightarrow H^1 ({\mathbb {R}}^N)\) of (39) with \(\lambda =\lambda _n\), \(\mu =\mu _n\) and \(u_n(0)={\bar{u}}_n\). By Lemma 6.1 we may assume that \({\bar{u}}_n \rightarrow {\bar{u}}_0\) in \(H^1({\mathbb {R}}^N)\). Therefore, \({\bar{u}}_0\in \partial U\cap D(\mathbf{A})\) and \(0=-\mathbf{A}{\bar{u}}_0+\widehat{\mathbf{F}}({\bar{u}}_0)\), a contradiction with the assumption. This proves the existence of \(\lambda _0>0\) such that, for all \(\lambda \in (0,\lambda _0]\), (40) holds.

Now, due to Proposition 5.4(iii), for each \(\lambda \in (0,\lambda _0]\), \({\varvec{\Psi }}_{\lambda T}^{(\lambda )}\) is an admissible homotopy in the sense of fixed point index theory for ultimately compact maps. Finally, by Proposition 2.1(iii), we get the desired equality of the indices. \(\square \)

As a consequence we get the following continuation principle.

Corollary 6.4

Suppose that an open bounded \(U\subset H^1({\mathbb {R}}^N)\) is such that (38) has no solution in \(\partial U,\) and for any \(\lambda \in (0,1)\) the problem

$$\begin{aligned} \left\{ \begin{array}{ll} {\frac{\partial u}{\partial t}} = \lambda \Delta u +\lambda f(t,x,u),&{}\quad x\in {\mathbb {R}}^N, \ t>0,\\ u(x,0) = u(x,T), &{}\quad x\in {\mathbb {R}}^N, \end{array} \right. \end{aligned}$$
(41)

has no solution \(u:[0,+\infty )\rightarrow H^1({\mathbb {R}}^N)\) with \(u(\cdot , 0)\in \partial U\). Then

$$\begin{aligned} \mathrm {Ind}({\varvec{\Phi }}_T, U) = \lim _{t\rightarrow 0^+} \mathrm {Ind}( \widehat{\varvec{\Phi }}_{t}, U) \end{aligned}$$

where \({\varvec{\Phi }}_T\) is the translation along trajectories operator for (1).

Proof

Let \(\lambda _0>0\) be as in Theorem 6.3. Since there are no solutions to (41), we infer that

$$\begin{aligned} {\varvec{\Phi }}_{\lambda T}^{(\lambda )} ({\bar{u}}) \ne {\bar{u}} \quad \text{ for } \text{ any } {\bar{u}}\in \partial U,\ \lambda \in (0,1). \end{aligned}$$

Now by Proposition 5.4(iii) and the homotopy invariance of the index, for any \(\lambda \in (0,1]\), we get \(\mathrm {Ind}({\varvec{\Phi }}_T, U)= \mathrm {Ind}(\widetilde{\varvec{\Phi }}_T^{(1)}, U) = \mathrm {Ind}(\widetilde{\varvec{\Phi }}_{T}^{(\lambda )}, U) = \mathrm {Ind}({\varvec{\Phi }}_{\lambda T}^{(\lambda )}, U)\), where \(\widetilde{\varvec{\Phi }}_{T}^{(\lambda )}\) is the translation along trajectories operator for the parabolic equation in (41) with the parameter \(\lambda \) and the last equality comes from a time rescaling argument saying that \(\widetilde{\varvec{\Phi }}_{T}^{(\lambda )} = {\varvec{\Phi }}_{\lambda T}^{(\lambda )}\). Now an application of Theorem 6.3 completes the proof. \(\square \)

The rest of the section is devoted to methods of verification the a priori bounds conditions occurring in the above corollary and computation of fixed point index. We shall use a linearization approach and assume the following asymptotic property of f at zero

$$\begin{aligned} \lim _{u\rightarrow 0} \frac{f(t,x,u)}{u} = \alpha (t,x):= \alpha _0(t,x) +\alpha _\infty (t,x) \end{aligned}$$
(42)

and at infinity

$$\begin{aligned} \lim _{|u|\rightarrow \infty } \frac{f(t,x,u)}{u} = \omega (t,x) := \omega _0 (t,x) + \omega _\infty (t,x) \end{aligned}$$
(43)

for all \(x\in {\mathbb {R}}^N\) and \(t\ge 0\), where \(\alpha _\infty , \omega _\infty \in L^\infty ([0,+\infty )\times {\mathbb {R}}^N)\), and \(\sup _{t\ge 0} \Vert \alpha _0 (t,\cdot )\Vert _{L^p} <+\infty \), \(\sup _{t\ge 0} \Vert \omega _0 (t,\cdot )\Vert _{L^p}<+\infty \). We shall also assume that, for all \(t,s \ge 0\) and a.e. \(x \in {\mathbb {R}}^N\),

$$\begin{aligned} \begin{array}{l} |\alpha (t,x)-\alpha (s,x)|\le k_\alpha (x) |t-s|^\nu \quad \text{ and }\quad |\omega (t,x)-\omega (s,x)| \le k_\omega (x)|t-s|^\nu , \end{array} \end{aligned}$$

with \(k_\alpha , k_\omega \) of Rellich–Kato type and \(\nu \in (0,1)\).

Proposition 6.5

Suppose that f satisfies conditions (2)–(5) and (25).

  1. (i)

    If (43) holds, \({\mathrm {Ker}}\,(\Delta +{\widehat{\omega }})=\{0\},\) where \(\widehat{\omega }:{\mathbb {R}}^N\rightarrow {\mathbb {R}}\) is the time average function of \(\omega ,\) given by \({\widehat{\omega }} (x) := \frac{1}{T}\int _{0}^{T} \omega (t,x) \mathrm {\,d}t,\) and the linear equation

    $$\begin{aligned} {\frac{\partial u}{\partial t}} = \lambda \Delta u + \lambda \omega (t,x) u,\quad x\in {\mathbb {R}}^N, \ t>0, \end{aligned}$$
    (44)

    has no nonzero T-periodic solutions for \(\lambda \in (0,1],\) then there exists \(R>0\) such that,  for any \(\lambda \in (0,1]\) the problem (41) has no T-periodic solutions \(u:[0,+\infty ) \rightarrow H^1({\mathbb {R}}^N)\) with \(\Vert u(0)\Vert _{H^1} \ge R\).

  2. (ii)

    If \(f(t,x,0)=0\) for all \(t \ge 0\) and a.e. \(x\in {\mathbb {R}}^N,\) (42) holds,  \({\mathrm {Ker}}\,(\Delta +\widehat{\alpha })=\{0\}\) and the linear equation

    $$\begin{aligned} {\frac{\partial u}{\partial t}} = \lambda \Delta u + \lambda \alpha (t,x) u,\quad x\in {\mathbb {R}}^N, \ t>0, \end{aligned}$$
    (45)

    has no nonzero T-periodic solutions,  then there exists \(r>0\) such that,  for any \(\lambda \in (0,1]\) the problem (41) has no T-periodic solutions \(u:[0,+\infty )\rightarrow H^1({\mathbb {R}}^N)\) with \(0<\Vert u(0)\Vert _{H^1} \le r\).

Proof

(i) Suppose to the contrary, i.e. that for any \(n\ge 1\) there exist \(\lambda _n\in (0,1)\) and a time T-periodic solution \(u_n:[0, +\infty )\rightarrow H^1({\mathbb {R}}^N)\) of

$$\begin{aligned} \frac{\partial u}{\partial t} = \lambda _n \Delta u + \lambda _n f(t,x,u), \quad x\in {\mathbb {R}}^N, \ t>0 \end{aligned}$$

with \(\Vert u_n (0)\Vert _{H^1}\rightarrow +\infty \). This means that \(z_n\) given by \(z_n(t):=\rho _n^{-1}u_n(t)\), \(\rho _n:=1+\Vert u_n(0)\Vert _{H^1}\), is a T-periodic solution of

$$\begin{aligned} \frac{\partial z}{\partial t} = \lambda _n \Delta z+ \lambda _n \rho _n^{-1} f(t,x,\rho _n z), \quad x\in {\mathbb {R}}^N, \ t>0. \end{aligned}$$
(46)

It is also clear that \(v_n:[0,+\infty )\rightarrow H^1({\mathbb {R}}^N)\) given by \(v_n(t): = z_n (t/\lambda _n)\) satisfies

$$\begin{aligned} \frac{\partial v}{\partial t} = \Delta v+ \rho _n^{-1} f(t/\lambda _n, x,\rho _n v),\quad x\in {\mathbb {R}}^N,\ t>0, \end{aligned}$$
(47)

and that \(\rho _n\rightarrow +\infty \). Define \(g_n: [0,+\infty ) \times {\mathbb {R}}^N \times {\mathbb {R}}\rightarrow {\mathbb {R}}\) by

$$\begin{aligned} g_n(t,x,v):= \rho _n^{-1} f(t/\lambda _n, x,\rho _n v), \quad n\ge 1, \ t\ge 0, \ x\in {\mathbb {R}}^N, \ v \in {\mathbb {R}}. \end{aligned}$$

Since the functions \(g_n\), \(n\ge 1\), satisfy (3) with a common \(m_0\) and (4) with common l and \(\{v_n(0)\}_{n\ge 1}\) is bounded, by use of Lemma 4.1 and Remark 3.1, we obtain a constant \(R_0>0\) such that \(\Vert v_n(t)\Vert _{H^1}\le R_0\) for all \(n\ge 1\) and \(t\ge 0\). For a moment fix an arbitrary \(M>0\) and for any \(n\ge 1\) take an integer \(k_n\ge 1\) such that \(k_n\lambda _n T > M\). Observe that Lemma 5.1 gives, for a fixed \(\gamma \in (0,|{\bar{a}}|)\) and all \(m \ge 1\) and \(n\ge 1\),

$$\begin{aligned} \Vert (1-\chi _m) v_n(0) \Vert _{L^2}^2= & {} \Vert (1-\chi _m) v_n (k_n \lambda _n T)\Vert _{L^2}^2 \le R_0^2 e^{-2\gamma k_n \lambda _n T} +\alpha _m \\\le & {} R_0^2 e^{-2\gamma M}+ \alpha _m \end{aligned}$$

with \(\alpha _m \rightarrow 0^+\) as \(m\rightarrow +\infty \). Since \(M>0\) is arbitrary we see that \(\Vert (1-\chi _m) v_n(0)\Vert _{L^2}\le \sqrt{\alpha _m}\) for \(m, n\ge 1\). Due to the Rellich–Kondrachov for any \(m \ge 1\), the set \(\{\chi _m v_n(0)\}_{n\ge 1}\) is relatively compact in \(L^2({\mathbb {R}}^N)\). Therefore, \(\{ v_n(0) \}_{n\ge 1}\) is relatively compact in \(L^2({\mathbb {R}}^N)\), since \(\alpha _m \rightarrow 0^+\) as \(m\rightarrow +\infty \). As a bounded sequence in \(H^1({\mathbb {R}}^N)\), \((v_n(0))\) contains a subsequence convergent in \(L^2({\mathbb {R}}^N)\) to some \({\bar{v}}_0\in H^1({\mathbb {R}}^N)\). Therefore, we may assume that \(v_n(0)\rightarrow {\bar{v}}_0\) in \(L^2({\mathbb {R}}^N)\). Moreover, we may suppose that \(\lambda _n\rightarrow \lambda _0\), as \(n\rightarrow +\infty \) for some \(\lambda _0 \in [0,1]\).

First consider the case when \(\lambda _0 \in (0,1]\). Let \(f_n: {\mathbb {R}}^N\times {\mathbb {R}}\rightarrow {\mathbb {R}}\), \(n\ge 1\), be given by

$$\begin{aligned} f_n(t,x,z):=\rho _n^{-1} f(t,x,\rho _n z ), \quad \text{ for } \text{ all } t\ge 0, \ z\in {\mathbb {R}} \text{ and } \text{ a.e. } x\in {\mathbb {R}}. \end{aligned}$$

Note that (43) and (3) yield

$$\begin{aligned} \lim _{n\rightarrow +\infty } f_n ( t, x, z)= \omega (t,x) z,\quad \text{ for } \text{ all } t\ge 0, \, z\in {\mathbb {R}} \text{ and } \text{ a.e. } x\in {\mathbb {R}}, \end{aligned}$$

and \(\Vert f_n(t,\cdot ,0)\Vert _{L^2}=\rho _n^{-1} \Vert f (t,\cdot , 0)\Vert _{L^2} \le \rho _n^{-1} \Vert m_0\Vert _{L^2}\rightarrow 0\), as \(n \rightarrow +\infty \). It allows us to apply Proposition 4.3 to (47). As a result we infer that \(z_n (t) \rightarrow z_0 ( t)\) in \(H^1({\mathbb {R}}^N)\) uniformly with respect to t from compact subsets of \((0,+\infty )\), where \(z_0:[0,+\infty )\rightarrow H^1({\mathbb {R}}^N)\) is a T-periodic solution of

$$\begin{aligned} \frac{\partial z}{\partial t} = \lambda _0 \Delta z + \lambda _0 \omega (t,x)z. \end{aligned}$$

Since \(\Vert z(0)\Vert _{H^1}=\Vert v(0)\Vert _{H^1}\ne 0\), we get a nontrivial T-periodic solution of (44) with \(\lambda =\lambda _0\), a contradiction proving the desired assertion.

In the situation when \(\lambda _0=0\), we apply Proposition 4.4 to (47) to see that \(v_n(t)\rightarrow {\widehat{v}}(t)\) uniformly with respect to t from compact subsets of \([0,+\infty )\), where \({\widehat{v}}:[0,+\infty )\rightarrow H^1({\mathbb {R}}^N)\) is a nontrivial solution of

$$\begin{aligned} \frac{\partial v}{\partial t} = \Delta v + {\widehat{\omega }} (x)v, \quad x\in {\mathbb {R}}^N, \ t>0. \end{aligned}$$

Now observe that, for any \(t>0\) and \(k_n:=[t/\lambda _n T]\), \(n\ge 1\), one has

$$\begin{aligned} v_n(0)=v_n(k_n\lambda _n T) \rightarrow {\widehat{v}} (t) \ quad\text{ in } H^1({\mathbb {R}}^N), \text{ as } n\rightarrow +\infty . \end{aligned}$$

Hence \({\widehat{v}} \equiv {\bar{v}}_0\) and, as a consequence,

$$\begin{aligned} 0= \Delta {\bar{v}}_0(x) + {\widehat{\omega }} (x) {\bar{v}}_0 (x), \quad x\in {\mathbb {R}}^N, \end{aligned}$$

which contradicts the assumption and completes the proof of (i).

To see (ii), suppose that assertion does not hold. Then there exist \(\lambda _n\in (0,1)\) and a T-periodic solutions \(u_n:[0, +\infty )\rightarrow H^1({\mathbb {R}}^N)\) of

$$\begin{aligned} \frac{\partial u}{\partial t} = \lambda _n \Delta u + \lambda _n f(t,x,u), \quad x\in {\mathbb {R}}^N, \ t>0 \end{aligned}$$

with \(\Vert u_n (0)\Vert _{H^1}>0\), \(n\ge 1\), and \(\Vert u_n (0)\Vert _{H^1}\rightarrow 0^+\) as \(n \rightarrow +\infty \). Put \(z_n:=\frac{u_n}{\rho _n}\) and let \(v_n(t):=z_n (t/\lambda _n)\) with \(\rho _n:=\Vert u_n(0)\Vert _{H^1}\). Then, for each \(n \ge 1\), \(z_n\) is a solution of

$$\begin{aligned} \frac{\partial z}{\partial t} = \lambda _n \Delta z+ \lambda _n \rho _n^{-1} f(t, x,\rho _n z),\quad x\in {\mathbb {R}}^N,\ t>0, \end{aligned}$$

and \(v_n\) is a solution of

$$\begin{aligned} \frac{\partial v}{\partial t} = \Delta v+ \rho _n^{-1} f(t/\lambda _n, x,\rho _n v),\quad x\in {\mathbb {R}}^N,\ t>0. \end{aligned}$$

The rest of the proof goes along the lines of the proof for (i). \(\square \)

Remark 6.6

Let us remark that the nonexistence of solutions for (44) or (45) may be also verified if \(\alpha \) or \(\omega \) are time dependent. Assume that there exists \({\bar{\omega }}_\infty >0\) such that \(\omega _\infty (t,x) \le - \bar{\omega }_\infty \) for all \(t\ge 0\) and a.e. \(x\in {\mathbb {R}}^N\) and

$$\begin{aligned} \sup _{t \in [0,T]}\Vert \omega _0(\cdot ,t)\Vert _{L^p}< \left\{ \begin{array}{lcll} \frac{p^{1/2p}{\bar{\omega }}_\infty ^{1-1/2p}}{2^{1/2p}},&{}&{} \text { if } N=1,\ p>2, \\ \frac{p^{1/p}{\bar{\omega }}_\infty ^{(1-1/p)}}{4^{1/p}},&{}&{} \text { if } N=2,\ p>2, \\ \frac{{\bar{\omega }}_\infty ^{1-N/2p}}{(N/2p)^{N/2p} C(N)^{N/p}},&{}&{} \text { if } N\ge 3, N\le p<\infty , \end{array} \right. \end{aligned}$$
(48)

where \(C(N)>0\) is the constant in the Sobolev inequality \(\Vert u\Vert _{L^{\frac{2N}{N-2}}}\le C(N) \Vert \nabla u\Vert _{L^2}\), \(u\in H^1({\mathbb {R}}^N)\). Suppose that u is a nonzero T-periodic solution of (44) for some \(\lambda \in (0,1)\). Then , for all \(t>0\),

$$\begin{aligned} \frac{\mathrm {\,d}}{\mathrm {\,d}t} \frac{1}{2\lambda } \Vert u(t)\Vert _{L^2}^2\! =\! -\! \int _{{\mathbb {R}}^N}\! |\nabla u(t)|^2\mathrm {\,d}x \!+\! \int _{{\mathbb {R}}^N} \!\omega _\infty (t,x)|u(t)|^2 \mathrm {\,d}x + \int _{{\mathbb {R}}^N}\omega _0(t,x)|u(t)|^2 \mathrm {\,d}x.\nonumber \\ \end{aligned}$$
(49)

Assume first that \(N=1\). Then, by use of the Hölder inequality,

$$\begin{aligned} \int _{0}^{T}\left( \Vert \nabla u(t)\Vert _{L^2}^2 +{\bar{\omega }}_\infty \Vert u(t)\Vert _{L^2}^2 \right) \mathrm {\,d}t&\le \int _0^T \Vert \omega _0(t,\cdot )\Vert _{L^p}\Vert u(t)\Vert _{L^2}^{2-2/p}\Vert u(t)\Vert _{L^\infty }^{2/p}\mathrm {\,d}t, \\&\le 2^{1/p} \int _0^T \Vert \omega _0(t,\cdot )\Vert _{L^p}\Vert \nabla u(t)\Vert _{L^2}^{1/p}\Vert u(t)\Vert _{L^2}^{2-1/p}\mathrm {\,d}t, \end{aligned}$$

where the latter inequality follows by the fact that \(\Vert u\Vert _{L^\infty }^2\le 2 \Vert \nabla u\Vert _{L^2}\Vert u\Vert _{L^2}\) for \(u \in H^1({\mathbb {R}}).\) In the Young inequality \(a b \le \frac{a^r}{\epsilon ^r r} + \frac{b^s \epsilon ^s}{s}\) where \(a,b\ge 0\), \(\epsilon >0\) and \(r\in (1,+\infty )\) such that \(\frac{1}{r}+\frac{1}{s}=1\), put \(a:=\Vert \omega _0 (\cdot ,t)\Vert _{L^p} \Vert \nabla u(t)\Vert _{L^2}^{1/p}\), \(b:= \Vert u(t)\Vert _{L^2}^{2-1/p}\) and \(r:=2p\) to obtain

$$\begin{aligned} \Vert \omega _0(t,\cdot )\Vert _{L^p}\Vert \nabla u(t)\Vert _{L^2}^{1/p}\Vert u(t)\Vert _{L^2}^{2-1/p} \le \frac{\Vert \omega _0(t,\cdot )\Vert _{L^p}^{2p}\Vert \nabla u(t)\Vert _{L^2}^2}{2p\epsilon ^{2p}}+\frac{\epsilon ^{2p/(2p-1)} \Vert u(t)\Vert _{L^2}^2}{2p/(2p-1)} \end{aligned}$$

for any \(\epsilon >0\) and fixed \(t \in [0,T]\). If we take \(\epsilon =\epsilon (t)\) so that \(2^{1/p}\frac{\Vert \omega _0(t,\cdot )\Vert _{L^p}^{2p}}{2p\epsilon ^{2p}}=1\), i.e. \(\epsilon (t):=\bigg (\frac{2^{(1-p)/p}}{p}\bigg )^{1/2p}\Vert \omega _0 (t,\cdot )\Vert _{L^p} \) and apply (48), then

$$\begin{aligned} {\bar{\omega }}_\infty \int _0^T \Vert u(t)\Vert _{L^2}^2\mathrm {\,d}t&\le 2^{1/p}\frac{2p-1}{2p}\int _0^T \epsilon (t)^{2p/(2p-1)}\Vert u(t)\Vert ^2_{L^2}\mathrm {\,d}t\\&\le 2^{1/p} \bigg (\frac{2^{(1-p)/p}}{p}\bigg )^{1/(2p-1)} \sup _{t \in [0,T]}\Vert \omega _0(t,\cdot )\Vert _{L^p}^{2p/(2p-1)}\int _0^T \Vert u(t)\Vert _{L^2}^2\mathrm {\,d}t \\&= \bigg (\frac{2}{p}\bigg )^{1/(2p-1)} \sup _{t \in [0,T]}\Vert \omega _0(t,\cdot )\Vert _{L^p}^{2p/(2p-1)}\int _0^T \Vert u(t)\Vert _{L^2}^2\mathrm {\,d}t \\&<{\bar{\omega }}_\infty \int _0^T \Vert u(t)\Vert _{L^2}^2\mathrm {\,d}t, \end{aligned}$$

a contradiction proving that (44) has no nontrivial T-periodic solutions. Assume now that \(N=2\). Then by (49) and the Hölder inequality it follows that

$$\begin{aligned} \int _{0}^{T}\!\!\left( \Vert \nabla u(t)\Vert _{L^2}^2 +{\bar{\omega }}_\infty \Vert u(t)\Vert _{L^2}^2 \right) \mathrm {\,d}t \!\le \! \int _{0}^{T}\!\! \Vert \omega _0(t,\cdot )\Vert _{L^p} \Vert u(t)\Vert _{L^{4}}^{4/p}\Vert u(t)\Vert _{L^2}^{2-4/p} \mathrm {\,d}t, \end{aligned}$$

which in view of the Sobolev inequality \(\Vert u\Vert _{L^{4}}^2\le 2\Vert u\Vert _{L^2}\Vert \nabla u\Vert _{L^2}\), \(u \in H^1({\mathbb {R}}^2)\) implies

$$\begin{aligned} \int _{0}^{T}\!\!\left( \Vert \nabla u(t)\Vert _{L^2}^2 +{\bar{\omega }}_\infty \Vert u(t)\Vert _{L^2}^2 \right) \!\mathrm {\,d}t \!&\le \!2^{2/p}\!\int _0^T \Vert \omega _0(t,\cdot )\Vert _{L^p} \Vert \nabla u(t)\Vert _{L^2}^{2/p}\Vert u(t)\Vert _{L^2}^{2-2/p} \mathrm {\,d}t. \end{aligned}$$
(50)

By use of the Young inequality, we obtain for any \(\epsilon >0\) and fixed \(t\in [0,T]\),

$$\begin{aligned} \Vert \omega _0(t,\cdot )\Vert _{L^p}\Vert \nabla u(t)\Vert _{L^2}^{2/p}\Vert u(t)\Vert _{L^2}^{2-2/p}\le \frac{\Vert \omega _0(t,\cdot )\Vert _{L^p}^p \Vert \nabla u(t)\Vert _{L^2}^2}{p\epsilon ^p}+ \frac{\epsilon ^{p/(p-1)}\Vert u(t)\Vert _{L^2}^2}{p/(p-1)}. \end{aligned}$$
(51)

Choose \(\epsilon =\epsilon (t)>0\) such that \(\frac{2^{2/p}}{p \epsilon ^p}\Vert \omega _0(\cdot ,t)\Vert _{L^p}^p=1\), i.e.

$$\begin{aligned} \epsilon (t):=\frac{2^{2/p^2}}{p^{1/p}}\Vert \omega _0(\cdot ,t)\Vert _{L^p}. \end{aligned}$$

Then, by applying (48), we have

$$\begin{aligned} {\bar{\omega }}_\infty \int _0^T \Vert u(t)\Vert _{L^2}^2&\le 2^{2/p}\frac{(p-1)}{p} \int _0^T \epsilon (t)^{p/(p-1)}\Vert u(t)\Vert _{L^2}^2 \mathrm {\,d}t \\&\le \bigg (\frac{4}{p}\bigg )^{1/(p-1)}\sup _{t \in [0,T]} \Vert \omega _0(t,\cdot )\Vert _{L^p}^{p/(p-1)}\int _0^T \Vert u(t) \Vert _{L^2}^2 \mathrm {\,d}t \\&<{\bar{\omega }}_\infty \int _0^T \Vert u(t)\Vert _{L^2}^2 \mathrm {\,d}t, \end{aligned}$$

which contradicts the existence of nonzero T-periodic solution for (44). Finally, for \(N \ge 3\), by use of the Hölder and the Sobolev inequalities we get

$$\begin{aligned}&\int _{0}^{T}\!\!\left( \Vert \nabla u(t)\Vert _{L^2}^2 +{\bar{\omega }}_\infty \Vert u(t)\Vert _{L^2}^2 \right) \mathrm {\,d}t \!\le \! \int _{0}^{T}\!\! \Vert \omega _0(t,\cdot )\Vert _{L^p} \Vert u(t) \Vert _{L^{2N/(N-2)}}^{N/p} \Vert u(t)\Vert _{L^2}^{2-N/p}\mathrm {\,d}t\\&\quad \le C(N)^{N/p} \int _{0}^{T} \Vert \omega _0 (\cdot ,t)\Vert _{L^p}\Vert \nabla u(t)\Vert _{L^2}^{N/p} \Vert u(t)\Vert _{L^2}^{2-N/p} \mathrm {\,d}t. \end{aligned}$$

In view of the Young inequality, for any \(\epsilon >0\) and fixed \(t\in [0,T]\),

$$\begin{aligned}&\Vert \omega _0(\cdot ,t)\Vert _{L^p}\Vert \nabla u(t)\Vert _{L^2}^{\frac{N}{p}} \Vert u(t)\Vert _{L^2}^{2-\!\frac{N}{p}}\!\!\! \le \! \frac{N/2p}{\epsilon ^{\frac{2p}{N}}} \Vert \omega _0(\cdot ,t)\Vert _{L^p}^{\frac{2p}{N}} \Vert \nabla u(t)\Vert _{L^2}^{2}\\&\quad +\! \left( \!1\!-\!\frac{N}{2p}\! \right) \epsilon ^{\frac{2p}{2p-N}}\!\Vert u(t)\Vert _{L^2}^{2} \end{aligned}$$

Take \(\epsilon =\epsilon (t)\) so that \(\frac{N}{2 p} \cdot \frac{C(N)^{N/p}}{\epsilon (t)^{\frac{2p}{N}}} \Vert \omega _0(\cdot ,t)\Vert _{L^p}^{\frac{2p}{N}} =1\), i.e. \(\epsilon (t)=(N/2p)^{N/2p} C(N)^{N^2/2p^2} \Vert \omega _0(\cdot ,t)\Vert _{L^p}\) and apply (48), then

$$\begin{aligned}&{\bar{\omega }}_\infty \int _{0}^{T} \Vert u(t)\Vert _{L^2}^{2}\mathrm {\,d}t \le C(N)^{N/p} \left( 1-\frac{N}{2p}\right) {\int _{0}^{T}}\epsilon (t)^{\frac{2p}{2p-N}}\Vert u(t)\Vert _{L^2}^{2} \mathrm {\,d}t \\&\quad \le (N/2p)^{N/(2p-N)} C(N)^{2N/(2p-N)} {\int _{0}^{T}}\Vert \omega _0(\cdot ,t) \Vert _{L^p}^{\frac{2p}{2p-N}}\Vert u(t)\Vert _{L^2}^{2} \mathrm {\,d}t\\&\quad \le (N/2p)^{N/(2p-N)} C(N)^{2N/(2p-N)} \sup _{t\in [0,T]} \Vert \omega _0 (\cdot ,t)\Vert _{L^p}^{\frac{2p}{2p-N}} \int _{0}^{T} \Vert u(t)\Vert _{L^2}^{2}\mathrm {\,d}t\\&\quad < {\bar{\omega }}_\infty \int _{0}^{T} \Vert u(t)\Vert _{L^2}^{2}\mathrm {\,d}t, \end{aligned}$$

a contradiction proving that (44) has no nontrivial T-periodic solutions.

7 Main results

Let us state the main results of the paper.

Theorem 7.1

Suppose that f satisfies conditions (2)–(5) and (7). If \({\mathrm {Ker}}\, (\Delta +{\widehat{\omega }}) = \{0 \},\) the problem

$$\begin{aligned} {\frac{\partial u}{\partial t}} = \lambda \Delta u + \lambda \omega (t,x) u,\quad x\in {\mathbb {R}}^N, \ t>0, \end{aligned}$$
(52)

has no nonzero T-periodic solutions for \(\lambda \in (0,1]\) and

$$\begin{aligned} \lim _{r\rightarrow +\infty } {\mathrm {esssup}}_{|x|>r}\, {\widehat{\omega }}_\infty (x)<0, \end{aligned}$$
(53)

then the equation

$$\begin{aligned} {\frac{\partial u}{\partial t}} = \Delta u + f(t,x,u),\quad x\in {\mathbb {R}}^N, \, t>0, \end{aligned}$$
(54)

admits a T-periodic solution

$$\begin{aligned} u\in C([0,+\infty ), H^2({\mathbb {R}}^N)) \cap C^1([0,+\infty ),L^2({\mathbb {R}}^N)). \end{aligned}$$

Theorem 7.2

Suppose that all the assumptions of Theorem 7.1 are satisfied and,  additionally,  that (42) holds,  \({\mathrm {Ker}}\, (\Delta +{\widehat{\alpha }}) = \{0 \},\) the equation

$$\begin{aligned} {\frac{\partial u}{\partial t}} = \lambda \Delta u + \lambda \alpha (t,x) u,\quad x\in {\mathbb {R}}^N, \ t>0, \end{aligned}$$
(55)

has no nonzero T-periodic solutions for \(\lambda \in (0,1]\) and

$$\begin{aligned} \lim _{r\rightarrow +\infty } \mathop {\mathrm {esssup}}\limits _{|x|>r}\, {\widehat{\alpha }}_\infty (x)<0. \end{aligned}$$

If \(m_+(\Delta +{\widehat{\alpha }})\not \equiv m_+(\Delta +{\widehat{\omega }})\) \(\text {mod}\) 2,  then the equation (54) admits a nontrivial T-periodic solution

$$\begin{aligned} u\in C([0,+\infty ), H^2({\mathbb {R}}^N))\cap C^1([0,+\infty ),L^2({\mathbb {R}}^N)). \end{aligned}$$

Below we provide the linearization scheme for computing the fixed point index of the Poincaré operator in the autonomous case.

Proposition 7.3

Assume that \(f:{\mathbb {R}}^N\times {\mathbb {R}}\rightarrow {\mathbb {R}}\) satisfies conditions (3), (5) and (25) (in their time-independent versions) and let \({\varvec{\Phi }}_t\) be the translation along trajectories for the autonomous equation

$$\begin{aligned} {\frac{\partial u}{\partial t}}= \Delta u + f(x,u),\quad x\in {\mathbb {R}}^N, \ t>0, \end{aligned}$$
  1. (i)

    If (43) holds and \({\mathrm {Ker}} (\Delta + \omega ) =\{ 0\},\) then there exists \(R_0>0\) such that \(-\Delta u(x)=f(x,u(x)),\ x\in {\mathbb {R}}^N,\) has no solutions \(u\in H^1({\mathbb {R}}^N)\) with \(\Vert u\Vert _{H^1} \ge R_0\) and there exists \({\bar{t}}>0\) such that,  for all \(t\in (0, {\bar{t}}],\) \({\varvec{\Phi }}_t ({\bar{u}})\ne {\bar{u}}\) for all \({\bar{u}}\in H^1({\mathbb {R}}^N) {\setminus } B_{H^1}(0,R_0)\) and,  for all and all \(t\in (0,{\bar{t}}]\) and \(R\ge R_0,\)

    $$\begin{aligned} \mathrm {Ind}({\varvec{\Phi }}_t, B_{H^1} (0,R) ) = (-1)^{m_+(\Delta +\omega ),} \end{aligned}$$

    where \(m_+(\Delta +\omega )\) is the total multiplicity of the positive eigenvalues of \(\Delta +\omega \).

  2. (ii)

    If (9) holds and \({\mathrm {Ker}} (\Delta +\alpha ) =\{ 0\},\) then there exists \(r_0>0\) such that \(-\Delta u(x)=f(x,u(x)),\ x\in {\mathbb {R}}^N,\) has no solutions with \(0<\Vert u\Vert _{H^1} \le r_0\) and there exists \({\bar{t}}>0\) such that,  for all \(t\in (0, {\bar{t}}],\) \({\varvec{\Phi }}_t ({\bar{u}})\ne \bar{u}\) for all \({\bar{u}}\in B_{H^1}(0,r_0){\setminus } \{0\}\) and,  for each \(t\in (0,{\bar{t}}],\)

    $$\begin{aligned} \mathrm {Ind}({\varvec{\Phi }}_t, B_{H^1} (0,r_0) ) = (-1)^{m_+(\Delta +\alpha ),} \end{aligned}$$

    where \(m_+(\Delta +\alpha )\) is the total multiplicity of the positive eigenvalues of \(\Delta +\alpha \).

Remark 7.4

Recall the known arguments on the spectrum of \(\Delta +\omega \) or, equivalently, of \(-\Delta - \omega \). To this end, define \({ \mathbf{B}}_0: D({\mathbf{B}}_0)\rightarrow L^2({\mathbb {R}}^N)\) with \(D({\mathbf{B}}_0):=H^1({\mathbb {R}}^N)\) by \({\mathbf{B}}_0 u:= \omega _0 u\) and \({\mathbf{B}}_\infty : L^2({\mathbb {R}}^N)\rightarrow L^2({\mathbb {R}}^N)\) by \({\mathbf{B}}_\infty u:=\omega _\infty u\). By [22], \(\mathbf{A}_\infty :=\mathbf{A} - {\mathbf{B}}_0 - {\mathbf{B}}_\infty \) is a \(C_0\) semigroup generator and its spectrum \(\sigma (\mathbf{A}_\infty )\) is contained in an interval \((-c, +\infty )\) with some \(c>0\). By the assumption (53), it follows from [23] that the essential spectrum

$$\begin{aligned} \sigma _{ess}(\mathbf{A}-{\mathbf{B}}_\infty ) \subset [d,+\infty ), \end{aligned}$$

where \(d=-\lim _{r\rightarrow +\infty } {\mathrm {esssup}}_{|x|>r}\, {\widehat{\omega }}_\infty (x)\). Moreover, it is known, that \(\mathbf {B}_0\) is relatively\((\mathbf {A}-\mathbf {B}_\infty )\)-compact – for the proof we refer to [24, Lem. 3.1], where the result is obtained under assumption \(N \ge 3\). However, a proper restatement, i.e. exploiting Sobolev embeddings \(H^1({\mathbb {R}}) \subset L^\infty ({\mathbb {R}})\) for \(N=1\) and \(H^1({\mathbb {R}}^2) \subset L^{4}({\mathbb {R}}^2)\) - in case \(N=2\) together with the Rellich–Kondrachov Theorem, leads to the same conclusion. Therefore, by use of the Weyl theorem on essential spectra, we obtain \(\sigma _{ess} (\mathbf{A}_\infty ) =\sigma _{ess} (\mathbf{A}-{\mathbf{B}_\infty }) \subset [{\bar{a}}, +\infty )\) (see e.g. [26]). Hence, by general characterizations of essential spectrum, we see that \(\sigma (\mathbf{A}_\infty )\cap (-\infty ,0)\) consists of isolated eigenvalues with finite dimensional eigenspaces (see [26]).

Proof of Proposition 7.3

(i) We start with an observation that there exists \(R_0>0\) such that the problem

$$\begin{aligned} 0=\Delta u + (1-\mu )f(x,u) + \mu \omega (x)u, \quad x\in {\mathbb {R}}^N, \end{aligned}$$
(56)

has no weak solutions in \(H^1({\mathbb {R}}^N){\setminus } B_{H^1}(0,R_0)\). To see this, suppose to the contrary that there exist a sequence \((\mu _n)\) in [0, 1] and solutions \({\bar{u}}_n\), \(n\ge 1\), of (56) with \(\mu =\mu _n\) such that \(\Vert \bar{u}_n\Vert _{H^1}\rightarrow +\infty \) as \(n\rightarrow +\infty \). Put \(\rho _n:=1+\Vert \bar{u}_n\Vert _{H^1}\) and observe that \({\bar{v}}_n:= \frac{{\bar{u}}_n}{\rho _n}\) are solutions of

$$\begin{aligned} 0=\Delta v + (1-\mu _n) \rho _n^{-1} f(x,\rho _n v)+\mu _n\omega (x) v, \quad x\in {\mathbb {R}}^N. \end{aligned}$$

Clearly

$$\begin{aligned} \rho _n^{-1} f(x,\rho _{n} v)\rightarrow \omega (x)v\quad \text{ as } n \rightarrow +\infty \text{ for } \text{ all } t\ge 0 \text{ and } \text{ a.a. } x\in {\mathbb {R}}^N. \end{aligned}$$

Hence, by use of Remark 6.2 we see that \(({\bar{u}}_n)\) contains a sequence convergent to some \({\bar{u}}_0\in H^1({\mathbb {R}}^N)\) being a weak nonzero solution of \(0=\Delta u + \omega (x) u, \,x\in {\mathbb {R}}^N,\) a contradiction proving that (56) has no solutions outside some ball \(B_{H^1}(0,R_0)\).

Now consider the equation

$$\begin{aligned} \frac{\partial u}{\partial t} = \Delta u + (1-\mu ) f(x,u) +\mu \omega (x) u, \quad x\in {\mathbb {R}}^N, \ t>0, \end{aligned}$$
(57)

where \(\mu \in [0,1]\) is a parameter. Let \({\varvec{\Psi }}_t : H^1({\mathbb {R}}^N)\times [0,1] \rightarrow H^1({\mathbb {R}}^N)\), \(t>0\), be the parameterized translation along trajectories operator for the above equation. In view of Theorem 6.3, there exists \({\bar{t}}>0\) such that

$$\begin{aligned} {\varvec{\Psi }}_t ({\bar{u}}, \mu )\ne {\bar{u}} \quad \text{ for } \text{ all } t\in (0, {\bar{t}}], {\bar{u}}\in \partial B_{H^1}(0,R_0). \end{aligned}$$

By Proposition 5.4(iii), the homotopy \({\varvec{\Psi }}_t\) is admissible in the sense of the fixed point theory for ultimately compact maps (see Sect. 2). Therefore, using the homotopy invariance one has, for \(t\in (0,{\bar{t}}]\),

$$\begin{aligned} \mathrm {Ind}({\varvec{\Phi }}_t, B_{H^1}(0,R_0)) = \mathrm {Ind}(e^{-t\mathbf{A}_\infty }, B_{H^1}(0,R_0)), \end{aligned}$$
(58)

where \(\mathbf{A}_\infty :=\mathbf{A} -{\mathbf{B}}_0 - {\mathbf{B}}_\infty \).

It is left to determine the fixed point index of \(e^{-t\mathbf{A}_\infty }\). We note that the set \(\sigma (\mathbf{A}_\infty ) \cap (-\infty ,0)\) is bounded and closed. Hence, in view of the spectral theorem (see [28]) there are closed subspaces \(X_-\) and \(X^+\) of \(L^2 ({\mathbb {R}}^N)\) such that \(X_-\oplus X^+=L^2 ({\mathbb {R}}^N)\), \(\dim X_-<+\infty \), \(\mathbf{A}_\infty (X_-)\subset X_-\), \(\mathbf{A}_\infty (D({\mathbf{A}_\infty })\cap X^+)\subset X^+\), \(\sigma (\mathbf{A}_\infty |_{X_-})=\sigma (\mathbf{A}_\infty )\cap (-\infty ,0)\), \(\sigma (\mathbf{A}_\infty |_{X^+})=\sigma (\mathbf{A}_\infty )\cap (0,+\infty )\). Define \( {{\varvec{\Theta }}}_t:H^1({\mathbb {R}}^N)\times [0,1] \rightarrow H^1({\mathbb {R}}^N)\) by

$$\begin{aligned} {{\varvec{\Theta }}}_t ({\bar{u}}, \mu ):= (1-\mu ) e^{-t \mathbf{A}_\infty }\bar{u} + \mu e^{-t \mathbf{A}_\infty } {\mathbf{P}}_-{\bar{u}}, \end{aligned}$$

where \({\mathbf{P}}_-:H^1({\mathbb {R}}^N)\rightarrow H^1({\mathbb {R}}^N)\) is the restriction of the projection onto \(X_-\cap H^1({\mathbb {R}}^N)\) in \(L^2({\mathbb {R}}^N)\). Since \(\dim X_-<+\infty \) we infer that \({\mathbf{P}}_-\) is continuous. W also claim that \({{\varvec{\Theta }}}_t\) is ultimately compact. To see this take a bounded set \(W\subset H^1({\mathbb {R}}^N)\) such that \(W = \overline{\mathrm {conv}}^{H^1} {{\varvec{\Theta }}}_t ( W\times [0,1]).\) This means that \(W \subset \overline{\mathrm {conv}}^{H^1} e^{-t\mathbf{A}_\infty }(W\cup {\mathbf{P}}_- W)\). Since \(W\cup {\mathbf{P}}_- W\) is bounded, Proposition 5.4(ii) implies that W is relatively compact in \(H^1({\mathbb {R}}^N)\), which proves the ultimate compactness of \({{\varvec{\Theta }}}_t\). Since \({\mathrm {Ker}} (I - {{\varvec{\Theta }}}_t (\cdot ,\mu ))=\{ 0\}\) for \(\mu \in [0,1]\), by the homotopy invariance and the restriction property of the Leray–Schauder fixed point index, one gets

$$\begin{aligned} \mathrm {Ind}(e^{-t\mathbf{A}_\infty }, B_{H^1} (0,R_0) )= & {} \mathrm {Ind}_{LS}(e^{-t \mathbf{A}_\infty }{{\mathbf{P}}_-} , B_{H^1}(0,R_0) )\\= & {} \mathrm {Ind}_{LS} ( e^{-t (\mathbf{A}_\infty |_{X_-})} , B_{H^1}(0,R_0)\cap X_-)\\= & {} (-1)^{\dim X_-}= (-1)^{m_+(\Delta +\omega )}. \end{aligned}$$

The latter equality comes from the fact that \(\sigma (\mathbf{A}_{\infty }|_{X_-})\subset (-\infty ,0) \) consists of isolated eigenvalues of finite dimensional eigenspaces. This ends the proof of (i) together with (58).

(ii) First we shall prove the existence of \(r_0 >0\) such that the problem

$$\begin{aligned} 0=\Delta u + (1-\mu )f(x,u) + \mu \alpha (x)u, \quad x\in {\mathbb {R}}^N, \end{aligned}$$
(59)

has no solutions in \(B_{H^1}(0, r_0){\setminus } \{0\}\). Suppose to the contrary that there exist a sequence \((\mu _n)\) in [0, 1] and solutions \({\bar{u}}_n:[0,+\infty )\rightarrow H^1({\mathbb {R}}^N)\), \(n\ge 1\), of (59) with \(\mu =\mu _n\) such that \(\Vert \bar{u}_n\Vert _{H^1}\rightarrow 0^+\) as \(n\rightarrow +\infty \) and \(\Vert {\bar{u}}_n\Vert _{H^1}\ne 0\), \(n\ge 1\). Put \(\rho _n:=\Vert {\bar{u}}_n\Vert _{H^1}\). Then \(\bar{v}_n:=\frac{{\bar{u}}_n}{\rho _n}\) are solutions of

$$\begin{aligned} 0=\Delta v + (1-\mu _n) \rho _n^{-1} f(x,\rho _n v)+\mu _n\alpha (x)v, \quad x\in {\mathbb {R}}^N. \end{aligned}$$

Observe that

$$\begin{aligned} \rho _n^{-1} f(x,\rho _{n} v)\rightarrow \alpha (x)v\quad \text{ as } n \rightarrow \infty \text{ for } \text{ a.a. } x\in {\mathbb {R}}^N. \end{aligned}$$

Using again Remark 6.2 one can see that \(({\bar{u}}_n)\) (up to a subsequence) converges to some nonzero solution of \(0=\Delta u + \alpha (x) u, \,x\in {\mathbb {R}}^N,\) a contradiction. Summing up, there is \(r_0 > 0\) such that (59) has no solutions \(u \in H^1({\mathbb {R}}^N)\) with \(0<\Vert u\Vert _{H^1} \le r_0\). The rest of the proof runs as before: by \({\varvec{\Psi }}_t : H^1({\mathbb {R}}^N)\times [0,1] \rightarrow H^1({\mathbb {R}}^N)\), \(t>0\) we denote the translation along trajectories operator for the equation

$$\begin{aligned} \frac{\partial u}{\partial t} = \Delta u + (1-\mu ) f(x,u) +\mu \alpha (x) u, \quad x\in {\mathbb {R}}^N, \, t>0, \, \mu \in [0,1], \end{aligned}$$
(60)

and, by applying Theorem 6.3 we obtain the existence of \({\bar{t}}>0\) such that

$$\begin{aligned} {\varvec{\Psi }}_t ({\bar{u}}, \mu )\ne {\bar{u}} \quad \text{ for } \text{ all } t\in (0, {\bar{t}}], {\bar{u}}\in \partial B_{H^1}(0,r_0). \end{aligned}$$

Next Proposition 5.4(iii) ensures the admissibility of \({\varvec{\Psi }}_t\) and by homotopy invariance, for \(t\in (0,{\bar{t}}]\), we have

$$\begin{aligned} \mathrm {Ind}({\varvec{\Phi }}_t, B_{H^1}(0,r_0)) = \mathrm {Ind}(e^{-t\mathbf{A}_0}, B_{H^1}(0,r_0)), \end{aligned}$$
(61)

where \(\mathbf{A}_0:=\mathbf{A}-\mathbf{C}_0 - \mathbf{C}_\infty \) and operators \(\mathbf{C}_i: D(\mathbf{C}_i) \rightarrow L^2({\mathbb {R}}^N)\) with \(D(\mathbf{C}_i) = H^1({\mathbb {R}}^N)\) are given by \(\mathbf{C}_i u:= \alpha _i u\), \(i \in \{0,\infty \}\). Now one can easily determine fixed point index of \(e^{-t\mathbf{A}_0}\) by arguing as in part (i) (with \(\mathbf{A}_\infty \) replaced by \(\mathbf{A}_0\) and \(B_{H^1}(0, R_0)\) replaced by \(B_{H^1}(0, r_0)\)) and, as a consequence, obtain that

$$\begin{aligned} \mathrm {Ind}(e^{-t\mathbf{A}_0}, B_{H^1}(0,r_0) )= & {} \mathrm {Ind}_{LS}(e^{-t \mathbf{A}_0}{{\mathbf{P}}_-} , B_{H^1}(0,r_0) )\\= & {} \mathrm {Ind}_{LS} ( e^{-t (\mathbf{A}_0|_{X_-})} , B_{H^1}(0,r_0)\cap X_-) = (-1)^{m_+ (\Delta +\alpha )}. \end{aligned}$$

This completes the proof. \(\square \)

Now we are ready to prove the main results.

Proof of Theorem 1.1

Let \({\varvec{\Phi }}_t\), \(t>0\), be the translation operator for (1). It is clear that

$$\begin{aligned} \lim _{|u|\rightarrow +\infty }\frac{{\widehat{f}}(x,u)}{u} = {\widehat{\omega }}(x), \quad \text{ for } \text{ any } x\in {\mathbb {R}}^N. \end{aligned}$$

Hence, by applying Proposition 7.3(i) we obtain \(R_0 > 0\) such that

$$\begin{aligned} \Delta u(x) + {\widehat{f}}(x,u(x))=0, \quad x\in {\mathbb {R}}^N, \end{aligned}$$

has no solutions in the set \(H^1({\mathbb {R}}^N){\setminus } B_{H^1}(0,R_0)\) and there exists \(t_0>0\) such that, for \(t\in (0,t_0]\),

$$\begin{aligned} \mathrm {Ind}(\widehat{\varvec{\Phi }}_t, B_{H^1}(0,R_0)) = (-1)^{m_+(\Delta +{\widehat{\omega }})}. \end{aligned}$$
(62)

Due to Proposition 6.5 and the assumption, increasing \(R_0\) if necessary, we can assume that (44) has no T-periodic solutions starting from \(H^1({\mathbb {R}}^N) {\setminus } B_{H^1}(0,R_0)\). Taking \(U:=B_{H^1}(0,R_0)\) and applying Corollary 6.4 we get

$$\begin{aligned} \mathrm {Ind}({\varvec{\Phi }}_T, B_{H^1}(0,R_0))= \lim _{t\rightarrow 0^+} \mathrm {Ind}(\widehat{\varvec{\Phi }}_t, B_{H^1}(0,R_0)), \end{aligned}$$

which along with (62) yields \(\mathrm {Ind}({\varvec{\Phi }}_T, B_{H^1}(0,R_0))=(-1)^{m_+(\Delta +{\widehat{\omega }})}\). This and the existence property of the fixed point index imply that there exists \({\bar{u}}\in B_{H^1}(0,R_0)\) such that \({\varvec{\Phi }}_T (\bar{u})={\bar{u}}\), i.e. there exists a T-periodic solution of (1). \(\square \)

Proof of Theorem 1.2

First use Proposition 7.3 to get \(R_0,r_0 >0\) such that

$$\begin{aligned} \lim _{t\rightarrow 0^+} \mathrm {Ind}(\widehat{\varvec{\Phi }}_t, B_{H^1}(0,R)) = (-1)^{m_+(\Delta +{\widehat{\omega }})} \quad \text{ if } \ R\ge R_0 \end{aligned}$$
(63)

and

$$\begin{aligned} \lim _{t\rightarrow 0^+} \mathrm {Ind}(\widehat{\varvec{\Phi }}_t, B_{H^1}(0,r)) = (-1)^{m_+(\Delta +{\widehat{\alpha }})} \quad \text{ if } \ 0 < r\le r_0. \end{aligned}$$
(64)

Now, due to Proposition 6.5 there exist \(R\ge R_0\) and \(r\in (0,r_0]\) such that, for any \(\lambda \in (0,1]\), (41) has no solutions with \(u(0) \in B_{H^1}(0,r) \cup \left( H^1({\mathbb {R}}^N) {\setminus } B_{H^1}(0,R)\right) \). Next we put \(U:= B_{H^1}(0,R) {\setminus } \overline{B_{H^1}(0,r)}\) and apply Corollary 6.4 to get

$$\begin{aligned} \mathrm {Ind}({{\varvec{\Phi }}}_T, U) = \lim _{t\rightarrow 0^+} \mathrm {Ind}(\widehat{\varvec{\Phi }}_t, U). \end{aligned}$$

This together with (63) and (64), by use of the additivity property of the fixed point index, yields

$$\begin{aligned} \mathrm {Ind}({{\varvec{\Phi }}}_T, U)= & {} \lim _{t\rightarrow 0^+} \mathrm {Ind}(\widehat{\varvec{\Phi }}_t, B_{H^1}(0,R)) - \lim _{t\rightarrow 0^+} \mathrm {Ind}(\widehat{\varvec{\Phi }}_t, B_{H^1}(0,r))\\= & {} (-1)^{m_+(\Delta +{\widehat{\omega }})}-(-1)^{m_+(\Delta +\widehat{\alpha })} \ne 0, \end{aligned}$$

which gives the existence of the fixed point of \({\varvec{\Phi }}_T\) in U. \(\square \)