Heat equation with an exponential nonlinear boundary condition in the half space

We consider the initial-boundary value problem for the heat equation in the half space with an exponential nonlinear boundary condition. We prove the existence of global-in-time solutions under the smallness condition on the initial data in the Orlicz space expL2(R+N)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm {exp}L^2({\mathbb {R}}^N_+)$$\end{document}. Furthermore, we derive decay estimates and the asymptotic behavior for small global-in-time solutions.


Introduction
We consider the initial-boundary value problem for the heat equation in the half space R N + = {x = (x , x N ) ∈ R N : x N > 0} with a nonlinear boundary condition ⎧ ⎨ x ∈ ∂R N + , t > 0, (1.1) Partial Differential Equations and Applications (2022) 3:36 where N ≥ 1, ∂ t = ∂/∂t, ∂ ν = −∂/∂ x N , and ϕ is the given initial data. Here f (u) is the nonlinearity which has an exponential growth at infinity with f (0) = 0. More precisely, the condition for the nonlinearity (see (1.9)) covers certain limiting cases which are critical with respect to the growth of the nonlinearity and the regularity of the initial data. In this paper, under a smallness condition on the initial data, we prove the existence of global-in-time solutions to problem (1.1). Furthermore, we derive some decay estimates and the asymptotic behavior of small global-in-time solutions. The nonlinear boundary value problem such as (1.1) can be physically interpreted as a nonlinear radiation law. The case of power nonlinearities f (u) = |u| p−1 u with p > 1, that is, has been extensively studied in many papers (see e.g. [5, 6, 11, 13, 17-22, 25, 26] and the references therein). It is well-known that problem (1.2) satisfies a scale invariance property, namely, for λ ∈ R + , if u is a solution to problem (1.2), then is also a solution to problem (1.2) with initial data ϕ λ (x) := λ 1/( p−1) ϕ(λx). In the study of the nonlinear boundary value problem (1.2), it seems that all function spaces invariant with respect to the scaling transformation (1.3) play an important role. In fact, for Lebesgue spaces, we can easily show that the norm of the space L q (R N + ) is invariant with respect to (1.3) if and only if q = q c := N ( p−1), and, for the given nonlinearity |u| p−1 u, the Lebesgue space L q c (R N + ) plays the role of critical space for the local well-posedness and the existence of global-in-time solutions to problem (1.2) (see e.g. [13,18,20]).
On the other hand, the case of the Cauchy problem with the power nonlinearity, that is, also satisfies a scale invariance property, namely, for λ ∈ R + , if u is a solution to problem (1.4), then u λ (x, t) := λ 2 p−1 u(λx, λ 2 t) (1.5) is also a solution to problem (1.4) with the initial data ϕ λ (x) := λ 2/( p−1) ϕ(λx). So we can easily show that the norm of the space L q (R N ) is invariant with respect to (1.5) if and only if q =q c := N ( p − 1)/2, and it is well-known that the Lebesgue space Lq c (R N ) plays the role of critical space for the well-posedness of problem (1.4) (see e.g. [3,12,27,30,31] and references therein). Furthermore, the scaling property (1.5) also holds for the nonlinear Schrödinger equation and it is well known that the Sobolev space H s c (R N ) with s c := N /2 − 2/( p − 1) plays the role of critical space for the well-posedness of problem (1.6) (see e.g. [4]). From these results, we have two critical growth rates of the nonlinearity, that is, p h := 1 + (2q)/N and p s := 1 + 4/(N − 2s), and these two critical exponents are connected by the Sobolev embedding,Ḣ s (R N ) → L q (R N ), where s and q satisfy 0 ≤ s < N /2 and 1/q = 1/2−s/N . The case s c = N /2 is a limiting case from the following points of view: (i) for s > N /2, H s (R N ) embeds into L ∞ (R N ); (ii) any power nonlinearity is subcritical, since H N /2 (R N ) embeds into any L q (R N ) space (for q ≥ 2); (iii) H N /2 (R N ) does not embed into L ∞ (R N ), and thanks to Trudinger's inequality [29] one knows that H N /2 (R N ) embeds into the Orlicz space expL 2 (R N ).
For this limiting case, Nakamura and Ozawa [24] consider the nonlinear Schrödinger equation with an exponential nonlinearity of asymptotic growth f (u) ∼ e u 2 and with a vanishing behavior at the origin, and they show the existence of global-in-time solutions under a smallness assumption of the initial data in H N /2 (R N ).
As a natural analogy to the results of [24], the third author of this paper and Ruf [28] and Ioku [14] consider the Cauchy problem of the semilinear heat equation with exponential nonlinearity of the form f (u) = |u| 4 N ue u 2 (1. 7) and the initial data ϕ belonging to the Orlicz space expL 2 (R N ) defined as (see also Definition 2.1). They consider the corresponding integral equation (1.8) and prove the existence of local/global-in-time (mild) solutions to this equation (1.8) under the smallness assumption of initial data in expL 2 (R N ). Furthermore, the authors of this paper and Ruf [10] show the equivalence between mild solutions (solution to the integral equation (1.8)) and weak solutions to the heat equation with the nonlinearity f (u) as in (1.7), and derive some decay estimates and the asymptotic behavior for small global-in-time solutions. The growth rate of (1.7) at infinity seems to be optimal in the framework of the Orlicz space expL 2 (R N ).
In fact, if f (u) ∼ e |u| r with r > 2, there exist some positive initial data ϕ ∈ expL 2 (R N ) such that problem (1.8) does not possess any classical local-in-time solutions (see [15]). For the fractional diffusion case and general power-exponential nonlinearities, see e.g. [8,10,23]. Furthermore, for ϕ ∈ expL 2 (R N ), which implies ϕ ∈ L p (R N ) for p ∈ [2, ∞), the decay rate of (1.7) near origin, that is, f (u) ∼ |u| 4/N u, is optimal in the framework of L 2 (R N ). See e.g. [3,30]. The above limiting case in R N appears from the relationship between p h and p s by the Sobolev embedding. For problem (1.2), we can easily show that the norm of the space H s (R N + ) is invariant with respect to (1.3) if and only if s =s c := N /2 − 1/( p − 1), and we have two critical growth rate of the nonlinearity, that is,p h = 1 + q/N andp s = 1 + 2/(N − 2s). These two exponents are also connected by the Sobolev embedding,Ḣ s (R N + ) → L q (R N + ), where s and q satisfy the same conditions as in the case of R N . This means that the same limiting case appears for problem (1.2). On the other hand, as far as we know, there are no results which treat the exponential nonlinearity for the nonlinear boundary problem (1.1).
Based on the above, in this paper, we assume that the nonlinearity f satisfies the following: there exist C f > 0 and λ > 0 such that Partial Differential Equations and Applications (2022) 3:36 This assumption covers the case which is one of the candidates for the optimal growth rate of the nonlinearity in the framework of the Orlicz space expL 2 (R N + ) and the optimal decay rate near origin in the framework of L 2 (R N + ) (see e.g. [18]). Following [10,14,28], for problem (1.1) with (1.9), we consider the corresponding integral equation, and prove the existence of global-in-time (mild) solutions under some smallness assumption of the initial data in expL 2 (R N + ). Furthermore, we obtain some decay estimates for the solutions in the following two cases ϕ ∈ expL 2 (R N + ) only (slowly decaying case), and ϕ ∈ expL 2 (R N + ) ∩ L p (R N + ) with p ∈ [1, 2) (rapidly decaying case). In particular, for the rapidly decaying case p = 1, we show that the global-in-time solutions with some suitable decay estimates behave asymptotically like suitable multiples of the Gauss kernel.
Before treating our main results, we introduce some notation and define a solution to problem (1.1). Throughout this paper we often identify R N −1 with ∂R N + . Let g N = g N (x, t) be the Gauss kernel on R N , that is, Let G = G(x, y, t) be the Green function for the heat equation on R N + with the homogenous Neumann boundary condition, that is, (1.11) where y * = (y , −y N ) for y = (y , y N ) ∈ R N + . Then, we define a (mild) solution to problem (1.1).
(i) In the case when N ≥ 2, we call u a solution to problem (1.1) ϕ in the weak * topology.
In the case when T = ∞, we call u a global-in-time solution to problem (1.1).
We recall that u(t) −→ t→0 ϕ in weak * topology if and only if for any ψ belonging to the predual space of expL 2 (R N + ) (see Sect. 2). In what follows, we denote by · expL 2 the norm of expL 2 := expL 2 (R N + ) defined by (2.14), for r ∈ [1, ∞], we write · L r := · L r (R N + ) and | · | L r := · L r (R N −1 ) for simplicity. Furthermore, for a function φ(x , Now we are ready to state the main results of this paper. First we show the existence of global-in-time solutions to problem (1.1) under the smallness assumption of the initial data in expL 2 . Theorem 1.1 Let N ≥ 1 and ϕ ∈ expL 2 . Suppose that f satisfies (1.9). Then there exist positive constants ε = ε(N ) > 0 and C = C(N ) > 0 such that, if ϕ expL 2 < ε, then there exists a unique global-in-time solution u to problem (1.1) satisfying where h(t) = min{t N /4 , 1}, and for any q ∈ [2, ∞), where is the gamma function On the other hand, if N = 1, then the boundary of R + is x = 0, namely, it is only one point. From these differences, we need to divide the proof into two cases, N ≥ 2 and N = 1, and we have two estimates as in (1.15).
(ii) Following [15], we denote by expL 2 . Then, by an argument similar to that in the proof of [15,Theorem 1.2], it seems likely to obtain the existence of local-in-time solutions to problem (1.1) for any ϕ ∈ expL 2 0 (R N + ) under the weaker condition where λ > 0 and C > 0. This has not been fully explored and it is left for further investigation.
From now, we focus on the unique solution u to problem (1.1) satisfying (1.14) and (1.15). The following result gives some decay estimates for the slowly decaying case, that is, ϕ ∈ expL 2 only. Theorem 1.2 Assume the same conditions as in Theorem 1.1. Furthermore, suppose that there exists a unique solution u to problem (1.1) satisfying (1.14) and (1.15). Then there exist some positive constants ε = ε(N ) > 0 and C = C(N ) > 0 such that, if ϕ expL 2 < ε, then the solution u satisfies Next we consider the rapidly decaying case, that is, ϕ ∈ expL 2 ∩ L p with p ∈ [1, 2). We can prove two kinds of results about decay estimates of solutions to problem (1.1). In Theorem 1.3, we only assume the smallness condition of the expL 2 norm of the initial data. This means that we can allow the L p norm of the same data to be large. On the other hand, under this mild assumption, we have an additional restriction about the range of L q spaces for the case N ≥ 3. In Theorem 1.4, under a stronger assumption, that is, a smallness assumption not only for the expL 2 but also for the L p norm of the initial data, we obtain better decay estimates, with no additional restrictions about the range of L q spaces even for the case N ≥ 3. In the following we denote for any r ≥ 1 · expL 2 ∩L r := max{ · expL 2 , · L r }.

Remark 1.3
By (1.9) the nonlinearity f (u) behaves like |u| 1+2/N for u → 0. So, for the case N ≥ 2, since it follows from (1.15) that u ∈ L ∞ loc (0, ∞; L q (∂R N + )) for q ≥ 2, the nonlinear term f (u) belongs to L p (∂R N + ) for p ≥ (2N )/(N + 2). For the case N = 2, this means that f (u) ∈ L p (∂R N + ) for all p ≥ 1, but this implies a true constraint for the case N ≥ 3. This is the reason why in Theorem 1.3 we have to introduce some parameters p 1 (and p 2 , p 3 , and p 4 in Lemmata 2.2, 5.1, and 5.6, respectively) which are meaningful only for the case N ≥ 3.
Finally we address the question of the asymptotic behavior of solutions to problem (1.1) when ϕ ∈ expL 2 ∩ L 1 . We show that global-in-time solutions with suitable decay properties behave asymptotically like suitable multiples of the Gauss kernel. (1.23)

Remark 1.4
For the case N ≥ 2, by (1.12) we see that On the other hand, for the case N = 1, by (1.13) we have The paper is organized as follows. In Sect. 2 we recall some properties of the kernel G and its associate semigroup. In Sect. 3, applying the Banach contraction mapping principle, we prove Theorem 1.1. In Sects. 4 and 5, modifying the arguments of [20], we derive decay estimates on the boundary, and prove Theorems 1.2, 1.3, and 1.4. In Sect. 6 we obtain the asymptotic behavior of solutions to problem (1.1).

Preliminaries
In this section we recall some properties of the kernel G = G(x, y, t) and its associate semigroup. Throughout this paper, by the letter C we denote generic positive constants that may have different values also within the same line.
We first recall the following properties of the kernel G (see e.g [13,20,22]): G(x, y, t)dy = 1 for any x ∈ R N + and t > 0; (ii) for any (x, t), (z, s) ∈ R N + × (0, ∞), it holds that Furthermore, it follows from (1.10) and (1.11) that We denote by S 1 (t)ϕ the unique bounded solution to the heat equation in R N + with the homogeneous Neumann boundary condition and the initial datum ϕ, that is, (2.4) and denote by e t ψ the unique bounded solution to the heat equation in R N −1 with the initial datum ψ, that is, In the case where N ≥ 2, we put for ψ ∈ L r (R N −1 ) with some r ∈ [1, ∞]. Since it holds that, for any r ∈ [1, ∞], (G 1 ) There exists a constant c 1 , which depends only on N , such that for ϕ ∈ L q (R N + ) and 1 ≤ q ≤ r ≤ ∞. Furthermore, there exists a constant c 2 , which depends only on N , such that, for the case N ≥ 2, (2.9) and, for the case N = 1, For any ψ ∈ L q (R N −1 ) and 1 ≤ q ≤ r ≤ ∞, it holds that Then, for any T > 0, S 1 (t)ϕ is bounded and smooth in R N + × (T , ∞).
We recall now the definition and the main properties of the Orlicz space expL 2 .

Definition 2.1
We define the Orlicz space expL 2 as where the norm is given by the Luxemburg type 14) The space expL 2 endowed with the norm u expL 2 is a Banach space, and admits as predual the Orlicz space defined by the complementary function of A(t) = e t 2 − 1, denoted bỹ A(t). This complementary function is a convex function such thatÃ(t) ∼ t 2 as t → 0 and A(t) ∼ t log 1/2 t as t → ∞. (see e.g. [2,Section 8].) Furthermore, it follows from (2.13) that and we have Next we recall the following property of the Gamma function.

Lemma 2.1 [10, Lemma 3.3]
For any q ≥ 1 and r ≥ 1, there exists a positive constant C > 0, which is independent of q and r , such that Applying this lemma, we prepare the following estimate for the nonlinear term f for the case N ≥ 2.
Proof For any k ∈ N ∪ {0}, we put Then, since it holds from N ≥ 2 and r ≥ p 2 with (2.20) that Applying Lemma 2.1 with the monotonicity property of the Gamma function (q) for q ≥ 3/2 (see, e.g. [1]), we see that This together with (2.23) implies that Therefore, taking a sufficiently small m < ε(r , λ) if necessary (e.g. m 2 ≤ 1/(4λr )), we get This implies (2.21), and the proof of Lemma 2.2 is complete.
Similarly to the case N ≥ 2, we prepare the following lemma, which is the one dimensional counterpart of Lemma 2.2.

Lemma 2.3
Let m > 0. Suppose that, for any q ∈ [2, ∞), the function u ∈ C(0, ∞) satisfies the condition Let f be the function satisfying the condition (1.9). Then there exists a positive constant

25)
and where C is independent of m and r .
Proof We first prove (2.25). For any k ∈ N ∪ {0}, let k be the constant defined by (2.22) with N = 1, namely, k = 2k + 3. Then, by (1.9) and (2.24) with q = k we have Since it holds from the monotonicity property of the Gamma function that Therefore, taking a sufficiently small m < ε(λ) if necessary, we get This implies (2.25).
Next we prove (2.26). For any k ∈ N ∪ {0}, put˜ k = 2k + 2. Then, similarly to the proof of (2.25), we have and then, by (2.24) with q = r and also q =˜ k and taking a sufficiently small m < ε(λ) if necessary, we have This implies (2.26), and the proof of Lemma 2.3 is complete.

Existence
In this section we prove Theorem 1.1. We first consider the case N ≥ 2. We introduce some notation. Let M > 0. Set Then (X M , d X ) is a complete metric space. For the proof of Theorem 1.1 we apply the Banach contraction mapping principle in X M to find a fixed point of where S 1 (t) is as in (2.4) and Here S 2 (t) is as in (2.6) and f satisfies (1.9). We remark that, for u ∈ X M , the function f (u) belongs to C(R N + ×(0, ∞)). Therefore, by Lemma 2.
More precisely, with an abuse of notation we denote by S 2 (t − s) f (u(s)) the operator S 2 (t − s) applied to the function f (u(x , 0, s)). In particular, we have Hence any fixed point of the integral operator satisfies the equation (1.12).
Furthermore, we have the following estimates for the function D[u].

Lemma 3.1 Let N ≥ 2 and u ∈ X M . Then there exists a positive constant ε
where C is independent of q and M. Furthermore, Proof We first prove (3.4). Let p 2 be the constant given in (2.20). Then, it holds that Since u ∈ X M , taking a sufficiently small ε 1 = ε 1 ( p 2 , λ) > 0 such that, for M < ε 1 , we can apply Lemma 2.2, and it holds that Substituting (3.6) to (3.5), we see that where B is the beta function, namely Substituting (3.9) to (3.8), we see that On the other hand, for fixed q ∈ [2, ∞), we put Then, it holds that p 2 ≤ q * < q and Since p 2 ≤ q * ≤ N , similarly to (3.6) again, taking a sufficiently small This together with (3.11) and (3.12) yields that where the constant C depends only on N since p 2 ≤ q * ≤ N . Thus, taking ε * = min{ε 1 , ε 2 , ε 3 } with (3.7), (3.10), and (3.13), we obtain (3.4).
Next we prove the continuity of D[u](x, t). Let T be an arbitrary positive constant. Then, it follows from (2.1) that for t ≥ T /2, we apply the same argument as in [9, Section 3, Chapter 1] to see that (3.14) Proof For any k ∈ N ∪ {0}, we put˜ Then, by (1.9) we recall that Since it follows from Hölder's inequality that . Furthermore, applying Lemma 2.1 with (3.15) and by the monotonicity property of the Gamma function, for k ≥ 1, we see that These together with (3.18) implies that Then, taking a sufficiently small ε * = ε * (N , λ) > 0 such that, for M < ε * , in a similar way as in Lemma 2.2, it holds that On the other hand, similarly to (3.12), by (2.12) with (q, r ) = ((2N )/3, 2N ), (3.3), and (3.16) we have Therefore, applying the same argument as in the proof of (3.19), for M < ε * , it holds that This implies that Combining (3.19) and (3.20), we have (3.14), thus Lemma 3.2 follows.

Remark 3.1
In the proof of Lemma 3.2, the estimate for sup t>0 t 1/(4N ) | · | L 2N is closed by itself. We need the term sup t>0 h(t) · L ∞ in the definition of the metric d X in order to ensure the uniform convergence of the Cauchy sequence so that the solution is continuous.
Now we are ready to complete the proof of Theorem 1.1 for the case N ≥ 2.
Proof of Theorem 1.1 (N ≥ 2). Let where c 1 and c 2 are constant given in (G 1 ). Then, by (2.8), (2.9), (2.17), and (2.18) we see that This together with property (G 3 ), Lemma 3.1, (3.2), and (3.21) yields that is a map on X M to itself. Furthermore, since it follows from (3.1) and (3.2) that for u, v ∈ X M , taking a sufficiently small ε 5 = ε 5 (N ) > 0 if necessary, for M < ε 5 , we can apply Lemma 3.2, and it holds that Then, applying the contraction mapping theorem ensures that there exists a unique u ∈ X M with is a complete metric space. Similarly to the proof of Theorem 1.1 for the case N ≥ 2, we apply the Banach contraction mapping principle in Y M to find a fixed point of Here g 1 is as in (1.10) and f satisfies (1.9). Applying Lemma 2.3, we have the following.
Since u ∈ Y M , taking a sufficiently small ε * = ε * (λ) > 0 such that, for M < ε * , we can apply Lemma 2.3, and it holds from (2.26) with r = 2 and (3.26) that Similarly, by (2.7) with (N , r ) = (1, ∞), (2.26), and (3.23), for any q ∈ [2, ∞), it holds that where C is independent of q and M. This implies that Then, we can take a sufficiently small ε * = ε * (λ) > 0 such that, for M < ε * , it holds that where D[v] is the function defined by (3.3). Since it follows from (1.14) and (2.17) that by (2.9) with q = r , for any q ∈ [2, ∞], we have Here c * is a constant independent of q and ϕ expL 2 . Furthermore, since it follows from the continuity of the function Then, we can take a sufficiently small ε > 0 such that, for ϕ expL 2 < ε, it follows from (4.2), (4.4), and (4.5) that On the other hand, we have the following.
We next consider the case N = 1. Let v be the function defined by (4.1). Then, it follows from (1.13) and (2.1) that the function v satisfies Here d * is a constant independent of ϕ expL 2 . Furthermore, similarly to (4.5), applying the same argument as in the proof of Lemma 3.3 with (1.15) and (4.1), we see that Then, we can take a sufficiently small ε > 0 such that, for ϕ expL 2 < ε, it follows from (4.27), (4.28), and (4.29) that On the other hand, we have the following, which is the one dimensional counterpart of Lemma 4.1. Let f be a function satisfying (1.9). Then, there exists ε * > 0, independent of T , such that, if A < ε * , then where C f is constant given in (1.9).

Rapidly decaying initial data
In this section we prove Theorems 1.3 and 1.4. Let L := ϕ expL 2 (5.1) We can assume, without loss of generality, that L < 1. Let p 1 be the constant given in (1.18). For ϕ L p 1 > 0, we denote where c 1 and c 2 are given in (G 1 ). Since we assume L < 1 and thanks to (1.17) we have Then we first show the following lemma, which is analogous to Lemma 4.1. N ≥ 2, T > 0, and p ∈ [1, 2). Furthermore let p 1 be the constant given in (1.18). Suppose that, for any q ∈ [p 1 , ∞], the function u ∈ C(R N + × (0, ∞)) satisfies

5)
where D is independent of q and K is the constant given in (5.2). Let f be a function satisfying (1.9). Then, forK as in (5.3), there exists a sufficiently large constant T 1 = T 1 (K , p 1 , λ, D) ≥ 1 such that if T ≥ T 1 it follows that, for any r ∈ [p 3 , ∞], where C f is given in (1.9) and Proof Let k ∈ N ∪ {0} and k be the constant given in (2.22). Since for any r ∈ [p 3 , ∞], by (1.9) and (5.5) we have for all t > 0. We can take a sufficiently large constant T 1 ≥ 1 such that, for all t > T 1 , it holds that It is enough to choose This together with (5.8) implies (5.6). Thus Lemma 5.1 follows.
Similarly, for the case N = 1, we have the following. where D > 0 and K is the constant given in (5.2). Let f be a function satisfying (1.9). Then, forK as in (5.3), there exists a sufficiently large constantT 1 =T 1 (K , p, λ, D) such that, if T ≥T 1 , then it follows that

11)
where C f is given in (1.9).
Next we prove (1.20) for small times.
where C * is independent of q, K , and T * .
For the case N ≥ 2, applying Lemmata 5.1 and 5.3, we show the decay estimate of |u(t)| L q .  F(N , p 1 ,K , λ) such that, if L < F and L is small enough, then, for any q ∈ [p 1 , ∞], where C depends only on N .
By Lemma 5.3, for any T * ≥ 1, there exists ε = ε( p 1 , T * ) such that, if L < ε, then where C * ≥ 1 is independent of q, K and T * . Let us fix T * large enough to be chosen later, put for all q ∈ [p 1 , ∞] and 0 < t < s .
Then, since T * ≥ 1, by (5.32) we have T ≥ 2T * ≥ 2. We prove T = ∞. The proof is by contradiction. We assume that T < ∞. Then, by (5.31) we see that On the other hand, by (2.9) with (q, r ) = ( p 1 , q) and (5.2) we have Let T 1 be the constant given in Lemma 5.1 with D = 2C * , and let us assume that Furthermore, let I 1 and I 2 be functions given in (4.16), and let p 2 be the constant given in (2.20). Then, for the term I 1 , since T ≥ 2T * , by (2.12) with (q, r ) = ( p 2 , q) we get

(5.36)
Since p 1 ≥ p 2 ≥ 1 and T ≥ 1, due to (1.15) and taking a sufficiently small L if necessary, we can apply Lemma 2.2 to the term A(T ), and we obtain (5.37) Furthermore, let p 3 be the constant given in (5.7). Then, since T * ≥ T 1 and it follows from p 1 < 2 that p 2 ≥ p 3 we can apply Lemma 5.1 to the term B(T ), and we have where C is independent of q, L, K , and T * . Moreover, for the term I 2 , since q ≥ p 1 ≥ p 3 and p 1 < 2, by (2.12) with q = r and (5.6) we see that where C is independent of q, L, K , and T * . This together with (4.16), (5.36), (5.37), and (5.38) implies that where D * is a constant independent of L, K , and T * . Since p 1 < 2, we can take a sufficiently large constant T * ≥ 1 so that This together with (5.35) implies that T * depends on λ,K , and p 1 but not on L. Then we can also take a sufficiently small constant L so that Combining (5.14), (5.34), and (5.44), we see that This contradicts (5.33), and we see T = ∞. In order to make clear the dependence of the choice we made on T * and L, we collect below all the conditions (5.35), (5.41), and (5.43) where T 1 satisfies (5.9) with D = 2C * , namely Here C * and D * are constants depending at most on N and p 1 . Then we can find a function F depending on N , p 1 ,K , and λ such that the conditions on L can be written as L < F(N , p 1 ,K , λ) and L small enough. Thus Lemma 5.4 follows.
Similarly, for the case N = 1, applying Lemmata 5.2 and 5.3, we have the following. where C is independent of p and K .
Proof Applying the same argument as in the proof of Lemma 5.4, we can prove this lemma. For reader's convenience, we give it here. Let u be a unique solution to problem (1.1) satisfying (1.14) and (1.15). Then, similarly to (5.31), we can easily show that u(0, t) ∈ C((0, ∞)).
(5.45) By Lemma 5.3, for any T * ≥ 1, there exists ε = ε( p, T * ) such that, if L < ε, then where C * ≥ 1 is independent of K and T * . Let us fix T * large enough to be chosen later, put Then, since T * ≥ 1, by (5.46) we have T ≥ 2T * ≥ 2. We prove T = ∞. The proof is by contradiction. We assume that T < ∞. Then, by (5.45) we see that On the other hand, by (2.10) with q = p and (5.2) we have LetT 1 be the constant given in Lemma 5.2 with D = 2C * , and let us assume Furthermore, since T ≥ 2T * , by (3.23) we put (5.50) Since p ≥ 1 and T ≥ 2T * , due to (1.15) and taking a sufficiently small L if necessary, we can apply Lemma 2.3 to the termĨ 1 (T ), and we obtaiñ Furthermore, since T * ≥T 1 and p < 2, for the termsĨ 2 (T ) andĨ 3 (T ), we can apply Lemma 5.2, and we havẽ where C is a constant independent of p, L, K , and T * . These together with (5.50) and (5.51) imply that where D * is a constant independent of L, K , and T * . Since p < 2, we can take a sufficiently large constant T * ≥ 1 so that which means This together with (5.49) implies that T * depends on λ,K , and p but not on L. Then we can also take a sufficiently small constant L so that This together with (5.22) and (5.48) implies This contradicts (5.47), and we see T = ∞. In order to make clear the dependence of the choice we made on T * and L, we collect below all the conditions (5.49), (5.54), and (5.56) whereT 1 satisfies (5.9) with (N , p 1 ) = (1, p) and D = 2C * , namelỹ Here C * and D * are constants depending at most on p. Then we can find a function F depending on p,K , and λ such that the condition L can be written as L < F( p,K , λ) and L small enough. Thus Lemma 5.5 follows.
Now we ready to prove Theorem 1.3. We first prove it for the case N ≥ 2.
For the nonlinear part, let J 1 and J 2 be functions given in (4.24), and let p 2 be the constant given in (2.20). Then, for the term J 1 , similarly to (5.36), by (2.11) with (q, r ) = ( p 2 , q) we put For the termÃ(t), since p 1 ≥ p 2 ≥ 1, by (1.15) and taking a sufficiently small L if necessary, we can apply Lemma 2.2, and we havẽ (5.59) Furthermore, let p 3 be the constant given in (5.7). Then, for the termB(t), since T ≥ T 1 , and p 2 ≥ p 3 , we can apply Lemma 5.1, and it follows from p 2 ≥ 1 that For p ∈ ( p 2 , 2) (which implies p 1 = p), since p 1 < 2, we can choose σ 1 ∈ (0, 1) satisfying Then, by (5.59) and (5.60) we havẽ This together with (5.4) and (5.58) implies that Choosing T large enough such that and L small enough such that thanks to (5.4) we get On the other hand, for p ≤ p 2 , namely p 1 = p 2 , we consider two cases, N = 2 and N ≥ 3. For the case N ≥ 3, since p 1 ∈ (1, 2), we can choose σ 2 ∈ (0, 1) satisfying Then, by (5.59) and (5.60) we see that This together with (5.4) and (5.58) implies that Choosing T large enough such that and L small enough such that thanks to (5.4) we get For the case N = 2, since p 1 = p 2 = 1 (which implies p = 1), by (5.59) and (5.60) again we see thatÃ This together with (5.4) and (5.58) implies that Choosing T large enough such that and L small enough such that thanks to (5.4) we get Therefore, by (5.61), (5.62), and (5.63), for N ≥ 2, we have Let us come back to the J 2 (t) term. Since T ≥ T 1 and q ≥ p 1 ≥ p 3 , we can apply Lemma 5.1, and by (2.11) with q = r and (5.6) we have Since p 1 < 2, we can choose σ 3 > 0 satisfying 0 < σ 3 < 1/ p 1 − 1/2, and we get Choosing T large enough such that Now, by density, let {ϕ n } ⊂ C ∞ 0 such that ϕ n → ϕ in L p 1 . Then, by (2.8), it holds that Since p 1 > 1, this proves that and so Thus the proof of Theorem 1.3 for the case N ≥ 2 is complete.
Next, applying the same argument as in the prof of Theorem 1.3 for the case N ≥ 2, we prove Theorem 1.3 for the case N = 1.
Proof of Theorem 1.3 (N = 1). Let u be a unique solution to problem (1.1) satisfying (1.14) and (1.15). Let T be a sufficiently large constant to be chosen later, which satisfies T ≥T 1 , whereT 1 is the constant given in Lemma 5.2 with D = C * . Suppose that L is sufficiently small so that Lemmata 5.3 and 5.5 hold. Then, it is enough to prove the decay estimate of D (t) L q for t ≥ 2T in order to obtain (1.20).
This implies that there is λ > 0 so that ϕ λ fulfills condition (2.4), even though its L p norm might be large.
In the end of this section we prove Theorem 1.4. In the following Lemmata, we assume u(t) L q bounded at the origin and decaying at infinity, and we can deduce that also f (u(t)) L r is bounded and decays at infinity for r ≥ p 3 , where p 3 is given in (5.7).