Nonlinear Neumann boundary value problem for semilinear heat equations with critical power nonlinearities

Abstract

We study the nonlinear Neumann boundary value problem for semilinear heat equation $\begin{matrix} \{\begin{matrix} \partial_{t} u - Δ u = λ | u |^{p}, t > 0, x \in R_{+}^{n}, \\ u (0, x) = ε u_{0} (x), x \in R_{+}^{n}, \\ - \partial_{x} u (t, x^{'}, 0) = γ | u |^{q} (t, x^{'}, 0), t > 0, x^{'} \in R^{n - 1} \end{matrix} \end{matrix}$ where $p = 1 + \frac{2}{n}$ , $q = 1 + \frac{1}{n}$ and $ε > 0$ is small enough. We investigate the life span of solutions for $λ, γ > 0$ . Also we study the global in time existence and large time asymptotic behavior of solutions in the case of $λ, γ < 0$ and $\int_{R_{+}^{n}} u_{0} (x) d x > 0$ .

Keywords

Nonlinear heat equation large time asymptotics initial-boundary value problem half line

1. Introduction

We consider the nonlinear Neumann boundary value problem for semilinear heat equations $\begin{matrix} (1.1) & \{\begin{matrix} \partial_{t} u - Δ u = λ f (u), t > 0, x \in R_{+}^{n}, \\ u (0, x) = ε u_{0} (x), x \in R_{+}^{n}, \\ - \frac{\partial u}{\partial x_{n}} = γ g (u), t > 0, x^{'} \in R^{n - 1}, \end{matrix} \end{matrix}$ where $ε > 0$ , $λ, γ \in R$ , $R_{+}^{n} = {x \in R^{n} : x_{n} > 0}$ , $x = (x^{'}, x_{n})$ , $x^{'} = (x_{1}, \dots, x_{n - 1})$ for $n ⩾ 2$ , $R_{+}^{1} = {x_{1} > 0}$ for $n = 1$ , the nonlinearities in the equation and in the Neumann type boundary condition are the following $\begin{matrix} (1.2) & f (u) = | u |^{p}, g (u) = | u |^{q} \end{matrix}$ or $\begin{matrix} (1.3) & f (u) = | u |^{p - 1} u, g (u) = | u |^{q - 1} u \end{matrix}$ and $\begin{matrix} \partial_{x_{n}} u (t, x^{'}, 0) = lim_{h \to 0 +} \frac{u (t, x^{'}, h) - u (t, x^{'}, 0)}{h} . \end{matrix}$ We study (1.1) with the critical powers of the nonlinearities $(p, q) = (1 + \frac{2}{n}, 1 + \frac{1}{n})$ .

The integral equation associated with (1.1) is written as follows $\begin{matrix} (1.4) & u (t) = ε S_{N} (t) u_{0} + (M_{1} u) (t) + (M_{2} u) (t) \end{matrix}$ (see Appendix A.1), where $\begin{array}{l} (M_{1} u) (t) = λ \int_{0}^{t} S_{N} (t - τ) f (u (τ)) d τ, \\ (M_{2} u) (t) = 2 γ \int_{0}^{t} \frac{1}{\sqrt{4 π (t - τ)}} e^{- \frac{x_{n}^{2}}{4 (t - τ)}} \tilde{S} (t - τ) g (u (τ)) |_{x_{n} = 0} d τ \end{array}$ with $\begin{array}{l} \tilde{S} (t) φ = \frac{1}{{(4 π t)}^{(n - 1) / 2}} \int_{R^{n - 1}} e^{- \frac{| x^{'} - y^{'} |^{2}}{4 t}} φ (y^{'}) d y^{'}, \\ S_{n N} (t) ψ = \frac{1}{\sqrt{4 π t}} \int_{0}^{\infty} (e^{- \frac{{(x_{n} - y_{n})}^{2}}{4 t}} + e^{- \frac{{(x_{n} + y_{n})}^{2}}{4 t}}) ψ (y_{n}) d y_{n}, \\ S_{N} (t) ϕ = S_{n N} (t) \tilde{S} (t) ϕ . \end{array}$ We also define the Green operator for the case of the Dirichlet type boundary conditions $\begin{matrix} S_{D} (t) ϕ = \frac{1}{\sqrt{4 π t}} \int_{0}^{\infty} (e^{- \frac{{(x_{n} - y_{n})}^{2}}{4 t}} - e^{- \frac{{(x_{n} + y_{n})}^{2}}{4 t}}) (\tilde{S} (t) ϕ) (y_{n}) d y_{n} . \end{matrix}$ When $γ = 0$ , then we can extend $u_{0}$ to all space $R^{n}$ as an even function with respect to $x_{n}$ , then (1.1) converts to the Cauchy problem $\begin{matrix} (1.5) & \{\begin{matrix} \partial_{t} u - Δ u = λ f (u), t > 0, x \in R^{n}, \\ u (0, x) = ε u_{0} (x), x \in R^{n}, \end{matrix} \end{matrix}$ which solution can be written by the Duhamel formula $\begin{matrix} (1.6) & u (t) = S (t) u_{0} + λ \int_{0}^{t} S (t - τ) f (u (τ)) d τ, \end{matrix}$ where the Green operator is $\begin{matrix} S (t) φ = \frac{1}{{(4 π t)}^{n / 2}} \int_{R^{n}} e^{- \frac{| x - y |^{2}}{4 t}} φ (y) d y . \end{matrix}$ When $f (v) = | v |^{p}$ , the large time asymptotics and the life span of solutions to (1.6) was studied in [3, 5, 10–13] in the case of $λ > 0$ . In [3], existence of blow up of solutions for $1 < p < 1 + \frac{2}{n}$ , $u_{0} > 0$ and global in time existence of solutions for $p > 1 + \frac{2}{n}$ , $0 < u_{0} ⩽ e^{- | x |^{2}}$ were shown. The case $p = 1 + \frac{2}{n}$ was studied in [5] for $n = 1, 2$ and in [10, 12] for general n and it was shown that solutions blow up in a finite time. In [11], the global existence, large time behavior or life span of solutions were studied for (1.6) with initial data $u_{0} (x) = {⟨ x ⟩}^{- a}$ . In particular, when $p = 1 + \frac{2}{n}$ , $a > n$ , it was shown that the upper bound of the existence time (life span) T is given by $\begin{matrix} T = exp (ε^{- \frac{2}{n}}) \end{matrix}$ for small $ε > 0$ and when $p > 1 + \frac{2}{n}$ , $a ⩾ \frac{2}{p - 1}$ , global in time existence of solutions was shown for small data. In [13], $L^{n (p - 1) / 2}$ global solutions were obtained if $u_{0} ⩾ 0$ and $u_{0} \in L^{n (p - 1) / 2}$ for $p > 1 + \frac{2}{n}$ .

In the case of $λ < 0$ , $p = 1 + \frac{2}{n}$ , $f (v) = | v |^{p - 1} v$ , (1.5) was studied in [6] and it was shown that the large time asymptotic behavior of solutions differs from that of the linear problem under the condition such that $\int_{R^{n}} u_{0} d x \neq 0$ . Therefore the power $p = 1 + \frac{2}{n}$ is considered as critical.

We note that the estimates $\begin{array}{l} {‖ (M_{1} u) (t) ‖}_{L^{1} (R_{+}^{n})} \\ ⩽ \int_{0}^{t} {‖ f (u (τ)) ‖}_{L^{1} (R_{+}^{n})} d τ \\ ⩽ \int_{0}^{t} {‖ u (τ) ‖}_{L^{p} (R_{+}^{n})}^{p} d τ ⩽ C \int_{0}^{t} {(1 + t)}^{- \frac{n}{2} (p - 1)} d τ \end{array}$ and $\begin{array}{l} {‖ (M_{2} u) (t) ‖}_{L^{1} (R_{+}^{n})} \\ ⩽ C \int_{0}^{t} {‖ u (τ, \cdot, 0) ‖}_{L^{q} (R^{n - 1})}^{q} d τ ⩽ C \int_{0}^{t} {(1 + t)}^{- \frac{1}{2} - \frac{n}{2} (q - 1)} d τ \end{array}$ are expected. Therefore if $p > 1 + \frac{2}{n}$ and $q > 1 + \frac{1}{n}$ , then the right hand sides of the above estimates are finite and we find that it is possible to apply the standard contraction mapping principle to show a global existence of small solutions for $λ, γ \in C$ .

Our purpose in the present paper is to consider $(p, q) = (1 + \frac{2}{n}, 1 + \frac{1}{n})$ and show a lower bound for the existence time of local solutions in the case of $λ, γ > 0$ and to prove the global in time existence and find the large time asymptotic behavior of solutions in case $λ, γ < 0$ . Our method can be applied to the case of $p = 1 + \frac{2}{n}$ and $q > 1 + \frac{1}{n}$ for $λ < 0$ , $γ \in C$ or $p > 1 + \frac{2}{n}$ and $q = 1 + \frac{1}{n}$ for $λ \in C$ , $γ < 0$ .

On the other hand, when $λ = 0$ , the integral equation (1.4) was studied in [2, 4, 8] and [9].

In [2], a system of heat equations coupled in the boundary conditions such that $\begin{matrix} \{\begin{matrix} u (t) = S_{N} (t) u_{0} + (M_{2} v) (t), \\ v (t) = S_{N} (t) v_{0} + ({\tilde{M}}_{2} v) (t), \end{matrix} \end{matrix}$ was studied, where $\begin{matrix} ({\tilde{M}}_{2} u) (t) = 2 γ \int_{0}^{t} \frac{1}{\sqrt{4 π (t - τ)}} e^{- \frac{x_{n}^{2}}{4 (t - τ)}} \tilde{S} (t - τ) | u |^{\tilde{q}} |_{x_{n} = 0} d τ . \end{matrix}$ We assume $γ > 0$ , $q \tilde{q} > 1$ $\begin{matrix} α = \frac{1}{2} \frac{q + 1}{q \tilde{q} - 1}, β = \frac{1}{2} \frac{\tilde{q} + 1}{q \tilde{q} - 1} . \end{matrix}$ Then they showed if $max (α, β) ⩾ \frac{n}{2}$ , all nonnegative solutions are nonglobal, whereas if $max (α, β) < \frac{n}{2}$ , small global solutions exist. When $q = \tilde{q}$ , $u_{0} = v_{0}$ , then we have a single equation $u (t) = S_{N} (t) u_{0} + (M_{2} u) (t)$ . In the case, their result says if $1 < q ⩽ 1 + \frac{1}{n}$ , all nonnegative solutions are nonglobal, while if $q > 1 + \frac{1}{n}$ , small global solutions exist (see [4]). Therefore the power $q = 1 + \frac{1}{n}$ is considered as critical. In [9], necessary and sufficient conditions for the solvability of the problem $u (t) = S_{N} (t) u_{0} + (M_{2} u) (t)$ were obtained, and the strongest singularity of the initial function for which we have existence of solutions was identified. In [8], the result of [9] was applied to get sharp estimates of the life span including the result of Theorem 1.1 when $λ = 0$ .

In order to state the first result, we define the function space of solutions $X_{T}$ as $\begin{array}{rcl} X_{T} & = & C ([0, T]; L^{1} (R_{+}^{n}) \cap L^{r} (R_{+}^{n})) \\ \cap C ((0, T]; W^{1, 1} (R_{+}^{n}) \cap W^{1, r} (R_{+}^{n})), \end{array}$ with the norm $\begin{array}{l} ‖ v ‖_{X_{T}} = sup_{t \in [0, T]} {‖ v (t) ‖}_{X}, \\ \begin{matrix} {‖ v (t) ‖}_{X} & = {⟨ t ⟩}^{\frac{n}{2} (1 - \frac{1}{r})} {‖ v (t) ‖}_{L^{r} (R_{+}^{n})} + {⟨ t ⟩}^{\frac{n}{2} (1 - \frac{1}{r})} t^{\frac{1}{2}} {‖ \nabla v (t) ‖}_{L^{r} (R_{+}^{n})} \\ + {‖ v (t) ‖}_{L^{1} (R_{+}^{n})} + t^{\frac{1}{2}} {‖ \nabla v (t) ‖}_{L^{1} (R_{+}^{n})}, \end{matrix} \end{array}$ where $r = 1 + \frac{1}{n}$ and $\begin{matrix} W^{1, s} (R_{+}^{n}) = {ϕ \in L^{s} (R_{+}^{n}); ‖ ϕ ‖_{L^{s} (R_{+}^{n})} + ‖ \nabla ϕ ‖_{L^{s} (R_{+}^{n})} < \infty}, 1 ⩽ s ⩽ \infty \end{matrix}$ Let $X_{T, ρ}$ be a closed ball in $X_{T}$ with a radius ρ and the center at the origin $\begin{matrix} X_{T, ρ} = {v \in X_{T}; ‖ v ‖_{X_{T}} ⩽ ρ} . \end{matrix}$

We state the local in time existence for the critical case.

Theorem 1.1.
We assume that $u_{0} \in L^{1} (R_{+}^{n}) \cap L^{1 + \frac{1}{n}} (R_{+}^{n})$ , then there exist an $ε > 0$ and positive constant c such that the integral equation ( 1.4 ) with ( 1.2 ) or ( 1.3 ) has a unique solution $\begin{array}{rcl} u & \in & C ([0, T]; L^{1} (R_{+}^{n}) \cap L^{1 + \frac{1}{n}} (R_{+}^{n})) \\ \cap C ((0, T]; W^{1, 1} (R_{+}^{n}) \cap W^{1, 1 + \frac{1}{n}} (R_{+}^{n})), \end{array}$ where $\begin{matrix} T ⩾ e^{c ε^{- \frac{1}{n}}} . \end{matrix}$
Remark 1.1.
By papers [5, 10–12] in the case of (1.2), we can find that the upper bound of the life span T is given by $e^{C ε^{- \frac{2}{n}}}$ . Indeed, since $(M_{2} u) (t) ⩾ 0$ , when $γ > 0$ , we have the estimate $\begin{matrix} u (t) ⩾ ε S_{N} (t) u_{0} + (M_{1} u) (t) . \end{matrix}$ Then if $λ > 0$ , $u_{0} > 0$ , using the above inequality we can obtain the life span of solutions in the same way as in the proofs of these papers. Therefore there exists a gap between $e^{c ε^{- \frac{1}{n}}}$ and $e^{C ε^{- \frac{2}{n}}}$ . However their proofs work well for the boundary problem since we have the integral inequality $\begin{matrix} u (t) ⩾ ε S_{N} (t) u_{0} + (M_{2} u) (t) \end{matrix}$ which is obtained by the conditions $λ, γ > 0$ . Therefore we will find (see Appendix A.2 below) that the upper bound for the existence time of solutions is given by $e^{C ε^{- \frac{1}{n}}}$ . This implies that the result of Theorem 1.1 is sharp (see also [8]).

In order to state the global existence result, we introduce the function space $\begin{matrix} X_{b, 0} = {ϕ \in L_{b}^{1} (R_{+}^{n}) \cap L_{b}^{1 + \frac{1}{n}} (R_{+}^{n}); ‖ ϕ ‖_{X_{b, 0}} < \infty}, \end{matrix}$ where $\begin{array}{l} ‖ ϕ ‖_{X_{b, 0}} = ‖ ϕ ‖_{L_{b}^{1} (R_{+}^{n})} + ‖ ϕ ‖_{L_{b}^{1 + \frac{1}{n}} (R_{+}^{n})}, \\ ‖ ϕ ‖_{L_{b}^{a} (R_{+}^{n})} = {‖ {⟨ \cdot ⟩}^{b} ϕ ‖}_{L^{a} (R_{+}^{n})}, b ⩾ 0, a ⩾ 1 . \end{array}$ Our basic function space $X_{b, \infty}$ is defined by $\begin{array}{rcl} X_{b, \infty} & = & C ([0, \infty); L_{b}^{1} (R_{+}^{n}) \cap L_{b}^{r} (R_{+}^{n})) \\ \cap C ((0, \infty); W_{b}^{1, 1} (R_{+}^{n}) \cap W_{b}^{1, r} (R_{+}^{n})), \end{array}$ with the norm $‖ v ‖_{X_{b, \infty}} = {sup}_{t \in [0, \infty)} ‖ v (t) ‖_{X_{b}}$ , where $\begin{array}{rcl} {‖ v (t) ‖}_{X_{b}} & = & {‖ v (t) ‖}_{X} + {⟨ t ⟩}^{- \frac{b}{2}} {‖ | \cdot |^{b} v (t) ‖}_{L^{1} (R_{+}^{n})} \\ + {⟨ t ⟩}^{- \frac{b}{2}} t^{\frac{1}{2}} {‖ | \cdot |^{b} \nabla v (t) ‖}_{L^{1} (R_{+}^{n})} + {⟨ t ⟩}^{- \frac{b}{2} + \frac{n}{2} (1 - \frac{1}{r})} {‖ | \cdot |^{b} v (t) ‖}_{L^{r} (R_{+}^{n})} \\ + {⟨ t ⟩}^{- \frac{b}{2} + \frac{n}{2} (1 - \frac{1}{r})} t^{\frac{1}{2}} {‖ | \cdot |^{b} \nabla v (t) ‖}_{L^{r} (R_{+}^{n})}, \end{array}$ where $r = 1 + \frac{1}{n}$ , $\begin{matrix} W_{b}^{1, s} (R_{+}^{n}) = {ϕ \in L_{b}^{s} (R_{+}^{n}); ‖ ϕ ‖_{L_{b}^{s} (R_{+}^{n})} + ‖ \nabla ϕ ‖_{L_{b}^{s} (R_{+}^{n})} < \infty}, 1 ⩽ s ⩽ \infty \end{matrix}$ and $X_{b, \infty, ρ}$ is the closed ball in $X_{b, \infty}$ with a radius ρ and the center at the origin. We also introduce the metric $‖ \cdot ‖_{Z_{δ, \infty}}$ in $X_{b, \infty}$ , where $‖ v ‖_{Z_{δ, \infty}} = {sup}_{t \in [0, \infty)} ‖ v (t) ‖_{Z_{δ}}$ and $\begin{matrix} {‖ v (t) ‖}_{Z_{δ}} = {⟨ t ⟩}^{- δ} {‖ v (t) ‖}_{X} \end{matrix}$ with $0 < δ < 1$ .

We now state our main result of this paper. Denote $θ = 2 \int_{R_{+}^{n}} u_{0} (x) d x$ .
Theorem 1.2.
We assume that $u_{0} \in X_{b, 0}$ , $0 < b < \frac{1}{n + 1}$ , $λ, γ < 0$ . We suppose $θ > 0$ for the case of ( 1.2 ) and $θ \neq 0$ for the case of ( 1.3 ). Then there exists an $ε > 0$ such that the integral equation ( 1.4 ) with ( 1.2 ) or ( 1.3 ) has a unique global solution $\begin{array}{rcl} u & \in & C ([0, \infty); L_{b}^{1} (R_{+}^{n}) \cap L_{b}^{r} (R_{+}^{n})) \\ \cap C ((0, \infty); W_{b}^{1, 1} (R_{+}^{n}) \cap W_{b}^{1, r} (R_{+}^{n})) . \end{array}$ Moreover, there exist positive constants $c_{1} > c_{2} > 0$ such that the estimate $\begin{matrix} \frac{c_{2} t^{- \frac{n}{2} (1 - \frac{1}{σ})}}{{(1 - γ | θ ε |^{\frac{1}{n}} log ⟨ t ⟩)}^{n}} ⩽ {‖ u (t) ‖}_{L^{σ} (R_{+}^{n})} ⩽ \frac{c_{1} t^{- \frac{n}{2} (1 - \frac{1}{σ})}}{{(1 - γ | θ ε |^{\frac{1}{n}} log ⟨ t ⟩)}^{n}} \end{matrix}$ holds for any $t ⩾ 0$ , where $1 ⩽ σ ⩽ 1 + \frac{1}{n}$ .
Remark 1.2.
In case $λ < 0$ and $γ = 0$ , our proof works well for (1.4) with (1.2) or (1.3) under the conditions of Theorem 1.2 and we have the time decay estimate for solutions such that $\begin{matrix} \frac{c_{2} t^{- \frac{n}{2} (1 - \frac{1}{σ})}}{{(1 - λ | θ ε |^{\frac{2}{n}} log ⟨ t ⟩)}^{\frac{n}{2}}} ⩽ {‖ u (t) ‖}_{L^{σ} (R_{+}^{n})} ⩽ \frac{c_{1} t^{- \frac{n}{2} (1 - \frac{1}{σ})}}{{(1 - λ | θ ε |^{\frac{2}{n}} log ⟨ t ⟩)}^{\frac{n}{2}}} \end{matrix}$ for some $c_{1} > c_{2} > 0$ . This fact implies that dissipative effect from the boundary is stronger than that of the nonlinear forcing term.
Remark 1.3.
In the case of (1.4) with (1.3), we have a-priori estimates for local solutions through the energy method. More precisely, we have $\begin{array}{l} \frac{1}{2} \frac{d}{d t} {‖ u (t) ‖}_{L^{2} (R_{+}^{n})}^{2} + {‖ \nabla u (t) ‖}_{L^{2} (R_{+}^{n})}^{2} \\ - γ {‖ u (t, 0) ‖}_{L^{q + 1} (R^{n - 1})}^{q + 1} - λ {‖ u (t) ‖}_{L^{p + 1} (R_{+}^{n})}^{p + 1} \\ = 0, \end{array}$ and $\begin{array}{l} \frac{1}{2} \frac{d}{d t} {‖ \nabla u (t) ‖}_{L^{2} (R_{+}^{n})}^{2} + {‖ \partial_{t} u (t) ‖}_{L^{2} (R_{+}^{n})}^{2} \\ - \frac{2 γ}{q + 1} \frac{d}{d t} {‖ u (t, 0) ‖}_{L^{q + 1} (R^{n - 1})}^{q + 1} - \frac{2 λ}{p + 1} \frac{d}{d t} {‖ u (t) ‖}_{L^{p + 1} (R_{+}^{n})}^{p + 1} = 0 . \end{array}$ Therefore we expect the global in time existence of solutions in the case of large data under some growth conditions on nonlinearities and regularity on the data. However it seems that it is difficult to get time decay estimates presented in Theorem 1.2 by the energy method only.

We organize our paper as follows. In Section 2, we prove some estimates of the evolution operator associated with homogeneous Neumann problem. Section 3 is devoted to the proof of Theorem 1.1. In Section 4, we translate the integral equation of the original problem to another one such that the integral of the nonlinearities over $R_{+}^{n}$ vanishes. In Lemma 4.1, we consider some estimates of nonlinear terms of the transformed integral equation. In order to get an additional time decay of solutions we prepare Lemma 4.2. We need Lemma 4.3 to apply the contraction mapping principle. Section 5 is devoted to the proof of Theorem 1.2. In Appendix A.1, we show that the integral representation of solutions satisfies the boundary and initial conditions. Appendix A.2 is devoted to the estimate of solutions by using the method of [5] which says that the result of Theorem 1.1 is sharp (see also [8]).
2. Preliminary estimates

2.1. Preliminary estimates for Theorem 1.1

We first consider the estimates of $S_{N} (t)$ . We extend $ϕ (x^{'}, x_{n})$ to $R^{n}$ as an even function with respect to $x_{n}$ such that $\begin{matrix} \tilde{ϕ} (x^{'}, x_{n}) = \{\begin{matrix} ϕ (x^{'}, x_{n}), & x_{n} ⩾ 0 \\ ϕ (x^{'}, - x_{n}), & x_{n} < 0 \end{matrix} \end{matrix}$ and define $S (t)$ by $\begin{matrix} S (t) φ = \frac{1}{{(4 π t)}^{n / 2}} \int_{R^{n}} e^{- \frac{| x - y |^{2}}{4 t}} φ (y) d y . \end{matrix}$ Then we can write $S_{N} (t) ϕ = S (t) \tilde{ϕ}$ . Also note that $\begin{array}{l} \int_{R_{+}^{n}} S_{N} (t) ϕ d x \\ = \int_{R_{+}^{n}} S (t) \tilde{ϕ} d x = \frac{1}{2} \int_{R^{n}} S (t) \tilde{ϕ} d x \\ = \frac{1}{2} \int_{R^{n}} \tilde{ϕ} d x = \int_{R_{+}^{n}} ϕ (x) d x . \end{array}$ Using the relation $‖ \tilde{ϕ} ‖_{L^{r} (R^{n})} = C ‖ ϕ ‖_{L^{r} (R_{+}^{n})}$ and applying Lemma 1.28 from book [7], p. 27, we get the following result.

Lemma 2.1.
The following estimates are true $\begin{array}{l} (2.1) & \begin{array}{l} {‖ S_{N} (t) ϕ ‖}_{L^{r} (R_{+}^{n})} ⩽ C ‖ ϕ ‖_{L^{r} (R_{+}^{n})}, \\ {‖ S_{N} (t) ϕ ‖}_{L^{r} (R_{+}^{n})} ⩽ C t^{- \frac{n}{2} (1 - \frac{1}{r})} ‖ ϕ ‖_{L^{1} (R_{+}^{n})}, \end{array} \end{array}$ and $\begin{array}{l} (2.2) & \begin{array}{l} {‖ \nabla S_{N} (t) ϕ ‖}_{L^{r} (R_{+}^{n})} ⩽ C t^{- \frac{1}{2}} ‖ ϕ ‖_{L^{r} (R_{+}^{n})}, \\ {‖ \nabla S_{N} (t) ϕ ‖}_{L^{r} (R_{+}^{n})} ⩽ C t^{- \frac{1}{2} - \frac{n}{2} (1 - \frac{1}{r})} ‖ ϕ ‖_{L^{1} (R_{+}^{n})} \end{array} \end{array}$ for $1 ⩽ r ⩽ \infty$ , $t > 0$ provided that the right-hand sides are finite.

As a consequence we obtain. Lemma 2.2.
Assume that $‖ ϕ ‖_{X_{0}} ⩽ ε$ . Then the estimate is true $\begin{matrix} {‖ S_{N} (\cdot) ϕ ‖}_{X_{\infty}} ⩽ C ε . \end{matrix}$
Proof.
By (2.1) and (2.2) from Lemma 2.1, we have the estimates $\begin{matrix} {⟨ t ⟩}^{\frac{n}{2} (1 - \frac{1}{r})} {‖ S_{N} (t) ϕ ‖}_{L^{r} (R_{+}^{n})} ⩽ C (‖ ϕ ‖_{L^{1} (R_{+}^{n})} + ‖ ϕ ‖_{L^{r} (R_{+}^{n})}) \end{matrix}$ and $\begin{matrix} {⟨ t ⟩}^{\frac{n}{2} (1 - \frac{1}{r})} t^{\frac{1}{2}} {‖ \nabla S_{N} (t) ϕ ‖}_{L^{r} (R_{+}^{n})} ⩽ C (‖ ϕ ‖_{L^{1} (R_{+}^{n})} + ‖ ϕ ‖_{L^{r} (R_{+}^{n})}) \end{matrix}$ for $1 ⩽ r ⩽ \infty$ , $t > 0$ . Therefore $‖ S_{N} (\cdot) ϕ ‖_{X_{\infty}} ⩽ C ‖ ϕ ‖_{X_{0}}$ . This completes the proof of the lemma. □

2.2. Preliminary estimates for Theorem 1.2

Denote $\begin{matrix} G (t) = \frac{1}{{(4 π t)}^{n / 2}} e^{- \frac{| x |^{2}}{4 t}} \end{matrix}$ and $\begin{matrix} θ = \int_{R^{n}} \tilde{ϕ} d x = 2 \int_{R_{+}^{n}} ϕ d x . \end{matrix}$ Since $S_{N} (t) ϕ = S (t) \tilde{ϕ}$ , and $‖ {⟨ \cdot ⟩}^{b} \tilde{ϕ} ‖_{L^{r} (R^{n})} = C ‖ {⟨ \cdot ⟩}^{b} ϕ ‖_{L^{r} (R_{+}^{n})}$ , applying Lemma 1.28 from book [7], p. 27, we find

Lemma 2.3.
The following estimates are true $\begin{array}{l} {‖ {⟨ \cdot ⟩}^{b} S_{N} (t) ϕ ‖}_{L^{r} (R_{+}^{n})} \\ ⩽ C {⟨ t ⟩}^{\frac{b}{2} - \frac{n}{2} (1 - \frac{1}{r})} ({‖ {⟨ \cdot ⟩}^{b} ϕ ‖}_{L^{1} (R_{+}^{n})} + {‖ {⟨ \cdot ⟩}^{b} ϕ ‖}_{L^{r} (R_{+}^{n})}), \\ {‖ | \cdot |^{b} (S_{N} (t) ϕ - θ G (t)) ‖}_{L^{r} (R_{+}^{n})} ⩽ C {⟨ t ⟩}^{\frac{b - a}{2}} t^{- \frac{n}{2} (1 - \frac{1}{r})} {‖ {⟨ \cdot ⟩}^{a} ϕ ‖}_{L^{1} (R_{+}^{n})} \end{array}$ and $\begin{matrix} {‖ | \cdot |^{b} \nabla (S_{N} (t) ϕ - θ G (t)) ‖}_{L^{r} (R_{+}^{n})} ⩽ C {⟨ t ⟩}^{\frac{b - a}{2}} t^{- \frac{1}{2} - \frac{n}{2} (1 - \frac{1}{r})} {‖ {⟨ \cdot ⟩}^{a} ϕ ‖}_{L^{1} (R_{+}^{n})}, \end{matrix}$ provided that the right hand sides are finite, where $1 ⩽ r ⩽ \infty$ , $t > 0$ , $b \in [0, a]$ , $0 < a < 1$ .

3. Proof of Theorem 1.1

3.1. Estimates for the nonlinear effects

In order to prove the result for the critical case, we need the estimates of $‖ M_{1} w ‖_{X_{T}}$ and $‖ M_{2} w ‖_{X_{T}}$ .

Lemma 3.1.
Assume that $w \in X_{T, ρ}$ . Then the estimates are valid $\begin{matrix} ‖ M_{1} w ‖_{X_{T}} ⩽ C ρ^{1 + \frac{2}{n}} log ⟨ T ⟩, ‖ M_{2} w ‖_{X_{T}} ⩽ C ρ^{1 + \frac{1}{n}} log ⟨ T ⟩ . \end{matrix}$
Proof.
By Lemma 2.1 $\begin{matrix} {‖ (M_{1} w) (t) ‖}_{L^{r} (R_{+}^{n})} ⩽ C \int_{0}^{t} {(t - τ)}^{- \frac{n}{2} (1 - \frac{1}{r})} {‖ w (τ) ‖}_{L^{1 + \frac{2}{n}} (R_{+}^{n})}^{1 + \frac{2}{n}} d τ, \end{matrix}$ where $1 ⩽ r ⩽ 1 + \frac{1}{n}$ . By the Gagliardo–Nirenberg–Sobolev interpolation inequality $\begin{matrix} ‖ ϕ ‖_{L^{1 + \frac{2}{n}} (R_{+}^{n})} ⩽ C ‖ ϕ ‖_{L^{1 + \frac{1}{n}} (R_{+}^{n})}^{1 - \frac{n^{2}}{(n + 1) (n + 2)}} ‖ \nabla ϕ ‖_{L^{1 + \frac{1}{n}} (R_{+}^{n})}^{\frac{n^{2}}{(n + 1) (n + 2)}} \end{matrix}$ and using estimates $\begin{matrix} {‖ w (t) ‖}_{L^{1 + \frac{1}{n}} (R_{+}^{n})} ⩽ C ρ {⟨ t ⟩}^{- \frac{n}{2 n + 2}} \end{matrix}$ and $\begin{matrix} {‖ \nabla w (t) ‖}_{L^{1 + \frac{1}{n}} (R_{+}^{n})}^{\frac{n}{n + 1}} ⩽ C ρ t^{- \frac{1}{2}} {⟨ t ⟩}^{- \frac{n}{2 n + 2}} \end{matrix}$ we find $\begin{array}{rcl} {‖ w (t) ‖}_{L^{1 + \frac{2}{n}} (R_{+}^{n})}^{1 + \frac{2}{n}} & ⩽ & C {‖ w (t) ‖}_{L^{1 + \frac{1}{n}} (R_{+}^{n})}^{1 + \frac{2}{n} - \frac{n}{n + 1}} {‖ \nabla w (t) ‖}_{L^{1 + \frac{1}{n}} (R_{+}^{n})}^{\frac{n}{n + 1}} \\ ⩽ & C ρ^{1 + \frac{2}{n}} t^{- \frac{n}{2 n + 2}} {⟨ t ⟩}^{- \frac{n + 2}{2 n + 2}} . \end{array}$ Therefore $\begin{matrix} {‖ (M_{1} w) (t) ‖}_{L^{r} (R_{+}^{n})} ⩽ C ρ^{1 + \frac{2}{n}} \int_{0}^{t} {(t - τ)}^{- \frac{n}{2} (1 - \frac{1}{r})} τ^{- \frac{n}{2 n + 2}} {⟨ τ ⟩}^{- \frac{n + 2}{2 n + 2}} d τ . \end{matrix}$ Hence for $t ⩽ 4$ , we find $\begin{matrix} {‖ (M_{1} w) (t) ‖}_{L^{r} (R_{+}^{n})} ⩽ C ρ^{1 + \frac{2}{n}} \int_{0}^{t} ({(t - τ)}^{- \frac{n + 2}{2 n + 2}} + τ^{- \frac{n + 2}{2 n + 2}}) d τ ⩽ C ρ^{1 + \frac{n}{2}} \end{matrix}$ if $1 ⩽ r ⩽ 1 + \frac{1}{n}$ , and for $4 < t ⩽ T$ , we have $\begin{array}{rcl} {‖ (M_{1} w) (t) ‖}_{L^{r} (R_{+}^{n})} & ⩽ & C ρ^{1 + \frac{2}{n}} t^{- \frac{n}{2} (1 - \frac{1}{r})} \int_{0}^{1} τ^{- \frac{n}{2 n + 2}} d τ \\ + C ρ^{1 + \frac{2}{n}} t^{- \frac{n}{2} (1 - \frac{1}{r})} \int_{1}^{\frac{t}{2}} τ^{- 1} d τ + C ρ^{1 + \frac{n}{2}} t^{- 1} \int_{\frac{t}{2}}^{t} {(t - τ)}^{- \frac{n}{2} (1 - \frac{1}{r})} d τ \\ ⩽ & C ρ^{1 + \frac{2}{n}} {⟨ t ⟩}^{- \frac{n}{2} (1 - \frac{1}{r})} log ⟨ t ⟩, \end{array}$ from which it follows that $\begin{matrix} {⟨ t ⟩}^{\frac{n}{2} (1 - \frac{1}{r})} {‖ (M_{1} w) (t) ‖}_{L^{r} (R_{+}^{n})} ⩽ C ρ^{1 + \frac{2}{n}} log ⟨ T ⟩ . \end{matrix}$ Similarly, we find $\begin{matrix} {‖ \nabla (M_{1} w) (t) ‖}_{L^{r} (R_{+}^{n})} ⩽ C ρ^{1 + \frac{2}{n}} \int_{0}^{t} {(t - τ)}^{- \frac{1}{2} - \frac{n}{2} (1 - \frac{1}{r})} τ^{- \frac{n}{2 n + 2}} {⟨ τ ⟩}^{- \frac{n + 2}{2 n + 2}} d τ . \end{matrix}$ Note that $\frac{1}{2} + \frac{n}{2} (1 - \frac{1}{r}) ⩽ \frac{2 n + 1}{2 n + 2} < 1$ for $1 ⩽ r ⩽ 1 + \frac{1}{n}$ , then as above we get $\begin{matrix} t^{\frac{1}{2}} {⟨ t ⟩}^{\frac{n}{2} (1 - \frac{1}{r})} {‖ \nabla (M_{1} w) (t) ‖}_{L^{r} (R_{+}^{n})} ⩽ C ρ^{1 + \frac{2}{n}} log ⟨ T ⟩ . \end{matrix}$ Next by Lemma 2.1 we get $\begin{array}{l} {‖ \frac{1}{\sqrt{4 π (t - τ)}} e^{- \frac{x_{n}^{2}}{4 (t - τ)}} \tilde{S} (t - τ) ϕ (τ) |_{x_{n} = 0} ‖}_{L^{r} (R_{+}^{n})} \\ ⩽ {‖ \frac{1}{\sqrt{4 π (t - τ)}} e^{- \frac{x_{n}^{2}}{4 (t - τ)}} ‖}_{L^{r} (R_{+})} {‖ \tilde{S} (t - τ) ϕ (τ) |_{x_{n} = 0} ‖}_{L^{r} (R^{n - 1})} \\ ⩽ C {(t - τ)}^{- \frac{n}{2} (1 - \frac{1}{r})} {‖ ϕ (τ, \cdot, 0) ‖}_{L^{1} (R^{n - 1})} . \end{array}$ Hence $\begin{array}{l} {‖ (M_{2} w) (t) ‖}_{L^{r} (R_{+}^{n})} \\ ⩽ C \int_{0}^{t} {(t - τ)}^{- \frac{n}{2} (1 - \frac{1}{r})} {‖ {| w (τ, \cdot, 0) |}^{1 + \frac{1}{n}} ‖}_{L^{1} (R^{n - 1})} d τ . \end{array}$ By the Gagliardo–Nirenberg–Sobolev interpolation inequality $\begin{matrix} {‖ {| ϕ (τ, \cdot, 0) |}^{1 + \frac{1}{n}} ‖}_{L^{1} (R^{n - 1})} ⩽ C ‖ ϕ ‖_{L^{1 + \frac{1}{n}} (R_{+}^{n})}^{\frac{1}{n}} ‖ \partial_{x_{n}} ϕ ‖_{L^{1 + \frac{1}{n}} (R_{+}^{n})} \end{matrix}$ and using estimates $\begin{matrix} {‖ w (t) ‖}_{L^{1 + \frac{1}{n}} (R_{+}^{n})} ⩽ C ρ {⟨ t ⟩}^{- \frac{n}{2 n + 2}} \end{matrix}$ and $\begin{matrix} {‖ \nabla w (t) ‖}_{L^{1 + \frac{1}{n}} (R_{+}^{n})} ⩽ C ρ t^{- \frac{1}{2}} {⟨ t ⟩}^{- \frac{n}{2 n + 2}} \end{matrix}$ we find $\begin{array}{l} {‖ {| w (τ, \cdot, 0) |}^{1 + \frac{1}{n}} ‖}_{L^{1} (R^{n - 1})} \\ ⩽ C {‖ w (τ) ‖}_{L^{1 + \frac{1}{n}} (R_{+}^{n})}^{\frac{1}{n}} {‖ \partial_{x_{n}} w (τ) ‖}_{L^{1 + \frac{1}{n}} (R_{+}^{n})} ⩽ C ρ^{1 + \frac{1}{n}} τ^{- \frac{1}{2}} {⟨ τ ⟩}^{- \frac{1}{2}} . \end{array}$ Therefore $\begin{matrix} {‖ (M_{2} w) (t) ‖}_{L^{r} (R_{+}^{n})} ⩽ C ρ^{1 + \frac{1}{n}} \int_{0}^{t} {(t - τ)}^{- \frac{n}{2} (1 - \frac{1}{r})} {⟨ τ ⟩}^{- \frac{1}{2}} τ^{- \frac{1}{2}} d τ . \end{matrix}$ As above we have $\frac{1}{2} + \frac{n}{2} (1 - \frac{1}{r}) < 1$ for $1 ⩽ r ⩽ 1 + \frac{1}{n}$ , hence $\begin{array}{l} {‖ (M_{2} w) (t) ‖}_{L^{r} (R_{+}^{n})} \\ ⩽ C ρ^{1 + \frac{1}{n}} \int_{0}^{t} ({(t - τ)}^{- \frac{1}{2} - \frac{n}{2} (1 - \frac{1}{r})} + τ^{- \frac{1}{2} - \frac{n}{2} (1 - \frac{1}{r})}) d τ ⩽ C ρ^{1 + \frac{1}{n}} \end{array}$ for $t ⩽ 4$ and for $4 < t ⩽ T$ $\begin{array}{rcl} {‖ (M_{2} w) (t) ‖}_{L^{r} (R_{+}^{n})} & ⩽ & C ρ^{1 + \frac{1}{n}} \int_{0}^{t} {(t - τ)}^{- \frac{n}{2} (1 - \frac{1}{r})} {⟨ τ ⟩}^{- \frac{1}{2}} τ^{- \frac{1}{2}} d τ \\ ⩽ & C ρ^{1 + \frac{1}{n}} t^{- \frac{n}{2} (1 - \frac{1}{r})} (\int_{0}^{2} τ^{- \frac{1}{2}} d τ + \int_{2}^{\frac{t}{2}} τ^{- 1} d τ) \\ + C ρ^{1 + \frac{1}{n}} t^{- 1} \int_{\frac{t}{2}}^{t} {(t - τ)}^{- \frac{n}{2} (1 - \frac{1}{r})} d τ ⩽ C ρ^{1 + \frac{1}{n}} {⟨ t ⟩}^{- \frac{n}{2} (1 - \frac{1}{r})} log ⟨ T ⟩ . \end{array}$ Similarly, we find $\begin{matrix} {‖ \nabla (M_{2} w) (t) ‖}_{L^{r} (R_{+}^{n})} ⩽ C ρ^{1 + \frac{1}{n}} \int_{0}^{t} {(t - τ)}^{- \frac{1}{2} - \frac{n}{2} (1 - \frac{1}{r})} {⟨ τ ⟩}^{- \frac{1}{2}} τ^{- \frac{1}{2}} d τ . \end{matrix}$ Then as above we get $\begin{matrix} t^{\frac{1}{2}} {⟨ t ⟩}^{\frac{n}{2} (1 - \frac{1}{r})} {‖ \nabla (M_{2} w) (t) ‖}_{L^{r} (R_{+}^{n})} ⩽ C ρ^{1 + \frac{1}{n}} log ⟨ T ⟩ . \end{matrix}$ This completes the proof of the lemma. □

In the same manner we find. Lemma 3.2.
Assume that $w_{1}, w_{2} \in X_{T, ρ}$ . Then the estimates are true $\begin{array}{l} ‖ M_{1} w_{1} - M_{1} w_{2} ‖_{X_{T}} ⩽ C ρ^{\frac{2}{n}} log ⟨ T ⟩ ‖ w_{1} - w_{2} ‖_{X_{T}}, \\ ‖ M_{2} w_{1} - M_{2} w_{2} ‖_{X_{T}} ⩽ C ρ^{\frac{1}{n}} log ⟨ T ⟩ ‖ w_{1} - w_{2} ‖_{X_{T}} . \end{array}$

3.2. Proof of Theorem 1.1

To prove the local in time existence of solutions, we consider $\begin{matrix} u = M w = ε S_{N} (t) u_{0} + M_{1} w + M_{2} w \end{matrix}$ then by Lemma 2.1 and Lemma 3.1 $\begin{matrix} ‖ M w ‖_{X_{T}} ⩽ C ε + C (1 + ρ^{\frac{1}{n}}) ρ^{1 + \frac{1}{n}} log ⟨ T ⟩ \end{matrix}$ and by Lemma 3.2 for $u_{j} = M w_{j}$ $\begin{matrix} ‖ M w_{1} - M w_{2} ‖_{X_{T}} ⩽ C ‖ w_{1} - w_{2} ‖_{X_{T}} ρ^{\frac{1}{n}} (1 + ρ^{\frac{1}{n}}) log ⟨ T ⟩ . \end{matrix}$ We put $ρ = ε$ , then Theorem 1.1 follows via the contraction mapping principle.

4. Transformed equation for Theorem 1.2

In this section, we consider the problem under the conditions such that $\begin{matrix} θ = 2 \int_{0}^{\infty} u_{0} (x) d x > 0, λ, γ < 0 . \end{matrix}$ Namely, we consider (1.1) with (1.2). We translate the equation (1.1) to another one by using the changing of the dependent variable $e^{φ (t)} u = v$ , $φ (0) = 0$ , which is initiated in [6]. By a direct calculation, we obtain $\begin{matrix} (4.1) & \{\begin{matrix} \partial_{t} v - Δ v = e^{- \frac{2}{n} φ} λ | v |^{1 + \frac{2}{n}} + φ^{'} v, t > 0, x \in R_{+}^{n}, \\ v (0, x) = ε u_{0} (x), x \in R_{+}^{n}, \\ - \partial_{x_{n}} v (t, x^{'}, 0) = γ e^{- \frac{1}{n} φ} | v |^{1 + \frac{1}{n}} (t, x^{'}, 0), t > 0, x^{'} \in R^{n - 1} . \end{matrix} \end{matrix}$ Integrating equation (4.1) in $R_{+}^{n}$ , we find $\begin{array}{l} \partial_{t} \int_{R_{+}^{n}} v (t, x) d x - \int_{R_{+}^{n}} \partial_{x_{n}}^{2} v (t, x) d x \\ = e^{- \frac{2}{n} φ (t)} λ \int_{R_{+}^{n}} | v |^{1 + \frac{2}{n}} (t, x) d x + φ^{'} (t) \int_{R_{+}^{n}} v (t, x) d x . \end{array}$ Using the boundary condition $- \partial_{x_{n}} v (t, x^{'}, 0) = γ e^{- \frac{1}{n} φ} | v |^{1 + \frac{1}{n}} (t, x^{'}, 0)$ , we get $\begin{array}{rcl} \int_{R_{+}^{n}} \partial_{x_{n}}^{2} v (t, x) d x & = & - \int_{R^{n - 1}} \partial_{x_{n}} v (t, x^{'}, 0) d x^{'} \\ = & e^{- \frac{1}{n} φ (t)} γ \int_{R^{n - 1}} | v |^{1 + \frac{1}{n}} (t, x^{'}, 0) d x^{'} . \end{array}$ Then it follows that $\begin{array}{rcl} \partial_{t} \int_{R_{+}^{n}} v (t, x) d x & = & e^{- \frac{2}{n} φ (t)} λ \int_{R_{+}^{n}} | v |^{1 + \frac{2}{n}} (t, x) d x \\ + e^{- \frac{1}{n} φ (t)} γ \int_{R^{n - 1}} | v |^{1 + \frac{1}{n}} (t, x^{'}, 0) d x^{'} + φ^{'} (t) \int_{R_{+}^{n}} v (t, x) d x . \end{array}$ If we now choose $φ (t)$ by the condition $\partial_{t} \int_{R_{+}^{n}} v (t, x) d x = 0$ , i.e. $\begin{array}{l} φ^{'} (t) \int_{R_{+}^{n}} v (t, x) d x \\ = - e^{- \frac{2}{n} φ (t)} λ {‖ v (t) ‖}_{L^{1 + \frac{2}{n}} (R_{+}^{n})}^{1 + \frac{2}{n}} - e^{- \frac{1}{n} φ (t)} γ {‖ v (t, \cdot, 0) ‖}_{L^{1 + \frac{1}{n}} (R^{n - 1})}^{1 + \frac{1}{n}}, \end{array}$ then we find $\begin{matrix} \int_{R_{+}^{n}} v (t, x) d x = ε \int_{R_{+}^{n}} u_{0} (x) d x = \frac{ε θ}{2} \end{matrix}$ and $\begin{array}{rcl} \partial_{t} v - Δ v & = & e^{- \frac{2}{n} φ} λ | v |^{1 + \frac{2}{n}} + φ^{'} v \\ = & λ e^{- \frac{2}{n} φ} (| v |^{1 + \frac{2}{n}} (t) - {‖ v (t) ‖}_{L^{1 + \frac{2}{n}} (R_{+}^{n})}^{1 + \frac{2}{n}} \frac{2 v (t)}{ε θ}) \\ - γ e^{- \frac{1}{n} φ} {‖ v (t, \cdot, 0) ‖}_{L^{1 + \frac{1}{n}} (R^{n - 1})}^{1 + \frac{1}{n}} \frac{2 v (t)}{ε θ} . \end{array}$ The corresponding integral equation is written as $\begin{array}{rcl} v (t) & = & ε S_{N} (t) u_{0} \\ + λ \int_{0}^{t} e^{- \frac{2}{n} φ (τ)} S_{N} (t - τ) (| v |^{1 + \frac{2}{n}} (τ) - {‖ v (τ) ‖}_{L^{1 + \frac{2}{n}} (R_{+}^{n})}^{1 + \frac{2}{n}} \frac{2 v (τ)}{ε θ}) d τ \\ + 2 γ \int_{0}^{t} e^{- \frac{1}{n} φ (τ)} (G_{n} (t - τ) \tilde{S} (t - τ) | v |^{1 + \frac{1}{n}} (τ, \cdot, 0) \\ - {‖ v (τ, \cdot, 0) ‖}_{L^{1 + \frac{1}{n}} (R^{n - 1})}^{1 + \frac{1}{n}} S_{N} (t - τ) \frac{v (τ)}{ε θ}) d τ, \end{array}$ where $\begin{matrix} G_{n} (t, x_{n}) = \frac{1}{\sqrt{4 π t}} e^{- \frac{x_{n}^{2}}{4 t}} . \end{matrix}$ We denote also $\begin{matrix} \tilde{G} (t, x^{'}) = \frac{1}{{(4 π t)}^{(n - 1) / 2}} e^{- \frac{| x^{'} |^{2}}{4 t}} \end{matrix}$ and $\begin{matrix} G (t, x) = \frac{1}{{(4 π t)}^{n / 2}} e^{- \frac{| x |^{2}}{4 t}} = \tilde{G} (t) G_{n} (t), \end{matrix}$ then we obtain the following integral equation $\begin{matrix} (4.2) & v (t) = ε S_{N} (t) u_{0} + M_{1} v + M_{2, 1} v + M_{2, 2} v, \end{matrix}$ where $\begin{array}{l} M_{1} v (t) = λ \int_{0}^{t} e^{- \frac{2}{n} φ (τ)} S_{N} (t - τ) (| v |^{1 + \frac{2}{n}} (τ) - {‖ v (τ) ‖}_{L^{1 + \frac{2}{n}} (R_{+}^{n})}^{1 + \frac{2}{n}} \frac{2 v (τ)}{ε θ}) d τ, \\ \begin{matrix} M_{2, 1} v & = 2 γ \int_{0}^{t} e^{- \frac{1}{n} φ (τ)} G_{n} (t - τ) \\ \times (\tilde{S} (t - τ) | v |^{1 + \frac{1}{n}} (τ, \cdot, 0) - {‖ v (τ, \cdot, 0) ‖}_{L^{1 + \frac{1}{n}} (R^{n - 1})}^{1 + \frac{1}{n}} \tilde{G} (t - τ)) d τ \end{matrix} \end{array}$ and $\begin{matrix} M_{2, 2} v = 2 γ \int_{0}^{t} e^{- \frac{1}{n} φ (τ)} {‖ v (τ, \cdot, 0) ‖}_{L^{1 + \frac{1}{n}} (R^{n - 1})}^{1 + \frac{1}{n}} (G (t - τ) - S_{N} (t - τ) \frac{v (τ)}{ε θ}) d τ . \end{matrix}$ Note that $\begin{matrix} \int_{R_{+}^{n}} S_{N} (t) u_{0} d x = \int_{R_{+}^{n}} u_{0} (x) d x = \frac{θ}{2} \end{matrix}$ and $\begin{matrix} \int_{R_{+}^{n}} v (t, x) d x = \frac{ε θ}{2} . \end{matrix}$ Indeed, integrating (4.2) over $R_{+}^{n}$ , we obtain $\begin{array}{rcl} \int_{R_{+}^{n}} v (t, x) d x & = & \frac{ε θ}{2} \\ + \frac{2 λ}{ε θ} \int_{0}^{t} e^{- \frac{2}{n} φ (τ)} {‖ v (τ) ‖}_{L^{1 + \frac{2}{n}} (R_{+}^{n})}^{1 + \frac{2}{n}} (\frac{ε θ}{2} - \int_{R_{+}^{n}} v (τ, x) d x) d τ \\ + \frac{2 γ}{ε θ} \int_{0}^{t} e^{- \frac{1}{n} φ (τ)} {‖ v (τ, \cdot, 0) ‖}_{L^{1 + \frac{1}{n}} (R^{n - 1})}^{1 + \frac{1}{n}} (\frac{ε θ}{2} - \int_{R_{+}^{n}} v (τ, x) d x) d τ, \end{array}$ since the third term on the right-hand side of (4.2) vanishes $\int_{R_{+}^{n}} M_{2, 1} v d x = 0$ . We put $\begin{matrix} F (t) = \int_{R_{+}^{n}} v (t, x) d x - \frac{ε θ}{2}, \end{matrix}$ then we can write the above equation as $F^{'} (t) = H (t) F (t)$ with initial condition $F (0) = 0$ , where $\begin{matrix} H (t) = - \frac{2 λ}{ε θ} e^{- \frac{2}{n} φ (t)} {‖ v (τ) ‖}_{L^{1 + \frac{2}{n}} (R_{+}^{n})}^{1 + \frac{2}{n}} - \frac{2 γ}{ε θ} e^{- \frac{1}{n} φ (t)} {‖ v (τ, \cdot, 0) ‖}_{L^{1 + \frac{1}{n}} (R^{n - 1})}^{1 + \frac{1}{n}} . \end{matrix}$ Integration in time yields $\begin{matrix} F (t) = F (0) e^{- \int_{0}^{t} H (t) d t} = 0 . \end{matrix}$ Therefore we have the conservation law such that $\begin{matrix} \int_{R_{+}^{n}} v (t, x) d x = \frac{ε θ}{2} \end{matrix}$ for all $t ⩾ 0$ . Below we will study equation (4.2) along with the equations for φ $\begin{array}{rcl} (4.3) & \begin{array}{l} \begin{matrix} \frac{d}{d t} φ (t) & = - \frac{2 λ}{ε θ} e^{- \frac{2}{n} φ (t)} {‖ v (t) ‖}_{L^{1 + \frac{2}{n}} (R_{+}^{n})}^{1 + \frac{2}{n}} \\ - \frac{2 γ}{ε θ} e^{- \frac{1}{n} φ (t)} {‖ v (t, \cdot, 0) ‖}_{L^{1 + \frac{1}{n}} (R^{n - 1})}^{1 + \frac{1}{n}}, \end{matrix} \\ φ (0) = 0 . \end{array} \end{array}$ In the case of nonlinearities (1.3), analogously we have $\begin{array}{l} M_{1} v (t) = λ \int_{0}^{t} e^{- \frac{2}{n} φ (τ)} S_{N} (t - τ) (| v |^{\frac{2}{n}} v (τ) - \int_{R_{+}^{n}} | v |^{\frac{2}{n}} v (τ, x) d x \frac{2 v (τ)}{ε θ}) d τ, \\ \begin{matrix} M_{2, 1} v & = 2 γ \int_{0}^{t} e^{- \frac{1}{n} φ (τ)} (\int_{R^{n - 1}} | v |^{\frac{1}{n}} v (τ, x^{'}, 0) d x^{'}) G_{n} (t - τ) \\ \times (\tilde{S} (t - τ) \frac{| v |^{\frac{1}{n}} v (τ, \cdot, 0)}{\int_{R^{n - 1}} | v |^{\frac{1}{n}} v (τ, x^{'}, 0) d x^{'}} - \tilde{G} (t - τ)) d τ \end{matrix} \end{array}$ and $\begin{array}{l} M_{2, 2} v \\ = 2 γ \int_{0}^{t} e^{- \frac{1}{n} φ (τ)} \int_{R^{n - 1}} | v |^{\frac{1}{n}} v (τ, x^{'}, 0) d x^{'} (G (t - τ) - S_{N} (t - τ) \frac{v (τ)}{ε θ}) d τ, \end{array}$ and also equations for φ $\begin{array}{l} \frac{d}{d t} φ (t) = - \frac{2 λ}{ε θ} e^{- \frac{2}{n} φ (t)} \int_{R_{+}^{n}} | v |^{\frac{2}{n}} v (t, x) d x - \frac{2 γ}{ε θ} e^{- \frac{1}{n} φ (t)} \int_{R^{n - 1}} | v |^{\frac{1}{n}} v (t, x^{'}, 0) d x^{'}, \\ φ (0) = 0, \end{array}$ instead of (4.3).

Next we prove the following result.

Lemma 4.1.
Assume that $0 < b < \frac{1}{n + 1}$ , $\begin{matrix} w \in X_{b, \infty, ε}, \int_{R_{+}^{n}} w (τ) d x = \frac{ε θ}{2} . \end{matrix}$ Also suppose that $e^{- \frac{1}{n} φ (t)} ⩽ C$ for any $t ⩾ 0$ . Then $\begin{matrix} M_{1} w \in X_{b, \infty, C ε^{1 + \frac{2}{n}}}, M_{2, 1} w + M_{2, 2} w \in X_{b, \infty, C ε^{1 + \frac{1}{n}}} . \end{matrix}$
Proof.
We let $f = | w |^{1 + \frac{2}{n}}$ (analogously $f = | w |^{\frac{2}{n}} w$ for the case of the nonlinearities (1.3)) and $\begin{matrix} g_{1} (τ) = g (τ) - \frac{2 w (τ)}{ε θ} \int_{R_{+}^{n}} f (τ, x) d x . \end{matrix}$ Then since $\hat{f_{1}} (t, 0) = \frac{1}{{(2 π)}^{n / 2}} \int_{R_{+}^{n}} f_{1} (τ, x) d x = 0$ , which is true by virtue of the condition $\int_{R_{+}^{n}} w (τ) d x = \frac{ε θ}{2}$ , we have $\begin{matrix} | x |^{β} S_{N} (t - τ) f_{1} (τ) = | x |^{β} (S_{N} (t - τ) f_{1} (τ) - G (t - τ) {(2 π)}^{\frac{n}{2}} \hat{f_{1}} (t, 0)) . \end{matrix}$ Then by Lemma 2.3 we obtain for $1 ⩽ r ⩽ 1 + \frac{1}{n}$ , $0 ⩽ β ⩽ a ⩽ b < 1$ $\begin{array}{l} {‖ | \cdot |^{β} S_{N} (t - τ) f_{1} (τ) ‖}_{L^{r} (R_{+}^{n})} \\ ⩽ C {⟨ t - τ ⟩}^{\frac{β - a}{2}} {(t - τ)}^{- \frac{n}{2} (1 - \frac{1}{r})} {‖ {⟨ \cdot ⟩}^{a} f_{1} (τ) ‖}_{L^{1} (R_{+}^{n})} \\ ⩽ C {⟨ t - τ ⟩}^{\frac{β - a}{2}} {(t - τ)}^{- \frac{n}{2} (1 - \frac{1}{r})} \\ \times ({‖ {⟨ \cdot ⟩}^{a} g (τ) ‖}_{L^{1} (R_{+}^{n})} + \frac{1}{ε} {‖ g (τ) ‖}_{L^{1} (R_{+}^{n})} {‖ {⟨ \cdot ⟩}^{a} w (τ) ‖}_{L^{1} (R_{+}^{n})}) \\ ⩽ C {⟨ t - τ ⟩}^{\frac{β - a}{2}} {(t - τ)}^{- \frac{n}{2} (1 - \frac{1}{r})} \\ \times ({‖ {⟨ \cdot ⟩}^{a} w (τ) ‖}_{L^{1 + \frac{2}{n}} (R_{+}^{n})} {‖ w (τ) ‖}_{L^{1 + \frac{2}{n}} (R_{+}^{n})}^{\frac{2}{n}} \\ + \frac{1}{ε} {‖ w (τ) ‖}_{L^{1 + \frac{2}{n}} (R_{+}^{n})}^{1 + \frac{2}{n}} {‖ {⟨ \cdot ⟩}^{a} w (τ) ‖}_{L^{1} (R_{+}^{n})}) . \end{array}$ Hence $\begin{array}{l} {‖ | \cdot |^{β} M_{1} w (t) ‖}_{L^{r} (R_{+}^{n})} \\ ⩽ C \int_{0}^{t} e^{- \frac{2}{n} φ (τ)} {‖ | \cdot |^{β} S_{N} (t - τ) f_{1} (τ) ‖}_{L^{r} (R_{+}^{n})} d τ \\ ⩽ C \int_{0}^{t} {⟨ t - τ ⟩}^{\frac{β - a}{2}} {(t - τ)}^{- \frac{n}{2} (1 - \frac{1}{r})} \\ \times ({‖ {⟨ \cdot ⟩}^{a} w (τ) ‖}_{L^{1 + \frac{2}{n}} (R_{+}^{n})} {‖ w (τ) ‖}_{L^{1 + \frac{2}{n}} (R_{+}^{n})}^{\frac{2}{n}} \\ (4.4) & + \frac{1}{ε} {‖ w (τ) ‖}_{L^{1 + \frac{2}{n}} (R_{+}^{n})}^{1 + \frac{2}{n}} {‖ {⟨ \cdot ⟩}^{a} w (τ) ‖}_{L^{1} (R_{+}^{n})} d τ) . \end{array}$ By the Gagliardo–Nirenberg–Sobolev interpolation inequality, we find $\begin{matrix} (4.5) & {‖ {⟨ \cdot ⟩}^{a} ϕ ‖}_{L^{1 + \frac{2}{n}} (R_{+}^{n})} ⩽ C {‖ {⟨ \cdot ⟩}^{a} ϕ ‖}_{L^{1 + \frac{1}{n}} (R_{+}^{n})}^{\frac{3 n + 2}{(n + 1) (n + 2)}} {‖ \nabla {⟨ \cdot ⟩}^{a} ϕ ‖}_{L^{1 + \frac{1}{n}} (R_{+}^{n})}^{\frac{n^{2}}{(n + 1) (n + 2)}} . \end{matrix}$ Then we compute $\begin{matrix} {‖ \nabla {⟨ \cdot ⟩}^{a} ϕ ‖}_{L^{1 + \frac{1}{n}} (R_{+}^{n})} ⩽ C {‖ {⟨ \cdot ⟩}^{a - 1} ϕ ‖}_{L^{1 + \frac{1}{n}} (R_{+}^{n})} + C {‖ {⟨ \cdot ⟩}^{a} \nabla ϕ ‖}_{L^{1 + \frac{1}{n}} (R_{+}^{n})} . \end{matrix}$ We extend ϕ to $R^{n}$ as an even function with respect to $x_{n}$ to see by the Sobolev embedding inequality (see Theorem 2.44 in [1]) $\begin{array}{rcl} {‖ {⟨ \cdot ⟩}^{a - 1} ϕ ‖}_{L^{1 + \frac{1}{n}} (R^{n})} & ⩽ & {‖ | \cdot |^{a - 1} ϕ ‖}_{L^{1 + \frac{1}{n}} (R^{n})} ⩽ C ‖ ϕ ‖_{L^{d} (R^{n})} \\ ⩽ & C {‖ | \nabla |^{1 - a} ϕ ‖}_{L^{1 + \frac{1}{n}} (R^{n})} ⩽ C ‖ \nabla ϕ ‖_{L^{1 + \frac{1}{n}} (R^{n})}^{1 - a} ‖ ϕ ‖_{L^{1 + \frac{1}{n}} (R^{n})}^{a} \end{array}$ with $\begin{matrix} \frac{1}{d} + \frac{1 - a}{n} = \frac{n}{n + 1} . \end{matrix}$ Therefore $\begin{array}{l} {‖ \nabla {⟨ \cdot ⟩}^{a} ϕ ‖}_{L^{1 + \frac{1}{n}} (R_{+}^{n})} \\ (4.6) & ⩽ C ‖ \nabla ϕ ‖_{L^{1 + \frac{1}{n}} (R_{+}^{n})}^{1 - a} ‖ ϕ ‖_{L^{1 + \frac{1}{n}} (R_{+}^{n})}^{a} + C {‖ {⟨ \cdot ⟩}^{a} \nabla ϕ ‖}_{L^{1 + \frac{1}{n}} (R_{+}^{n})} . \end{array}$ Using the estimate $\begin{matrix} {‖ {⟨ \cdot ⟩}^{a} \nabla^{k} w (t) ‖}_{L^{1 + \frac{1}{n}} (R_{+}^{n})} ⩽ C ε t^{- \frac{k}{2}} {⟨ t ⟩}^{\frac{a}{2} - \frac{n}{2 n + 2}}, \end{matrix}$ for $k = 0, 1$ and $0 ⩽ a ⩽ b$ , we obtain form (4.5) and (4.6) $\begin{matrix} (4.7) & {‖ {⟨ \cdot ⟩}^{a} w (t) ‖}_{L^{1 + \frac{2}{n}} (R_{+}^{n})} ⩽ C ε t^{- \frac{n^{2}}{2 (n + 1) (n + 2)}} {⟨ t ⟩}^{\frac{a}{2} - \frac{n}{2 n + 2}} . \end{matrix}$ By (4.4) and (4.7) we find $\begin{matrix} (4.8) & {‖ | \cdot |^{β} M_{1} w (t) ‖}_{L^{r} (R_{+}^{n})} ⩽ C ε^{1 + \frac{2}{n}} \int_{0}^{t} {(t - τ)}^{\frac{β - b}{2} - \frac{n}{2} (1 - \frac{1}{r})} τ^{- \frac{n}{2 n + 2}} {⟨ τ ⟩}^{\frac{b}{2} - \frac{n + 2}{2 n + 2}} d τ . \end{matrix}$ Then we get from (4.8) $\begin{array}{l} {‖ | \cdot |^{β} M_{1} w (t) ‖}_{L^{r} (R_{+}^{n})} \\ ⩽ C ε^{1 + \frac{2}{n}} (\int_{0}^{1} + \int_{1}^{t}) {⟨ t - τ ⟩}^{\frac{β - a}{2}} {(t - τ)}^{- \frac{n}{2} (1 - \frac{1}{r})} τ^{- \frac{n}{2 n + 2}} {⟨ τ ⟩}^{\frac{b}{2} - \frac{n + 2}{2 n + 2}} d τ \\ ⩽ C ε^{1 + \frac{2}{n}} + C ε^{1 + \frac{2}{n}} \int_{1}^{t} {⟨ t - τ ⟩}^{\frac{β - a}{2}} {(t - τ)}^{- \frac{n}{2} (1 - \frac{1}{r})} τ^{\frac{b}{2} - 1} d τ \\ (4.9) & ⩽ C ε^{1 + \frac{2}{n}} t^{- \frac{n}{2} (1 - \frac{1}{r}) + \frac{β}{2}} \end{array}$ for $t ⩾ 2$ . We also have for $0 < t ⩽ 2$ $\begin{array}{l} {‖ | \cdot |^{β} M_{1} w (t) ‖}_{L^{r} (R_{+}^{n})} \\ (4.10) & ⩽ C ε^{1 + \frac{2}{n}} \int_{0}^{t} ({(t - τ)}^{- \frac{2 n}{2 n + 2}} + τ^{- \frac{2 n}{2 n + 2}}) d τ ⩽ C ε^{1 + \frac{2}{n}} . \end{array}$ By (4.9) and (4.10) we obtain $\begin{matrix} {‖ | \cdot |^{β} M_{1} w (t) ‖}_{L^{r} (R_{+}^{n})} ⩽ C ε^{1 + \frac{2}{n}} {⟨ t ⟩}^{\frac{β}{2} - \frac{n}{2} (1 - \frac{1}{r})} \end{matrix}$ from which it follows that $\begin{array}{l} sup_{t > 0} {⟨ t ⟩}^{- \frac{β}{2}} {‖ | \cdot |^{β} M_{1} w (t) ‖}_{L^{1} (R_{+}^{n})} \\ (4.11) & + sup_{t > 0} {⟨ t ⟩}^{- \frac{β}{2} + \frac{n}{2 n + 2}} {‖ | \cdot |^{β} M_{1} w (t) ‖}_{L^{1 + \frac{1}{n}} (R_{+}^{n})} ⩽ C ε^{1 + \frac{2}{n}} . \end{array}$ In the same way as in the proof of (4.11) we have $\begin{array}{l} {⟨ t ⟩}^{- \frac{β}{2}} t^{\frac{1}{2}} {‖ | \cdot |^{β} \nabla M_{1} w (t) ‖}_{L^{1} (R_{+}^{n})} \\ + {⟨ t ⟩}^{- \frac{β}{2} + \frac{n}{2 n + 2}} t^{\frac{1}{2}} {‖ | \cdot |^{β} \nabla M_{1} w (t) ‖}_{L^{1 + \frac{1}{n}} (R_{+}^{n})} \\ (4.12) & ⩽ C ε^{1 + \frac{2}{n}} . \end{array}$ We take $β = b$ or $β = 0$ in (4.11) and (4.12), then we find that $\begin{matrix} (4.13) & M_{1} w \in X_{b, \infty, C ε^{1 + \frac{2}{n}}} . \end{matrix}$

We next consider $\begin{matrix} M_{2, 2} w (t) = 2 γ \int_{0}^{t} e^{- \frac{1}{n} φ (τ)} {‖ w (τ, \cdot, 0) ‖}_{L^{1 + \frac{1}{n}} (R^{n - 1})}^{1 + \frac{1}{n}} (G (t - τ) - S_{N} (t - τ) \frac{w (τ, x)}{ε θ}) d τ \end{matrix}$ or $\begin{array}{rcl} M_{2, 2} w (t) & = & 2 γ \int_{0}^{t} e^{- \frac{1}{n} φ (τ)} \int_{R^{n - 1}} {| w (τ, x^{'}, 0) |}^{\frac{1}{n}} w (τ, x^{'}, 0) d x^{'} \\ \times (G (t - τ) - S_{N} (t - τ) \frac{w (τ, x)}{ε θ}) d τ . \end{array}$ In what follows, we consider the case (1.2) since the case (1.3) can be treated in the same way. We write $\begin{matrix} (4.14) & {‖ w (τ, \cdot, 0) ‖}_{L^{1 + \frac{1}{n}} (R^{n - 1})}^{1 + \frac{1}{n}} = - \int_{R_{+}^{n}} \partial_{x_{n}} | w |^{1 + \frac{1}{n}} (τ, x) d x . \end{matrix}$ Then by Lemma 2.3 and using $\begin{matrix} {‖ {⟨ \cdot ⟩}^{a} \nabla^{k} w (τ) ‖}_{L^{r} (R_{+}^{n})} ⩽ C ε τ^{- \frac{k}{2}} {⟨ τ ⟩}^{\frac{a}{2} - \frac{n}{2} (1 - \frac{1}{r})} \end{matrix}$ for $k = 0, 1$ , $1 ⩽ r ⩽ 1 + \frac{1}{n}$ , $0 ⩽ a ⩽ b$ , we get $\begin{array}{l} {‖ | \cdot |^{β} M_{2, 2} w (t) ‖}_{L^{r} (R_{+}^{n})} \\ ⩽ \frac{C}{ε θ} \int_{0}^{t} {‖ \partial_{x_{n}} w (τ) ‖}_{L^{1 + \frac{1}{n}} (R_{+}^{n})} {‖ w (τ) ‖}_{L^{1 + \frac{1}{n}} (R_{+}^{n})}^{\frac{1}{n}} \\ \times {‖ | \cdot |^{β} (G (t - τ) ε θ - S_{N} (t - τ) w (τ, x)) ‖}_{L^{r} (R_{+}^{n})} d τ \\ ⩽ \frac{C}{ε θ} \int_{0}^{t} {⟨ t - τ ⟩}^{\frac{β - b}{2}} {(t - τ)}^{- \frac{n}{2} (1 - \frac{1}{r})} {‖ \partial_{x_{n}} w (τ) ‖}_{L^{1 + \frac{1}{n}} (R_{+}^{n})} \\ \times {‖ w (τ) ‖}_{L^{1 + \frac{1}{n}} (R_{+}^{n})}^{\frac{1}{n}} {‖ {⟨ \cdot ⟩}^{b} w (τ) ‖}_{L^{1} (R_{+}^{n})} d τ \\ ⩽ C ε^{1 + \frac{1}{n}} \int_{0}^{t} {⟨ t - τ ⟩}^{\frac{β - b}{2}} {(t - τ)}^{- \frac{n}{2} (1 - \frac{1}{r})} τ^{- \frac{1}{2}} {⟨ τ ⟩}^{\frac{b - 1}{2}} d τ . \end{array}$ We take $β = 0$ or $β = b$ to find that $\begin{array}{l} {⟨ t ⟩}^{\frac{n}{2} (1 - \frac{1}{r}) + \frac{b}{2}} {‖ | \cdot |^{b} M_{2, 2} w (t) ‖}_{L^{r} (R_{+}^{n})} \\ + {⟨ t ⟩}^{\frac{n}{2} (1 - \frac{1}{r})} {‖ M_{2, 2} w (t) ‖}_{L^{r} (R_{+}^{n})} \\ (4.15) & ⩽ C ε^{1 + \frac{1}{n}} . \end{array}$ In the same way as in the proof of (4.15) we have $\begin{array}{l} {⟨ t ⟩}^{- \frac{b}{2} + \frac{n}{2} (1 - \frac{1}{r})} t^{\frac{1}{2}} {‖ | \cdot |^{b} \nabla M_{2, 2} w (t) ‖}_{L^{r} (R_{+}^{n})} \\ + {⟨ t ⟩}^{\frac{1}{2} + \frac{n}{2} (1 - \frac{1}{r})} t^{\frac{1}{2}} {‖ \nabla M_{2, 2} w (t) ‖}_{L^{r} (R_{+}^{n})} \\ (4.16) & ⩽ C ε^{1 + \frac{1}{n}} . \end{array}$ By (4.15) and (4.16) $\begin{matrix} (4.17) & M_{2, 2} w \in X_{b, \infty, C ε^{1 + \frac{1}{n}}} . \end{matrix}$ We next consider the estimate of $M_{2, 1} w (t)$ . We again use Lemma 2.3 to obtain with $g (τ, x^{'}) = | w |^{1 + \frac{1}{n}} (τ, x^{'}, 0)$ $\begin{array}{l} {‖ | x_{n} |^{β} G_{n} (t - τ) (\tilde{S} (t - τ) g (τ) - \tilde{G} (t - τ) {‖ g (τ) ‖}_{L^{1} (R^{n - 1})}) ‖}_{L^{r} (R_{+}^{n})} \\ ⩽ C {(t - τ)}^{\frac{β}{2} - \frac{1}{2} (1 - \frac{1}{r})} \\ \times {‖ \tilde{S} (t - τ) g (τ) - \tilde{G} (t - τ) {‖ g (τ) ‖}_{L^{1} (R^{n - 1})} ‖}_{L^{r} (R^{n - 1})} \\ ⩽ C {(t - τ)}^{\frac{β}{2} - \frac{1}{2} (1 - \frac{1}{r})} {⟨ t - τ ⟩}^{- \frac{b}{2}} {(t - τ)}^{- \frac{n - 1}{2} (1 - \frac{1}{r})} \\ \times {‖ {| x^{'} |}^{b} g (τ) ‖}_{L^{1} (R^{n - 1})} \\ (4.18) & ⩽ C {⟨ t - τ ⟩}^{\frac{β - b}{2}} {(t - τ)}^{- \frac{n}{2} (1 - \frac{1}{r})} {‖ {| x^{'} |}^{b} g (τ) ‖}_{L^{1} (R^{n - 1})} \end{array}$ and $\begin{array}{l} {‖ {| x^{'} |}^{β} G_{n} (t - τ) (\tilde{S} (t - τ) g (τ) - \tilde{G} (t - τ) {‖ g (τ) ‖}_{L^{1} (R^{n - 1})}) ‖}_{L^{r} (R_{+}^{n})} \\ ⩽ C {(t - τ)}^{- \frac{1}{2} (1 - \frac{1}{r})} \\ \times {‖ {| x^{'} |}^{β} (\tilde{S} (t - τ) f (τ) - \tilde{G} (t - τ) {‖ g (τ) ‖}_{L^{1} (R^{n - 1})}) ‖}_{L^{r} (R^{n - 1})} \\ ⩽ C {(t - τ)}^{- \frac{1}{2} (1 - \frac{1}{r})} {⟨ t - τ ⟩}^{\frac{β - b}{2}} {(t - τ)}^{- \frac{n - 1}{2} (1 - \frac{1}{r})} \\ \times {‖ {| x^{'} |}^{b} g (τ) ‖}_{L^{1} (R^{n - 1})} \\ (4.19) & ⩽ C {⟨ t - τ ⟩}^{\frac{β - b}{2}} {(t - τ)}^{- \frac{n}{2} (1 - \frac{1}{r})} {‖ {| x^{'} |}^{b} g (τ) ‖}_{L^{1} (R^{n - 1})} . \end{array}$ By (4.14) $\begin{array}{rcl} {‖ {| x^{'} |}^{b} g (τ) ‖}_{L^{1} (R^{n - 1})} & ⩽ & C {‖ {| x^{'} |}^{b} \int_{0}^{\infty} \partial_{x_{n}} | w |^{1 + \frac{1}{n}} (τ, x^{'}, x_{n}) d x_{n} ‖}_{L^{1} (R^{n - 1})} \\ ⩽ & C {‖ | \cdot |^{b} \nabla w (τ) ‖}_{L^{1 + \frac{1}{n}} (R^{n - 1})} {‖ w (τ) ‖}_{L^{1 + \frac{1}{n}} (R^{n - 1})}^{\frac{1}{n}} ⩽ C ε^{1 + \frac{1}{n}} τ^{- \frac{1}{2}} {⟨ τ ⟩}^{- \frac{1}{2} + \frac{b}{2}} . \end{array}$ Therefore by (4.18), (4.19) and the fact that $| x |^{a} ⩽ C | x_{n} |^{a} + C | x^{'} |^{a}$ , it follows that $\begin{array}{l} {‖ | \cdot |^{β} M_{2, 1} w (t) ‖}_{L^{r} (R^{n})} \\ ⩽ C \int_{0}^{t} {‖ | \cdot |^{β} G_{n} (t - τ) (\tilde{S} (t - τ) g (τ) - \tilde{G} (t - τ) \int_{R^{n - 1}} g (τ, x^{'}) d x^{'}) ‖}_{L^{r} (R^{n})} d τ \\ ⩽ C ε^{1 + \frac{1}{n}} \int_{0}^{t} {⟨ t - τ ⟩}^{\frac{β - b}{2}} {(t - τ)}^{- \frac{n}{2} (1 - \frac{1}{r})} τ^{- \frac{1}{2}} {⟨ τ ⟩}^{- \frac{1}{2} + \frac{b}{2}} d τ \end{array}$ and similarly, $\begin{array}{l} {‖ | \cdot |^{β} \nabla M_{2, 1} w (t) ‖}_{L^{r} (R_{+}^{n})} \\ ⩽ C \int_{0}^{t} {‖ | \cdot |^{β} \nabla G_{n} (t - τ) (\tilde{S} (t - τ) g (τ) - \tilde{G} (t - τ) \int_{R^{n - 1}} g (τ, x^{'}) d x^{'}) ‖}_{L^{r} (R_{+}^{n})} d τ \\ ⩽ C ε^{1 + \frac{1}{n}} \int_{0}^{t} {⟨ t - τ ⟩}^{\frac{β - b}{2}} {(t - τ)}^{- \frac{1}{2} - \frac{n}{2} (1 - \frac{1}{r})} τ^{- \frac{1}{2}} {⟨ τ ⟩}^{- \frac{1}{2} + \frac{b}{2}} d τ . \end{array}$ We take $β = 0$ or $β = b$ , to find that $\begin{array}{l} {⟨ t ⟩}^{\frac{n}{2} (1 - \frac{1}{r}) + \frac{b}{2}} {‖ | \cdot |^{b} M_{2, 1} w (t) ‖}_{L^{r} (R_{+}^{n})} + {⟨ t ⟩}^{\frac{n}{2} (1 - \frac{1}{r})} {‖ M_{2, 1} w (t) ‖}_{L^{r} (R_{+}^{n})} \\ + t^{\frac{1}{2}} {⟨ t ⟩}^{\frac{n}{2} (1 - \frac{1}{r}) + \frac{b}{2}} {‖ | \cdot |^{b} \nabla M_{2, 1} w (t) ‖}_{L^{r} (R_{+}^{n})} \\ + t^{\frac{1}{2}} {⟨ t ⟩}^{\frac{n}{2} (1 - \frac{1}{r})} {‖ \nabla M_{2, 1} w (t) ‖}_{L^{r} (R_{+}^{n})} ⩽ C ε^{1 + \frac{1}{n}} \end{array}$ from which it follows that $\begin{matrix} (4.20) & M_{2, 1} w \in X_{b, \infty, C ε^{1 + \frac{1}{n}}} . \end{matrix}$ We have the result of the lemma by (4.13), (4.17) and (4.20). □

We consider the estimate of $φ (t)$ which is defined by $\begin{array}{l} (4.21) & \begin{array}{l} \frac{d}{d t} φ (t) = - \frac{λ}{ε θ} e^{- \frac{2}{n} φ (t)} f (t) - \frac{γ}{ε θ} e^{- \frac{1}{n} φ (t)} h (t), \\ φ (0) = 0, \end{array} \end{array}$ with $\begin{matrix} f (t) = {‖ w (t) ‖}_{L^{1 + \frac{2}{n}} (R_{+}^{n})}^{1 + \frac{2}{n}}, h (t) = {‖ w (t, 0) ‖}_{L^{1 + \frac{1}{n}} (R^{n - 1})}^{1 + \frac{1}{n}} \end{matrix}$ or $\begin{array}{l} (4.22) & \begin{array}{l} \frac{d}{d t} φ (t) = - \frac{λ}{ε θ} e^{- \frac{2}{n} φ (t)} f (t) - \frac{γ}{ε θ} e^{- \frac{1}{n} φ (t)} h (t), \\ φ (0) = 0, \end{array} \end{array}$ with $\begin{matrix} f (t) = \int_{R_{+}^{n}} | w |^{\frac{2}{n}} w d x, h (t) = \int_{R^{n - 1}} | w |^{\frac{1}{n}} w (t, x^{'}, 0) d x^{'} \end{matrix}$ Multiplying both sides of (4.21) by $g^{2} = e^{\frac{2}{n} φ (t)}$ , we have the nonlinear ordinary differential equations $\begin{matrix} (4.23) & \frac{1}{2} \frac{d}{d t} g^{2} (t) = g (t) f (t) + h (t), g (0) = 1 \end{matrix}$ Local in time of existence can be obtained in the standard way. A-priori estimate of local solution of $g (t)$ is obtained by the inequality $\begin{matrix} \frac{d}{d t} g^{2} (t) ⩽ g {(t)}^{2} + f {(t)}^{2} + 2 h (t) \end{matrix}$ which implies global existence of solutions. Namely, for any $w \in X_{b, \infty, ε}$ , we have a global solution $g = e^{\frac{1}{n} φ (t)} \in C ([0, \infty))$ . Next we prove the two-sided estimate for $e^{- \frac{1}{n} φ (t)}$ .
Lemma 4.2.
We assume that $w \in X_{b, \infty, C ε}$ , $λ < 0$ , $γ < 0$ , $u_{0} \in X_{b, 0}$ , $0 < b < \frac{1}{n + 1}$ and $\begin{matrix} θ = \int_{R^{n}} u_{0} (x) d x > 0 . \end{matrix}$ Furthermore, we suppose that $\begin{matrix} {‖ w - ε S (\cdot) u_{0} ‖}_{X_{b, \infty}} ⩽ C ε^{1 + \frac{1}{n}} . \end{matrix}$ Then there exist an $ε > 0$ and positive constants $c_{1}$ , $c_{2}$ such that the solution $φ (t)$ of ( 4.21 ) satisfies the estimates $\begin{matrix} (4.24) & \frac{1}{2 + c_{1} | ε θ |^{\frac{1}{n}} log ⟨ t ⟩} ⩽ e^{- \frac{1}{n} φ (t)} ⩽ \frac{2}{1 + c_{2} | ε θ |^{\frac{1}{n}} log ⟨ t ⟩} \end{matrix}$ for all $t > 0$ .
Proof.
We start with (4.23). We find that there exists a unique solution $g \in C ([0, \infty); R)$ . Since $w \in X_{b, \infty, C ε}$ , we have $\begin{matrix} | f (t) | ⩽ C ε^{\frac{1}{n}} {⟨ t ⟩}^{- 1}, | h (t) | ⩽ C ε^{\frac{2}{n}} {⟨ t ⟩}^{- 1} \end{matrix}$ which implies that $\begin{matrix} (4.25) & | g^{2} (t) - 1 | ⩽ C ε^{\frac{1}{n}} \int_{0}^{t} | g (τ) | {⟨ τ ⟩}^{- 1} d τ + C ε^{\frac{2}{n}} \int_{0}^{t} {⟨ τ ⟩}^{- 1} d τ . \end{matrix}$ Hence $\begin{matrix} g^{2} (t) ⩽ 1 + C ε^{\frac{1}{n}} (sup_{τ \in [0, t]} | g (τ) |) log ⟨ t ⟩ + C ε^{\frac{2}{n}} log ⟨ t ⟩ \end{matrix}$ namely, $\begin{matrix} (4.26) & {(sup_{τ \in [0, t]} | g (τ) |)}^{2} ⩽ 2 + C {(ε^{\frac{1}{n}} log ⟨ t ⟩)}^{2} \end{matrix}$ any $t ⩾ 0$ , from which it follows that $\begin{matrix} sup_{t \in [0, 4]} | g (t) | ⩽ 2 + C ε^{\frac{1}{n}} . \end{matrix}$ We apply the estimate to (4.25) to get $\begin{matrix} (4.27) & 2 + C ε^{\frac{1}{n}} ⩾ g^{2} (t) ⩾ 1 - C ε^{\frac{1}{n}} \end{matrix}$ for any $t \in [0, 4]$ . We assume that $t > 4$ . By a direct calculation we obtain $\begin{array}{l} f (ε G (t) θ) \\ = - γ | ε θ |^{\frac{1}{n}} {‖ G (t, 0) ‖}_{L^{1 + \frac{1}{n}} (R^{n - 1})}^{1 + \frac{1}{n}} \\ = - γ | ε θ |^{\frac{1}{n}} {(4 π)}^{- \frac{n}{2} (1 + \frac{1}{n})} \int_{R^{n - 1}} \frac{1}{t^{\frac{n}{2} (1 + \frac{1}{n})}} e^{- \frac{| x^{'} |^{2} (1 + \frac{1}{n})}{4 t}} d x^{'} \\ = - γ | ε θ |^{\frac{1}{n}} {(4 π)}^{- \frac{n}{2} (1 + \frac{1}{n})} t^{- 1} \int_{R^{n - 1}} e^{- \frac{| x^{'} |^{2} (1 + \frac{1}{n})}{4}} d x^{'} \\ = - γ | ε θ |^{\frac{1}{n}} {(4 π)}^{- \frac{n}{2} (1 + \frac{1}{n})} {(\frac{4 π}{(1 + \frac{1}{n})})}^{\frac{n - 1}{2}} t^{- 1} \\ = C | ε θ |^{\frac{1}{n}} t^{- 1} \end{array}$ and $\begin{array}{l} h (ε G (t) θ) \\ = - λ | ε θ |^{\frac{2}{n}} {‖ G (t) ‖}_{L^{1 + \frac{2}{n}} (R^{n})}^{1 + \frac{2}{n}} \\ = - \frac{λ}{2} | ε θ |^{\frac{2}{n}} {(4 π)}^{- \frac{n}{2} (1 + \frac{2}{n})} \int_{R^{n}} \frac{1}{t^{\frac{n}{2} (1 + \frac{2}{n})}} e^{- \frac{| x |^{2} (1 + \frac{2}{n})}{4 t}} d x \\ = - \frac{λ}{2} | ε θ |^{\frac{2}{n}} {(4 π)}^{- \frac{n}{2} (1 + \frac{2}{n})} t^{- 1} \int_{R^{n}} e^{- \frac{| x |^{2} (1 + \frac{2}{n})}{4}} d x \\ = - \frac{λ}{2} | ε θ |^{\frac{2}{n}} {(4 π)}^{- \frac{n}{2} (1 + \frac{2}{n})} {(\frac{4 π}{(1 + \frac{2}{n})})}^{\frac{n}{2}} t^{- 1} \\ = C | ε θ |^{\frac{2}{n}} t^{- 1} . \end{array}$ Hence we have $\begin{array}{l} g (t) f (w (t)) + h (w (t)) \\ = g (t) f (ε G (t) θ) + h (ε G (t) θ) \\ + g (t) (f (w (t)) - f (ε G (t) θ)) + (h (w (t)) - h (ε G (t) θ)) \\ = C ε^{\frac{1}{n}} t^{- 1} (g (t) + ε^{\frac{1}{n}}) \\ + g (t) (f (w (t)) - f (ε G (t) θ)) + (h (w (t)) - h (ε G (t) θ)) \end{array}$ which implies there exists an $ε > 0$ such that $\begin{array}{l} g (t) \frac{d}{d t} g (t) \\ ⩾ C | ε θ |^{\frac{1}{n}} t^{- 1} (g (t) + {(ε θ)}^{\frac{1}{n}}) \\ + g (t) (f (w (t)) - f (ε G (t) θ)) + (h (w (t)) - h (ε G (t) θ)) \\ ⩾ C | ε θ |^{\frac{1}{n}} t^{- 1} (g (t) + | ε θ |^{\frac{1}{n}}) - C | ε θ |^{\frac{2}{n}} t^{- 1} g (t) - C | ε θ |^{\frac{4}{n}} t^{- 1} \\ ⩾ C | ε θ |^{\frac{1}{n}} t^{- 1} (g (t) + | ε θ |^{\frac{1}{n}}) ⩾ C | ε θ |^{\frac{1}{n}} t^{- 1} g (t) . \end{array}$ Integrating in time, we obtain with (4.27) $\begin{matrix} (4.28) & g (t) ⩾ g (1) + C | ε θ |^{\frac{1}{n}} log t ⩾ \frac{1}{2} + C | ε θ |^{\frac{1}{n}} log t \end{matrix}$ Therefore we find that there exist an $ε > 0$ and $c > 0$ such that $\begin{matrix} g {(t)}^{- 1} = e^{- \frac{1}{n} φ (t)} ⩽ \frac{2}{1 + c | ε θ |^{\frac{1}{n}} log t} \end{matrix}$ for $t ⩾ 1$ . We also have by (4.27) $\begin{matrix} e^{- \frac{1}{n} φ (t)} ⩽ \frac{3}{2} \end{matrix}$ for $t ⩽ 4$ . Hence for any $t ⩾ 0$ , there exists $c_{2} > 0$ such that $\begin{matrix} e^{- \frac{1}{n} φ (t)} ⩽ \frac{2}{1 + c_{2} | ε θ |^{\frac{1}{n}} log ⟨ t ⟩} . \end{matrix}$ By (4.26), there exists $c_{1} > 0$ such that $\begin{matrix} e^{- \frac{1}{n} φ (t)} ⩾ \frac{1}{2 + c_{1} | ε θ |^{\frac{1}{n}} log ⟨ t ⟩} . \end{matrix}$ This completes the proof of the lemma. □

We define $v^{(0)} = ε S (t) u_{0}$ and $\begin{matrix} \frac{d}{d t} e^{\frac{1}{n} φ^{(0)} (t)} = - \frac{2 γ}{ε θ n} {‖ v^{(0)} (t, \cdot, 0) ‖}_{L^{1 + \frac{1}{n}} (R^{n - 1})}^{1 + \frac{1}{n}} - \frac{2 λ}{ε θ n} e^{- \frac{1}{n} φ^{(0)} (t)} {‖ v^{(0)} (t) ‖}_{L^{1 + \frac{2}{n}} (R_{+}^{n})}^{1 + \frac{2}{n}} \end{matrix}$ and the sequence $(v^{(k)}, φ^{(k)}), k ⩾ 1$ as follows: $\begin{matrix} v^{(k)} (t) = M v^{(k - 1)} = ε S (t) u_{0} + M_{1} v^{(k - 1)} + M_{2, 1} v^{(k - 1)} + M_{2, 2} v^{(k - 1)} \end{matrix}$ and $\begin{array}{l} \begin{matrix} \frac{d}{d t} e^{\frac{1}{n} φ^{(k)} (t)} & = - \frac{2 γ}{ε θ n} {‖ v^{(k)} (t, \cdot, 0) ‖}_{L^{1 + \frac{1}{n}} (R^{n - 1})}^{1 + \frac{1}{n}} \\ - \frac{2 λ}{ε θ n} e^{- \frac{1}{n} φ^{(k)}} {‖ v^{(k)} (t) ‖}_{L^{1 + \frac{2}{n}} (R_{+}^{n})}^{1 + \frac{2}{n}}, \end{matrix} \\ φ^{(k)} (0) = 0 . \end{array}$ We only treat the case of (1.2), for simplicity. From Lemma 4.2, we find that for any $v^{(k)} \in X_{b, \infty, C ε}$ , there exists a unique $φ^{(k)} \in C ([0, \infty))$ satisfying the estimate (4.24). Therefore $φ^{(k)}$ is defined by $v^{(k)}$ and so the mapping M is well defined. Since $e^{- \frac{1}{n} φ^{(k)} (t)} ⩽ C$ for $v^{(k)} \in X_{b, \infty, C ε}$ , by Lemma 2.3 and Lemma 4.1 we have the estimate $\begin{array}{rcl} {‖ M v^{(k)} ‖}_{X_{b, \infty}} & ⩽ & ε ({‖ {⟨ \cdot ⟩}^{b} u_{0} ‖}_{L^{1} (R^{n})} + {‖ {⟨ \cdot ⟩}^{b} u_{0} ‖}_{L^{1 + \frac{1}{n}} (R^{n})}) + C ε^{1 + \frac{1}{n}} + C ε^{1 + \frac{2}{n}} \\ ⩽ & 2 ε ({‖ {⟨ \cdot ⟩}^{b} u_{0} ‖}_{L^{1} (R^{n})} + {‖ {⟨ \cdot ⟩}^{b} u_{0} ‖}_{L^{1 + \frac{1}{n}} (R^{n})}) \end{array}$ for small $ε > 0$ . We consider the difference between $v^{(k + 1)}$ and $v^{(k)}$ $\begin{array}{rcl} v^{(k + 1)} - v^{(k)} & = & M v^{(k)} - M v^{(k - 1)} \\ = & M_{1} v^{(k)} - M_{1} v^{(k - 1)} + M_{2, 1} v^{(k)} - M_{2, 1} v^{(k - 1)} + M_{2, 2} v^{(k)} - M_{2, 2} v^{(k - 1)}, \end{array}$ where $\begin{array}{l} M_{1} v^{(k)} (t) = λ \int_{0}^{t} e^{- \frac{2}{n} φ^{(k)} (τ)} S_{N} (t - τ) (f^{(k)} (τ) - {‖ f^{(k)} (τ) ‖}_{L^{1} (R_{+}^{n})} \frac{2 v^{(k)} (τ)}{ε θ}) d τ, \\ M_{2, 1} v^{(k)} (t) \\ = \frac{2 γ}{ε θ} \int_{0}^{t} e^{- \frac{1}{n} φ^{(k)} (τ)} (ε θ G_{n} (t - τ) (\tilde{S} (t - τ) g^{(k)} (τ) - \tilde{G} (t - τ) {‖ g^{(k)} (τ) ‖}_{L^{1} (R^{n - 1})})) d τ, \\ M_{2, 2} v^{(k)} (t) = \frac{2 γ}{ε θ} \int_{0}^{t} e^{- \frac{1}{n} φ^{(k)} (τ)} {‖ g^{(k)} (τ) ‖}_{L^{1} (R^{n - 1})} (G (t - τ) ε θ - S (t - τ) v^{(k)} (τ)) d τ, \\ f^{(k)} (t) = {| v^{(k)} (t) |}^{1 + \frac{2}{n}}, g^{(k)} (t) = {| v^{(k)} (t) |}^{1 + \frac{1}{n}} |_{x_{n} = 0} . \end{array}$ Lemma 4.3.
Assume that $v^{(k)}, v^{(k - 1)} \in X_{b, \infty, C ε}$ . Then $\begin{matrix} {‖ M v^{(k)} - M v^{(k - 1)} ‖}_{Z_{δ, \infty}} ⩽ C ε^{\frac{1}{n}} {‖ v^{(k)} - v^{(k - 1)} ‖}_{Z_{δ, \infty}} . \end{matrix}$
Proof.
We only consider the case of (1.2), for simplicity. The case of (1.3) is treated in the same way. We have $\begin{array}{l} M_{1} v^{(k)} - M_{1} v^{(k - 1)} \\ = λ \int_{0}^{t} e^{- \frac{2}{n} φ^{(k)} (τ)} S_{N} (t - τ) (f^{(k)} (τ) - f^{(k - 1)} (τ)) d τ \\ + λ \int_{0}^{t} (e^{- \frac{2}{n} φ^{(k)} (τ)} - e^{- \frac{2}{n} φ^{(k - 1)} (τ)}) S_{N} (t - τ) f^{(k - 1)} (τ) d τ \\ - \frac{2 λ}{ε θ} \int_{0}^{t} e^{- \frac{2}{n} φ^{(k)} (τ)} S (t - τ) ({‖ f^{(k)} (τ) ‖}_{L^{1} (R_{+}^{n})} v^{(k)} (τ) - {‖ f^{(k - 1)} (τ) ‖}_{L^{1} (R_{+}^{n})} v^{(k - 1)} (τ)) d τ \\ - \frac{2 λ}{ε θ} \int_{0}^{t} (e^{- \frac{2}{n} φ^{(k)} (τ)} - e^{- \frac{2}{n} φ^{(k - 1)} (τ)}) S_{N} (t - τ) {‖ f^{(k - 1)} (τ) ‖}_{L^{1} (R_{+}^{n})} v^{(k - 1)} (τ) d τ \end{array}$ from which with Lemma 2.1 $\begin{array}{l} {‖ M_{1} v^{(k)} (t) - M_{1} v^{(k - 1)} (t) ‖}_{L^{r} (R_{+}^{n})} \\ ⩽ C \int_{0}^{t} {(t - τ)}^{- \frac{n}{2} (1 - \frac{1}{r})} {‖ f^{(k)} (τ) - f^{(k - 1)} (τ) ‖}_{L^{1} (R_{+}^{n})} d τ \\ + C \int_{0}^{t} | e^{- \frac{2}{n} φ^{(k)} (τ)} - e^{- \frac{2}{n} φ^{(k - 1)} (τ)} | {(t - τ)}^{- \frac{n}{2} (1 - \frac{1}{r})} {‖ f^{(k - 1)} (τ) ‖}_{L^{1} (R_{+}^{n})} d τ \\ + C ε^{- 1} \int_{0}^{t} {(t - τ)}^{- \frac{n}{2} (1 - \frac{1}{r})} {‖ f^{(k)} (τ) - f^{(k - 1)} (τ) ‖}_{L^{1} (R_{+}^{n})} {‖ v^{(k)} (τ) ‖}_{L^{1} (R_{+}^{n})} d τ \\ + C ε^{- 1} \int_{0}^{t} {(t - τ)}^{- \frac{n}{2} (1 - \frac{1}{r})} {‖ f^{(k - 1)} (τ) ‖}_{L^{1} (R_{+}^{n})} {‖ v^{(k)} (τ) - v^{(k - 1)} (τ) ‖}_{L^{1} (R_{+}^{n})} d τ \\ + C ε^{- 1} \int_{0}^{t} | e^{- \frac{2}{n} φ^{(k)} (τ)} - e^{- \frac{2}{n} φ^{(k - 1)} (τ)} | \\ \times {(t - τ)}^{- \frac{n}{2} (1 - \frac{1}{r})} {‖ f^{(k - 1)} (τ) ‖}_{L^{1} (R_{+}^{n})} {‖ v^{(k - 1)} (τ) ‖}_{L^{1} (R_{+}^{n})} d τ . \end{array}$ Since $\begin{array}{rcl} M_{2} v^{(k)} (t) & = & M_{2, 1} v^{(k)} (t) + M_{2, 2} v^{(k)} (t) \\ = & \frac{2 γ}{ε θ} \int_{0}^{t} e^{- \frac{1}{n} φ^{(k)} (τ)} 2 ε θ G_{n} (t - τ) \tilde{S} (t - τ) g^{(k)} (τ) d τ \\ - \frac{2 γ}{ε θ} \int_{0}^{t} e^{- \frac{1}{n} φ^{(k)} (τ)} \int_{R^{n - 1}} g^{(k)} (τ) d x^{'} S (t - τ) v^{(k)} (τ) d τ \end{array}$ we have $\begin{array}{l} {‖ M_{2} v^{(k)} (t) - M_{2} v^{(k - 1)} (t) ‖}_{L^{r} (R_{+}^{n})} \\ ⩽ C \int_{0}^{t} {(t - τ)}^{- \frac{n}{2} (1 - \frac{1}{r})} {‖ g^{(k)} (τ, \cdot, 0) - g^{(k - 1)} (τ, \cdot, 0) ‖}_{L^{1} (R^{n - 1})} d τ \\ + C \int_{0}^{t} | e^{- \frac{1}{n} φ^{(k)} (τ)} - e^{- \frac{1}{n} φ^{(k - 1)} (τ)} | {(t - τ)}^{- \frac{n}{2} (1 - \frac{1}{r})} {‖ g^{(k - 1)} (τ, \cdot, 0) ‖}_{L^{1} (R^{n - 1})} d τ \\ + C ε^{- 1} \int_{0}^{t} {(t - τ)}^{- \frac{n}{2} (1 - \frac{1}{r})} {‖ g^{(k)} (τ, \cdot, 0) - g^{(k - 1)} (τ, \cdot, 0) ‖}_{L^{1} (R^{n - 1})} {‖ v^{(k)} (τ) ‖}_{L^{1} (R_{+}^{n})} d τ \\ + C ε^{- 1} \int_{0}^{t} {(t - τ)}^{- \frac{n}{2} (1 - \frac{1}{r})} {‖ g^{(k - 1)} (τ, \cdot, 0) ‖}_{L^{1} (R^{n - 1})} {‖ v^{(k)} (τ) - v^{(k - 1)} (τ) ‖}_{L^{1} (R_{+}^{n})} d τ \\ + C ε^{- 1} \int_{0}^{t} | e^{- \frac{1}{n} φ^{(k)} (τ)} - e^{- \frac{1}{n} φ^{(k - 1)} (τ)} | {(t - τ)}^{- \frac{n}{2} (1 - \frac{1}{r})} \\ \times {‖ g^{(k - 1)} (τ, \cdot, 0) ‖}_{L^{1} (R^{n - 1})} {‖ v^{(k - 1)} (τ) ‖}_{L^{1} (R_{+}^{n})} d τ . \end{array}$ By the mean value theorem we find that there exists $0 < ξ^{'} < 1$ such that $\begin{array}{rcl} | e^{- \frac{1}{n} φ^{(k)} (t)} - e^{- \frac{1}{n} φ^{(k - 1)} (t)} | & = & | (1 - e^{\frac{1}{n} (φ^{(k)} (t) - φ^{(k - 1)} (t))}) e^{- \frac{1}{n} φ^{(k)} (t)} | \\ = & | \frac{1}{n} (φ^{(k)} (t) - φ^{(k - 1)} (t)) e^{- \frac{1}{n} (1 - ξ^{'}) φ^{(k)} (t) - \frac{1}{n} ξ^{'} φ^{(k - 1)} (t)} | . \end{array}$ Therefore by Lemma 4.2 $\begin{matrix} | e^{- \frac{1}{n} φ^{(k)} (t)} - e^{- \frac{1}{n} φ^{(k - 1)} (t)} | ⩽ C | φ^{(k)} (t) - φ^{(k - 1)} (t) |, \end{matrix}$ which implies $\begin{array}{l} {‖ M_{1} v^{(k)} (t) - M_{1} v^{(k - 1)} (t) ‖}_{L^{r} (R_{+}^{n})} \\ ⩽ C \int_{0}^{t} {(t - τ)}^{- \frac{n}{2} (1 - \frac{1}{r})} {‖ v^{(k)} (τ) - v^{(k - 1)} (τ) ‖}_{L^{1 + \frac{2}{n}} (R_{+}^{n})} \\ \times ({‖ v^{(k)} (τ) ‖}_{L^{1 + \frac{2}{n}} (R_{+}^{n})}^{\frac{2}{n}} + {‖ v^{(k - 1)} (τ) ‖}_{L^{1 + \frac{2}{n}} (R_{+}^{n})}^{\frac{2}{n}}) d τ \\ + C \int_{0}^{t} | φ^{(k)} (τ) - φ^{(k - 1)} (τ) | {(t - τ)}^{- \frac{n}{2} (1 - \frac{1}{r})} {‖ v^{(k - 1)} (τ) ‖}_{L^{1 + \frac{2}{n}} (R_{+}^{n})}^{1 + \frac{2}{n}} d τ \\ + C ε^{- 1} \int_{0}^{t} {(t - τ)}^{- \frac{n}{2} (1 - \frac{1}{r})} {‖ v^{(k - 1)} (τ) ‖}_{L^{1 + \frac{2}{n}} (R_{+}^{n})}^{1 + \frac{2}{n}} {‖ v^{(k)} (τ) - v^{(k - 1)} (τ) ‖}_{L^{1} (R_{+}^{n})} d τ \\ + C ε^{- 1} \int_{0}^{t} | φ^{(k)} (τ) - φ^{(k - 1)} (τ) | {(t - τ)}^{- \frac{n}{2} (1 - \frac{1}{r})} {‖ v^{(k - 1)} (τ) ‖}_{L^{1 + \frac{2}{n}} (R_{+}^{n})}^{1 + \frac{2}{n}} {‖ v^{(k - 1)} (τ) ‖}_{L^{1} (R_{+}^{n})} d τ \\ ⩽ C ε^{\frac{2}{n}} {‖ v^{(k)} - v^{(k - 1)} ‖}_{Z_{δ, \infty}} \int_{0}^{t} {(t - τ)}^{- \frac{n}{2} (1 - \frac{1}{r})} {⟨ τ ⟩}^{- 1 + δ} d τ \\ + C ε^{1 + \frac{2}{n}} (sup_{t \in [0, \infty)} {⟨ t ⟩}^{- δ} | φ^{(k)} (t) - φ^{(k - 1)} (t) |) \int_{0}^{t} {(t - τ)}^{- \frac{n}{2} (1 - \frac{1}{r})} {⟨ τ ⟩}^{- 1 + δ} d τ \\ ⩽ C ε^{\frac{2}{n}} {⟨ t ⟩}^{- \frac{n}{2} (1 - \frac{1}{r}) + δ} {‖ v^{(k)} - v^{(k - 1)} ‖}_{Z_{δ, \infty}} \\ + C ε^{1 + \frac{2}{n}} {⟨ t ⟩}^{- \frac{n}{2} (1 - \frac{1}{r}) + δ} (sup_{t \in [0, \infty)} {⟨ t ⟩}^{- δ} | φ^{(k)} (t) - φ^{(k - 1)} (t) |) . \end{array}$ Also $\begin{array}{l} {‖ M_{2} v^{(k)} (t) - M_{2} v^{(k - 1)} (t) ‖}_{L^{r} (R_{+}^{n})} \\ ⩽ C \int_{0}^{t} {(t - τ)}^{- \frac{n}{2} (1 - \frac{1}{r})} {‖ v^{(k)} (τ, \cdot, 0) - v^{(k - 1)} (τ, \cdot, 0) ‖}_{L^{1 + \frac{1}{n}} (R^{n - 1})} \\ \times ({‖ v^{(k)} (τ, \cdot, 0) ‖}_{L^{1 + \frac{1}{n}} (R^{n - 1})}^{\frac{1}{n}} + {‖ v^{(k - 1)} (τ, \cdot, 0) ‖}_{L^{1 + \frac{1}{n}} (R^{n - 1})}^{\frac{1}{n}}) d τ \\ + C \int_{0}^{t} ‖ φ^{(k)} (τ) - φ^{(k - 1)} (τ) ‖ {(t - τ)}^{- \frac{n}{2} (1 - \frac{1}{r})} {‖ v^{(k - 1)} (τ, \cdot, 0) ‖}_{L^{1 + \frac{1}{n}} (R^{n - 1})}^{1 + \frac{1}{n}} d τ \\ + C ε^{- 1} {‖ v^{(k)} - v^{(k - 1)} ‖}_{Z_{δ, \infty}} \int_{0}^{t} {(t - τ)}^{- \frac{n}{2} (1 - \frac{1}{r})} {⟨ τ ⟩}^{δ} {‖ v^{(k - 1)} (τ, \cdot, 0) ‖}_{L^{1 + \frac{1}{n}} (R^{n - 1})}^{1 + \frac{1}{n}} d τ . \end{array}$

Since for any $s ⩾ 1$ $\begin{array}{rcl} {‖ v (t, \cdot, 0) ‖}_{L^{s} (R^{n - 1})}^{s} & = & - \int_{0}^{\infty} \partial_{x_{n}} \int_{R^{n - 1}} {| v (t, x^{'}, x_{n}) |}^{s} d x^{'} d x_{n} \\ ⩽ & C {‖ \partial_{x_{n}} v (t) ‖}_{L^{s} (R_{+}^{n})} {‖ v (t) ‖}_{L^{s} (R_{+}^{n})}^{s - 1} \end{array}$ we have $\begin{array}{rcl} {‖ v (t, \cdot, 0) ‖}_{L^{1 + \frac{1}{n}} (R^{n - 1})} & ⩽ & C {‖ \partial_{x_{n}} v (t) ‖}_{L^{1 + \frac{1}{n}} (R_{+}^{n})}^{\frac{n}{n + 1}} {‖ v (t) ‖}_{L^{1 + \frac{1}{n}} (R_{+}^{n})}^{\frac{1}{n + 1}} \\ ⩽ & C ε {⟨ t ⟩}^{- \frac{n}{2 n + 2}} t^{- \frac{n}{2 n + 2}} \end{array}$ for $v \in X_{b, \infty, C ε}$ . We also have $\begin{array}{rcl} {‖ v (t, \cdot, 0) ‖}_{L^{1 + \frac{1}{n}} (R^{n - 1})} & ⩽ & C {‖ \partial_{x_{n}} v (t) ‖}_{L^{1 + \frac{1}{n}} (R_{+}^{n})}^{\frac{n}{n + 1}} {‖ v (t) ‖}_{L^{1 + \frac{1}{n}} (R_{+}^{n})}^{\frac{1}{n + 1}} \\ ⩽ & C {⟨ t ⟩}^{δ - \frac{n}{2 n + 2}} t^{- \frac{n}{2 n + 2}} ‖ v ‖_{Z_{δ, \infty}} . \end{array}$ Therefore $\begin{array}{l} {‖ M_{2} v^{(k)} (t) - M_{2} v^{(k - 1)} (t) ‖}_{L^{r} (R_{+}^{n})} \\ ⩽ C ε^{\frac{1}{n}} {‖ v^{(k)} - v^{(k - 1)} ‖}_{Z_{δ, \infty}} \int_{0}^{t} {(t - τ)}^{- \frac{n}{2} (1 - \frac{1}{r})} {⟨ τ ⟩}^{δ - \frac{1}{2}} τ^{- \frac{1}{2}} d τ \\ + C ε^{1 + \frac{1}{n}} (sup_{t \in [0, \infty)} {⟨ t ⟩}^{- δ} | φ^{(k)} (t) - φ^{(k - 1)} (t) |) \int_{0}^{t} {(t - τ)}^{- \frac{n}{2} (1 - \frac{1}{r})} {⟨ τ ⟩}^{δ - \frac{1}{2}} τ^{- \frac{1}{2}} d τ \\ ⩽ C ε^{\frac{1}{n}} {⟨ t ⟩}^{- \frac{n}{2} (1 - \frac{1}{r}) + δ} {‖ v^{(k)} - v^{(k - 1)} ‖}_{Z_{δ, \infty}} \\ + C ε^{1 + \frac{1}{n}} {⟨ t ⟩}^{- \frac{n}{2} (1 - \frac{1}{r}) + δ} (sup_{t \in [0, \infty)} {⟨ t ⟩}^{- δ} | φ^{(k)} (t) - φ^{(k - 1)} (t) |) . \end{array}$ We consider the estimate of $| φ^{(k)} (t) - φ^{(k - 1)} (t) |$ . We have $\begin{array}{rcl} \frac{d}{d t} φ^{(k)} (t) & = & - \frac{2 λ}{ε θ} e^{- \frac{2}{n} φ^{(k)} (t)} \int_{R^{n}} g^{(k)} (t, x) d x \\ - \frac{2 γ}{ε θ} e^{- \frac{1}{n} φ^{(k)} (t)} \int_{R^{n - 1}} f^{(k)} (t, x^{'}, 0) d x^{'} \end{array}$ from which it follows that $\begin{array}{l} | φ^{(k)} (t) - φ^{(k - 1)} (t) | \\ ⩽ C ε^{- 1} \int_{0}^{t} {‖ f^{(k)} (τ) - f^{(k - 1)} (τ) ‖}_{L^{1} (R_{+}^{n})} d τ \\ + C ε^{- 1} \int_{0}^{t} | φ^{(k)} (τ) - φ^{(k - 1)} (τ) | {‖ f^{(k - 1)} (τ) ‖}_{L^{1} (R_{+}^{n})} d τ \\ + C ε^{- 1} \int_{R^{n}} {‖ g^{(k)} (τ, \cdot, 0) - g^{(k - 1)} (τ, \cdot, 0) ‖}_{L^{1} (R^{n - 1})} d x \\ + C ε^{- 1} \int_{0}^{t} | φ^{(k)} (τ) - φ^{(k - 1)} (τ) | {‖ g^{(k - 1)} (τ, \cdot, 0) ‖}_{L^{1} (R^{n - 1})} d τ \\ ⩽ C ε^{- 1} \int_{0}^{t} ({‖ v^{(k)} (τ) ‖}_{L^{1 + \frac{2}{n}}}^{\frac{2}{n}} + {‖ v^{(k - 1)} (τ) ‖}_{L^{1 + \frac{2}{n}} (R_{+}^{n})}^{\frac{2}{n}}) {‖ v^{(k)} (τ) - v^{(k - 1)} (τ) ‖}_{L^{1 + \frac{2}{n}} (R_{+}^{n})} d τ \\ + C ε^{- 1} \int_{0}^{t} | φ^{(k)} (τ) - φ^{(k - 1)} (τ) | {‖ v^{(k - 1)} (τ) ‖}_{L^{1 + \frac{2}{n}} (R_{+}^{n})}^{1 + \frac{2}{n}} d τ \\ + C ε^{- 1} \int_{R^{n}} ({‖ v^{(k)} (τ, \cdot, 0) ‖}_{L^{1 + \frac{1}{n}} (R^{n - 1})}^{\frac{1}{n}} + {‖ v^{(k - 1)} (τ, \cdot, 0) ‖}_{L^{1 + \frac{1}{n}} (R^{n - 1})}^{\frac{1}{n}}) \\ \times {‖ v^{(k)} (τ, \cdot, 0) - v^{(k - 1)} (τ, \cdot, 0) ‖}_{L^{1 + \frac{1}{n}} (R^{n - 1})} d τ \\ + C ε^{- 1} \int_{0}^{t} | φ^{(k)} (τ) - φ^{(k - 1)} (τ) | {‖ v^{(k - 1)} (τ, \cdot, 0) ‖}_{L^{1 + \frac{1}{n}} (R^{n - 1})}^{1 + \frac{1}{n}} d τ . \end{array}$ Hence $\begin{array}{l} | φ^{(k)} (t) - φ^{(k - 1)} (t) | \\ ⩽ C ε^{- 1 + \frac{2}{n}} {‖ v^{(k)} - v^{(k - 1)} ‖}_{Z_{δ, \infty}} \int_{0}^{t} {⟨ τ ⟩}^{δ - \frac{n + 2}{2 n + 2}} τ^{- \frac{n}{2 n + 2}} d τ \\ + C ε^{\frac{2}{n}} (sup {⟨ t ⟩}^{- δ} | φ^{(k)} (t) - φ^{(k - 1)} (t) |) \int_{0}^{t} {⟨ τ ⟩}^{δ - \frac{n + 2}{2 n + 2}} τ^{- \frac{n}{2 n + 2}} d τ \\ + C ε^{- 1 + \frac{1}{n}} {‖ v^{(k)} - v^{(k - 1)} ‖}_{Z_{δ, \infty}} \int_{0}^{t} {⟨ τ ⟩}^{δ - \frac{1}{2}} τ^{- \frac{1}{2}} d τ \\ + C ε^{\frac{1}{n}} (sup {⟨ t ⟩}^{- δ} | φ^{(k)} (t) - φ^{(k - 1)} (t) |) \int_{0}^{t} {⟨ τ ⟩}^{δ - \frac{1}{2} δ} τ^{- \frac{1}{2}} d τ \end{array}$ from which it follows that $\begin{matrix} sup_{t} {⟨ t ⟩}^{- δ} | φ^{(k)} (t) - φ^{(k - 1)} (t) | ⩽ C ε^{- 1 + \frac{1}{n}} {‖ v^{(k)} - v^{(k - 1)} ‖}_{Z_{δ, \infty}} . \end{matrix}$ Therefore $\begin{array}{l} {‖ M v^{(k)} (t) - M v^{(k - 1)} (t) ‖}_{L^{r} (R_{+}^{n})} \\ ⩽ {‖ M_{1} v^{(k)} (t) - M_{1} v^{(k - 1)} (t) ‖}_{L^{r} (R_{+}^{n})} + {‖ M_{2} v^{(k)} (t) - M_{2} v^{(k - 1)} (t) ‖}_{L^{r} (R_{+}^{n})} \\ ⩽ C ε^{\frac{1}{n}} {⟨ t ⟩}^{δ - \frac{n}{2} (1 - \frac{1}{r})} {‖ v^{(k)} - v^{(k - 1)} ‖}_{Z_{δ, \infty}} . \end{array}$ In the same way, $\begin{matrix} {‖ \nabla (M v^{(k)} (t) - M v^{(k - 1)} (t)) ‖}_{L^{r} (R_{+}^{n})} ⩽ C ε^{\frac{1}{n}} t^{- \frac{1}{2}} {⟨ t ⟩}^{δ - \frac{n}{2} (1 - \frac{1}{r})} {‖ v^{(k)} - v^{(k - 1)} ‖}_{Z_{δ, \infty}} . \end{matrix}$ Thus we have the result of the lemma. □

5. Proof of Theorem 1.2

We now consider the linearized version of problem (4.2) which is written as $\begin{matrix} (5.1) & v (t) = M w = ε S (t) u_{0} + M_{1} w + M_{2, 1} w + M_{2, 2} w \end{matrix}$ with the condition such that $\begin{matrix} \int_{R_{+}^{n}} w (t) d x = \frac{ε θ}{2} . \end{matrix}$ We assume that $w \in X_{b, \infty, C ε}$ , then by Lemma 4.1, we have $\begin{matrix} ‖ M w ‖_{X_{b, \infty}} ⩽ 10 ε ‖ u_{0} ‖_{X_{b, 0}} + C ε^{1 + \frac{1}{n}} \end{matrix}$ and for $w_{1}, w_{2} \in X_{b, \infty, C ε}$ we find by Lemma 4.3 $\begin{matrix} ‖ M w_{1} - M w_{2} ‖_{Z_{δ, \infty}} ⩽ C ε^{\frac{1}{n}} ‖ w_{1} - w_{2} ‖_{Z_{δ, \infty}} . \end{matrix}$ Therefore we have the result by the contraction mapping principle.

Footnotes

Acknowledgements

We would like to thank anonymous referee for useful comments and letting us know some references which are related to nonlinear Neumann boundary conditions. The work of N.H. is partially supported by JSPS KAKENHI Grant Numbers JP20K03680, JP19H05597. The work of E.I.K. is partially supported by CONACYT 252053-F. The work of P.I.N. is partially supported by CONACYT project 283698-F and PAPIIT project IN103221. The work of T.O. is partially supported by JSPS KAKENHI Grant Number JP19H05597.

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