1 Introduction and Statement of Results

In the present chapter, we prove the global-in-time existence of the solutions of the Cauchy problem for the semilinear Klein–Gordon equation in the FLRW (Friedmann–Lema\(\hat {\mathrm {i}}\)tre–Robertson–Walker) space–time of the contracting universe for the self-interacting scalar field.

The metric g in the FLRW space–time of the contracting universe in the Lama\(\hat {\mathrm{i}} \)tre–Robertson coordinates (see, e.g., [10]) is defined as follows, g 00 = g 00 = −1, g 0j = g 0j = 0, g ij(x, t) = e −2t σ ij(x), i, j = 1, 2, …, n, where \(\sum _{j=1}^n\sigma ^{ij} (x) \sigma _{jk} (x)=\delta _{ik} \), and δ ij is Kronecker’s delta. The metric σ ij(x) describes the time slices. The covariant Klein–Gordon equation in that space–time in the coordinates is

$$\displaystyle \begin{aligned} \psi _{tt} - \frac{e^{2t}}{\sqrt{|\det \sigma ( x)| }} \sum_{i,j=1}^n \frac{\partial }{\partial x^i}\left( \sqrt{|\det \sigma ( x)| } \sigma ^{ij} (x)\frac{\partial }{\partial x^j} \psi \right) - n \psi_t + m^2 \psi = F(\psi ) \,. \end{aligned} $$
(1.1)

It is obvious that the properties of this equation and of its solutions are not time invertible. In the present chapter, we are interested in the Cauchy problem, which, in fact, is not equivalent to the time backward problem for the equation with the reflected time t →−t:

$$\displaystyle \begin{aligned} \psi _{tt} - \frac{e^{-2t}}{\sqrt{|\det \sigma ( x)| }} \sum_{i,j=1}^n \frac{\partial }{\partial x^i}\left( \sqrt{|\det \sigma ( x)| } \sigma ^{ij} (x)\frac{\partial }{\partial x^j} \psi \right) + n \psi_t + m^2 \psi = F(\psi ) \,. \end{aligned} $$
(1.2)

The last equation is the semilinear Klein–Gordon equation in the de Sitter space–time. Equation (1.2) is well investigated, and the conditions for the existence of small data global-in-time solutions for some important σ are discovered [1, 2, 5,6,7,8, 11, 18, 19].

Equation (1.1) is a special case of the equation:

$$\displaystyle \begin{aligned} \begin{array}{rcl} {} &\displaystyle &\displaystyle \psi _{tt} - e^{2 t} A(x,\partial_x) \psi - n \psi _t + m^2 \psi = F(\psi )\,, \end{array} \end{aligned} $$
(1.3)

where \( A(x,\partial _x) = \sum _{|\alpha |\leq 2} a_{\alpha }(x)\partial _x^\alpha \) is a second-order elliptic partial differential operator. We also assume that the mass m can be a complex number, \(m^2 \in {\mathbb C} \).

In the present chapter, we study also the class of equations containing, in particular, the Higgs boson equation with the Higgs potential, that is the equation:

$$\displaystyle \begin{aligned} \psi _{tt} - e^{2t} A(x,D)\psi - n\psi_t = \mu^2 \psi - \lambda \psi ^3 , \end{aligned} $$
(1.4)

with λ > 0 and μ > 0, while n = 3.

To formulate the main theorem of this chapter, we need a characterization of the nonlinear term F. We want to stress here that the explicit form of F is not used merely. There are estimates of the form \(\|F(\psi )\|{ }_{X} < C \| \psi \|{ }_{X'}^{\alpha }\| \psi \|{ }_{X'{}'}\), for some function spaces X, X′, and X′′. Furthermore, since we prove that for small data the solution is bounded in \(L^{p'}\)-norm, we are only concerned with the behavior of F at the origin. Let \(B^{s,q}_{p}\) denote the Besov space.

Condition (\(\mathcal L\))

The smooth in x function F = F(x, ψ) is said to be Lipschitz continuous with exponent α ≥ 0 in the space \(B^{s,q}_{p}\) if there is a constant C ≥ 0 such that

$$\displaystyle \begin{aligned} \| F(x,\psi_1 (x))- F(x,\psi_2(x) ) \|{}_{B^{s,q}_{p}} \leq C \| \psi_1 - \psi_2 \|{}_{B^{s,q}_{p'}} \Big( \| \psi_1 \|{}^{\alpha} _{B^{s,q}_{p'}} + \| \psi_2 \|{}^{\alpha} _{B^{s,q}_{p'}} \Big) \end{aligned}$$

for all \(\psi _1,\psi _2 \in B^{s,q}_{p'}\) , where 1∕p + 1∕p′ = 1.

The polynomial in ψ functions F(x, ψ) = ±|ψ|α+1 and F(ψ) = ±|ψ|α ψ are important examples of the Lipschitz continuous with exponent α > 0 in the Lebesgue spaces \(L^p({\mathbb R}^n)\) and the Sobolev space \(H_{(s)}({\mathbb R}^n) \), s > n∕2, functions.

Define also the metric space:

$$\displaystyle \begin{aligned} X({R,B^{s,q}_{p},\gamma}) := \left\{ \psi \in C([0,\infty) ; B^{s,q}_{p} \; \Big| \; \parallel \psi \parallel _X := \sup_{t \in [0,\infty) } e^{\gamma t} \parallel \psi (x ,t) \parallel _{B^{s,q}_{p}} \le R \right\}\,, \end{aligned}$$

where \(\gamma \in {\mathbb R}\), with the metric \( d(\psi _1,\psi _2) := \sup _{t \in [0,\infty ) } e^{\gamma t} \parallel \psi _1 (x , t) - \psi _2 (x ,t) \parallel _{B^{s,q}_{p}}\).

We study the Cauchy problem through the integral equation. To determine that integral equation, we appeal to the operator:

$$\displaystyle \begin{aligned} G:={\mathcal K}\circ {\mathcal E}{\mathcal E} \end{aligned}$$

(\({\mathcal E}{\mathcal E}\) stands for the evolution equation) that is designed as follows. For the function f(x, t), we define

$$\displaystyle \begin{aligned} v(x,t;b):= {\mathcal E}{\mathcal E} [f](x,t;b)\,, \end{aligned}$$

where the function v(x, t;b) is a solution to the Cauchy problem:

$$\displaystyle \begin{aligned} \begin{array}{rcl} {} &\displaystyle &\displaystyle \partial_t^2 v - A(x,D)v =0, \quad x \in {\mathbb R}^n, \quad t \geq 0, \end{array} \end{aligned} $$
(1.5)
$$\displaystyle \begin{aligned} \begin{array}{rcl} {} &\displaystyle &\displaystyle v(x,0;b)=f(x,b)\,, \quad v_t(x,0;b)= 0\,, \quad x \in {\mathbb R}^n\,, \end{array} \end{aligned} $$
(1.6)

while the integral transform \({\mathcal K}\) is introduced by:

$$\displaystyle \begin{aligned} \begin{array}{rcl} {\mathcal K}[v] (x,t) &\displaystyle := &\displaystyle 2 e^{\frac{n}{2}t}\int_{ 0}^{t} db \int_{ 0}^{ e^{t}- e^{b}} dr \, e^{-\frac{n}{2}b} v(x,r ;b) E(r,t; 0,b;M) \,. \end{array} \end{aligned} $$

Here, the principal square root \(M:=(n^2/4-m^2)^{\frac {1}{2}}\) is the main parameter that controls estimates and solvability of the integral equation. The last will be obtained by means of the integral transform. The kernel E(r, t;0, b;M) was introduced in [16, 20] (see also (2.2)). Hence,

$$\displaystyle \begin{aligned} G[f] (x,t) = 2 e^{\frac{n}{2}t}\int_{ 0}^{t} db \int_{ 0}^{ e^{t}- e^{b}} dr \, e^{-\frac{n}{2}b}\,{\mathcal E}{\mathcal E} [f](x,r ;b) E(r,t; 0,b;M) \,. \end{aligned}$$

Obviously, the Cauchy problem for Eq. (1.3) leads to the following integral equation:

$$\displaystyle \begin{aligned} \varPhi (x,t) = \varPhi _0(x,t) + G[ e^{-\varGamma \cdot }F(\cdot ,\varPhi ) ] (x,t) \end{aligned} $$
(1.7)

where Γ = 0. Φ 0 is generated by the initial value problem (1.8), (1.9) with F ≡ 0 .

We define the solution of the Cauchy problem through the last integral equation. For the real numbers γ and Γ, we define

$$\displaystyle \begin{aligned} \begin{array}{rcl} I(t):= e^{ t ( \frac{n}{2}+ {\Re M} + \gamma )} \int_{ 0}^{t } e^{-( \frac{n}{2}+{\Re M} + \gamma (\alpha +1)+\varGamma )b}\,db\,. \end{array} \end{aligned} $$

The main result of this chapter is the following theorem. The next theorem states the existence of global-in-time solution for small initial data in Sobolev spaces.

Theorem 1

Assume that A(x, ∂ x) is the Laplace operator on \({\mathbb R}^n\) , and the nonlinear term F(Φ) is Lipschitz continuous in the space \(H_{(s)}({\mathbb R}^n) \) , s > n∕2 ≥ 1, F(x, 0) ≡ 0, and α > 0.

(GS) Assume also that \( \Re M >0\) and that \(M=\Re M\) if \(\Re M=1/2\) , one of the following three conditions is fulfilled:

$$\displaystyle \begin{aligned} \begin{array}{rcl} &\displaystyle (i) &\displaystyle \frac{n}{2}+\Re M + \gamma (\alpha +1)+\varGamma >0 ,\quad \frac{n}{2}+\max\{\frac{1}{2},\Re M \}+ \gamma \leq 0 ,\\ &\displaystyle (ii) &\displaystyle \frac{n}{2}+\Re M+ \gamma (\alpha +1)+\varGamma = 0 ,\quad \frac{n}{2}+\max\{\frac{1}{2},\Re M \}+ \gamma < 0 ,\\ &\displaystyle (iii) &\displaystyle \frac{n}{2}+\Re M + \gamma (\alpha +1)+\varGamma < 0,\quad \frac{n}{2}+\max\{\frac{1}{2},\Re M \}+\gamma \leq 0 ,\quad \gamma \alpha +\varGamma \geq 0 . \end{array} \end{aligned} $$

Then, there exists ε 0 > 0 such that, for every given functions \(\varphi _0 ,\varphi _1 \in H_{(s)}({\mathbb R}^n) \) , satisfying estimate:

$$\displaystyle \begin{aligned} \| \varphi_0 \|{}_{H_{(s)} ({\mathbb R}^n)} + \|\varphi_1 \|{}_{ H_{(s)} ({\mathbb R}^n)} \leq \varepsilon, \qquad \varepsilon < \varepsilon_0\,, \end{aligned}$$

there exists a solution \(\varPhi \in C ([0,\infty );H_{(s)}({\mathbb R}^n))\) of the Cauchy problem:

$$\displaystyle \begin{aligned} \begin{array}{rcl} {} &\displaystyle &\displaystyle \varPhi _{tt} - n \varPhi _t - e^{2 t} A(x,\partial_x) \varPhi + m^2 \varPhi = e^{-\varGamma t} F(x,\varPhi )\,, \end{array} \end{aligned} $$
(1.8)
$$\displaystyle \begin{aligned} \begin{array}{rcl} {} &\displaystyle &\displaystyle \varPhi (x,0)= \varphi_0 (x)\, , \quad \varPhi _t(x,0)=\varphi_1 (x) \,. \end{array} \end{aligned} $$
(1.9)

The solution Φ(x, t) belongs to the space \( X({2\varepsilon ,H_{(s)} ({\mathbb R}^n), \gamma }) \) , that is:

$$\displaystyle \begin{aligned} \begin{array}{rcl} \sup_{t \in [0,\infty)} e^{ \gamma t} \|\varPhi (\cdot ,t) \|{}_{H_{(s)} ({\mathbb R}^n)} \leq 2\varepsilon\, . \end{array} \end{aligned} $$

(LS) If \(\Re M > 0\) , \( \frac {n}{2}+\max \{\frac {1}{2},\Re M \}+ \gamma \leq 0\) , and neither of the three conditions (i)–(iii) is fulfilled, then the lifespan T ls of the solution can be estimated from below as follows:

$$\displaystyle \begin{aligned} T_{ls} \geq \mathcal{I} \left(C_0(M,n,\alpha ,\gamma, \varGamma ) ^{-1} \left( \| \varphi_0 \|{}_{ H_{(s)} ({\mathbb R}^n)} + \|\varphi_1 \|{}_{H_{(s)} ({\mathbb R}^n)} \right) ^{-\alpha }\right) \,. \end{aligned}$$

with some constant C 0(M, n, α, γ, Γ) when \(\| \varphi _0 \|{ }_{ H_{(s)} ({\mathbb R}^n)} + \|\varphi _1 \|{ }_{ H_{(s)} ({\mathbb R}^n)} \) is sufficiently small. Here, \(\mathcal {I}\) is the function inverse to I = I(t).

The theorem covers the equations with F(Φ) = ±|Φ|α+1 and F(Φ) = ±|Φ|α Φ. The last theorem implies also the existence of the energy class solution. The sharpness of the condition on α is an interesting open problem that will not be discussed here. In particular, the theorem states an estimate for the lifespan T ls of the solution of the Higgs boson equation (1.4) in the contracting universe.

In order to prove theorem, we establish the estimates in the Besov spaces for the linear equation. For the equation without source term, these estimates for large time t imply the limitation for the rate of growth as follows:

$$\displaystyle \begin{aligned} \begin{array}{rcl} \| \varPhi (x,t) \|{}_{X'} &\displaystyle \le &\displaystyle \|\varphi _0 \|{}_{X} e^{(\frac{n}{2}+a+\Re M)t} \begin{cases} 1 \quad \mathrm{if} \quad \Re \, M>1/2 \\ t^{\mathrm{sgn}| \Im M|}+ e^{(\frac{1}{2}-\Re M)t } \quad \mathrm{if } \quad \Re \,M \leq 1/2 \end{cases} \\ &\displaystyle &\displaystyle \,+ \|\varphi _1 \|{}_{X} e^{(\frac{n}{2}+a+\Re M)t}\,, \end{array} \end{aligned} $$

where if \(X=B^{s,q}_{p}\), then \(X'=B^{s',q}_{p'}\), \(a:=s-s'-2n\left ( 1/p-1/2\right ) \), 1∕p + 1∕p′ = 1, while \(X'=L^{p'}\) if X = L p. In the case of Sobolev spaces, \(X=X'=H_{(s)}({\mathbb R}^n)\), p = 2.

The integral transform \({\mathcal K}\) allows us to avoid consideration in the phase space and to apply immediately the well-known decay estimates for the solution of the wave equation (operator \({\mathcal E}{\mathcal E} \)) (see, e.g., [3]).

Ebert and do Nascimento [4] study the long-time behavior of the energy of solutions for a class of linear equations with time-dependent mass and speed of propagation. They introduce a classification of the potential term, which clarifies whether the solution behaves like the solution to the wave equation or Klein–Gordon equation. For the equation:

$$\displaystyle \begin{aligned} u_{tt} - e^{2t} \varDelta u +m^2 u = |u|{}^p , \end{aligned}$$

with n ≤ 4, m > 0, \(2\leq p\leq \frac {n}{[n-2]_+} \) they establish the existence of energy class solution for small data. Their proof is based on the splitting of the phase space into pseudo-differential and hyperbolic zones. That method of zones was invented for the hyperbolic operators with multiple characteristics (see [15]) and then modified and successfully used to study equations in the unbounded time domain (see [9, 12,13,14], and references therein).

In the next section, we will give outline of the proof of Theorem 1. The complete proof and sharpness of the obtained results will be published in the forthcoming paper.

2 Outline of the Proof of Theorem 1

The following partial Liouville transform (change of unknown function) \( u= e^{-\frac {n}{2}t}\psi \), \(\psi = e^{\frac {n}{2}t}u\), eliminates the term with time derivative of Eq. (1.8). We obtain

$$\displaystyle \begin{aligned} u _{tt} - e^{2 t} A(x,\partial_x) u + \left( m^2 -\frac{n^2}{4} \right) u = e^{(-\frac{n}{2}-\varGamma )t} F(e^{\frac{n}{2}t}u )\,, \end{aligned}$$

which can be written as follows:

$$\displaystyle \begin{aligned} u _{tt} - e^{2 t} A(x,\partial_x) u - M^2 u = e^{(-\frac{n}{2}-\varGamma )t} F(e^{\frac{n}{2}t}u )\,, \end{aligned}$$

where \( M=\left ( n^2 - {4}m^2\right )^{\frac {1}{2}}/2\). We consider the linear part of the equation:

$$\displaystyle \begin{aligned} u_{tt} - e^{2t} A(x,D) u - M^2 u= - e^{-\frac{n}{2}t}V'(e^{\frac{n}{2}t}u ), \end{aligned} $$
(2.1)

with \(M \in {\mathbb C} \). Equation (2.1) covers two important cases. The first one is the Higgs boson equation, which has V ′(ϕ) = λϕ 3 and M 2 = n 2∕4 + μ 2 with λ > 0, μ > 0, and n = 3. This includes also equation of tachyonic scalar fields living on the de Sitter universe. The second case is the case of the small physical mass (the light scalar field), that is \(0 \leq m \le \frac {n }{2}\). For the last case, \( M = \sqrt { n^2 -4m^2}/2\).

We introduce the kernel functions E(x, t;x 0, t 0;M), K 0(z, t;M), and K 1(z, t;M) (see also [16, 20]). First, for \(M \in {\mathbb C} \) we define the function:

$$\displaystyle \begin{aligned} \begin{array}{rcl} {} E(x,t;x_0,t_0;M) &\displaystyle = &\displaystyle 4 ^{-M} e^{ -M(t_0+t) } \Big((e^{t }+e^{t_0})^2 - (x - x_0)^2\Big)^{-\frac{1}{2}+M } \\ &\displaystyle &\displaystyle \times F\Big(\frac{1}{2}-M ,\frac{1}{2}-M ;1; \frac{ ( e^{t}-e^{t_0 })^2 -(x- x_0 )^2 }{( e^{t}+e^{t_0 })^2 -(x- x_0 )^2 } \Big) . \end{array} \end{aligned} $$
(2.2)

Next, we define also the kernels K 0(z, t;M) and K 1(z, t;M) by:

$$\displaystyle \begin{aligned} \begin{array}{rcl} K_0(z,t;M) &\displaystyle := &\displaystyle - \left[ \frac{\partial }{\partial b} E(z,t;0,b;M) \right]_{b=0} \quad \mathrm{and} \quad K_1(z,t;M) := E(z ,t;0,0;M) \,. \end{array} \end{aligned} $$

The solution u = u(x, t) to the Cauchy problem:

$$\displaystyle \begin{aligned} u_{tt} - e^{2t}A(x,D) u -M^2 u= f ,\quad \quad u(x,0)= \varphi_0 (x)\, , \quad u_t(x,0)=\varphi_1 (x)\,, \end{aligned}$$

with \( f \in C^\infty ({\mathbb R}^{n+1})\) and with φ 0, \( \varphi _1 \in C_0^\infty ({\mathbb R}^n) \), n ≥ 2, is given in [17] by the next expression:

$$\displaystyle \begin{aligned} \begin{array}{rcl} {} u(x,t) &\displaystyle = &\displaystyle 2 \int_{ 0}^{t} db \int_{ 0}^{ e^{t}- e^{b}} dr \, v(x,r ;b) E(r,t; 0,b;M) \\ &\displaystyle &\displaystyle + e ^{-\frac{t}{2}} v_{\varphi_0} (x, \phi (t)) + \, 2\int_{ 0}^{1} v_{\varphi_0} (x, \phi (t)s) K_0(\phi (t)s,t;M)\phi (t)\, ds \\ &\displaystyle &\displaystyle +\, 2\int_{0}^1 v_{\varphi _1 } (x, \phi (t) s) K_1(\phi (t)s,t;M) \phi (t)\, ds , \quad x \in {\mathbb R}^n, \,\, t>0\,, \end{array} \end{aligned} $$
(2.3)

where the function v(x, t;b) is a solution to the Cauchy problem (1.5) and (1.6), while ϕ(t) := e t − 1. Here, for \(\varphi \in C_0^\infty ({\mathbb R}^n)\) and for \(x \in {\mathbb R}^n\), the function v φ(x, ϕ(t)s) coincides with the value v(x, ϕ(t)s) of the solution v(x, t) of the Cauchy problem for Eq. (1.5) with the first initial datum φ(x), while the second datum is zero. Thus, for the solution Φ of the Cauchy problem:

$$\displaystyle \begin{aligned} \varPhi _{tt} - n \varPhi _t - e^{2 t} \bigtriangleup \varPhi + m^2\varPhi = f ,\quad \varPhi (x,0)= \varphi _0(x) , \quad \varPhi _t(x,0)=\varphi _1(x), \end{aligned}$$

due to the relation \(u = e^{-\frac {n}{2}t}\varPhi \), we obtain from (2.3)

$$\displaystyle \begin{aligned} \begin{array}{rcl} \varPhi (x,t) &\displaystyle = &\displaystyle 2 e^{\frac{n}{2}t}\int_{ 0}^{t} db \int_{ 0}^{ e^{t}- e^{b}} dr \, e^{-\frac{n}{2}b} v(x,r ;b) E(r,t; 0,b;M) + e^{\frac{n-1}{2}t} v_{\varphi_0} (x, \phi (t)) \\ &\displaystyle &\displaystyle + \, e^{\frac{n}{2}t}\int_{ 0}^{1} v_{\varphi_0} (x, \phi (t)s)\big(2 K_0(\phi (t)s,t;M)- nK_1(\phi (t)s,t;M)\big)\phi (t)\, ds \\ &\displaystyle &\displaystyle +\, 2e^{\frac{n}{2}t}\int_{0}^1 v_{\varphi _1 } (x, \phi (t) s) K_1(\phi (t)s,t;M) \phi (t)\, ds , \quad x \in {\mathbb R}^n, \,\, t>0\, , \end{array} \end{aligned} $$

where the function v(x, t;b) is a solution to the Cauchy problem (1.5) and (1.6), while the function v φ(x, ϕ(t)s) coincides with the value v(x, ϕ(t)s) of the solution v(x, t) of the Cauchy problem for Eq. (1.5) with the initial datum φ(x), while the second datum is zero.

\( B^{s,q}_{p}- B^{s',q}_{p'}\) Estimates for Equation Without Source

Let φ j = φ(2−j ξ), j > 0, and \(\varphi _0=1-\sum _{j=1}^\infty \varphi _j \), where \(\varphi \in C_0^\infty ({\mathbb R}^n) \) with φ ≥ 0 and supp\(\,\varphi \subseteq \{ \xi \in {\mathbb R}^n\,;\, 1/2 <|\xi |< 2\}\), is that \( \sum _{-\infty }^\infty \varphi (2^{-j} \xi ) =1, \,\, \xi \neq 0\,. \) The norm \(\|g\|{ }_{B^{s,q}_{p}} \) of the Besov space \( B^{s,q}_{p}\) is defined as follows \( \|v\|{ }_{B^{s,q}_{p}}= \left (\sum _{j=0}^\infty \left ( 2^{js} \|{\mathcal F}^{-1}\left ( \varphi _j\hat {v} \right ) \|{ }_{p} \right )^q \right ) ^{1/q}\), where \(\hat {v} \) is the Fourier transform of v.

Theorem 2

Assume that A(x, ∂ x) is the Laplace operator on \({\mathbb R}^n\) and that s, s′≥ 0, q ≥ 1, 1 ≤ p ≤ 2, 1∕p + 1∕p′ = 1, and δ = 1∕p − 1∕2, (n + 1)δ ≤ s − s′, − 1 < s − s′− 2nδ. Denote a := s − s′− 2nδ. The solution Φ = Φ(x, t) of the Cauchy problem:

$$\displaystyle \begin{aligned} \varPhi _{tt} - n \varPhi _t - e^{2 t} A(x,D) \varPhi + m^2\varPhi = 0\,, \quad \varPhi (x,0)= \varphi_0 (x)\, , \quad \varPhi _t(x,0)=\varphi_1 (x)\,, \end{aligned} $$
(2.4)

with \( \Re M>0\) satisfies the following estimate:

$$\displaystyle \begin{aligned} \begin{array}{rcl} \| \varPhi (x,t) \|{}_{B^{s',q}_{p'}} &\displaystyle \lesssim &\displaystyle \|\varphi _0 \|{}_{B^{s,q}_{p}} e^{\frac{n}{2}t}\Bigg( e^{-\frac{1}{2}t}(e^{t}-1)^{a} + (e^{t }-1)^{a+1}\Bigg[ e^{-\Re Mt } (e^{t }+1)^{2 \Re M-1}\\ &\displaystyle &\displaystyle + (e^t+1)^{ \Re M-1} \begin{cases} 1 \quad \mathrm{if} \quad \Re \, M>1/2 \\ t^{1-\mathrm{sgn}|\frac{1}{2}-\Re M|}+ e^{(\frac{1}{2}-\Re M)t } \quad \mathrm{if} \quad \Re \,M \leq 1/2 \end{cases} \Bigg] \Bigg) \\ &\displaystyle &\displaystyle \,+ \|\varphi _1 \|{}_{B^{s,q}_{p}}e^{\frac{n}{2}t} e^{-\Re Mt }(e^{t }-1)^{a+1} (e^{t }+1)^{2 \Re M-1} , \, \,\, \mathit{\mbox{ for all}}\,\,\, t>0. \end{array} \end{aligned} $$

Corollary 1

For large t, the solution Φ = Φ(x, t) of the Cauchy problem (2.4) satisfies the following estimate:

\( B^{s,q}_{p}- B^{s',q}_{p'}\) Estimates for Equation with Source

Theorem 3

Let Φ = Φ(x, t) be a solution of the Cauchy problem:

$$\displaystyle \begin{aligned} \varPhi _{tt} + n \varPhi _t - e^{2 t} \bigtriangleup \varPhi + m^2\varPhi = f\,, \quad \varPhi (x,0)= 0\, , \quad \varPhi _t(x,0)=0\,. \end{aligned}$$

Then, solution Φ = Φ(x, t) for \( \Re M >0 \) satisfies the following estimate:

$$\displaystyle \begin{aligned} \begin{array}{rcl} \| \varPhi (x ,t) \|{}_{B^{s',q}_{p'}} &\displaystyle \leq &\displaystyle C_M e^{ t \left( \frac{n}{2} +\Re M +s-s'-n\left(\frac{1}{p} -\frac{1}{p'} \right)\right)} \\ &\displaystyle &\displaystyle \times \int_{ 0}^{t} e^{-( \frac{n}{2}+\Re M )b} \|f(x,b)\|{}_{B^{s,q}_{p}} \,db \,. \end{array} \end{aligned} $$

for all t > 0, provided that s, s′≥ 0, q ≥ 1, 1 ≤ p ≤ 2, 1∕p + 1∕p′ = 1, and δ = 1∕p − 1∕2, (n + 1)δ ≤ s − s′, − 1 < s − s′− 2nδ.

In order to complete the proof of Theorem 1, we appeal to the integral equation (1.7). Using Theorems 2 and 3 and Banach fixed-point theorem, we prove the existence of a unique solution of the integral equation (1.7) and obtain an estimate of the lifespan. Assumptions (i)–(iii) imply that corresponding operator is a contraction, and thus Banach fixed-point theorem is applicable.