Asymptotic stability of stationary solutions for the Kirchhoff equation

Abstract

This paper considers nonlinear Kirchhoff equation with Kelvin–Voigt damping. This model is used to describe the transversal motion of a stretched string. The existence of smooth stationary solutions of nonlinear Kirchhoff equation has been studied widely. In the present contribution, we prove that a class of stationary solutions is asymptotic stable by overcoming the “loss of derivative” phenomenon causing from the Kirchhoff operator. The key point is to find a suitable weighted function when we carry out the energy estimate for the linearized equation.

Keywords

Kirchhoff equation Global existence Asymptotic stability Ground state Kelvin–Voigt damping

1. Introduction and main result

G. Kirchhoff [19] proposed a model equation (integro-differential equation) to describe the transversal motion of the elastic string, which now is called Kirchhoff equation $\begin{array}{l} u_{t t} - (1 + \int_{Ω} | \nabla u |^{2} d x) △ u = 0, t \in R, x \in Ω \subset R^{d}, \end{array}$ which is a Hamiltonian structure, that is, we define the energy $\begin{array}{l} H (t; u) : = \frac{1}{2} \int_{Ω} (| \partial_{t} u |^{2} + | \nabla u |^{2}) d x + \frac{1}{4} {(\int_{Ω} | \nabla u |^{2} d x)}^{2}, \end{array}$ then it holds $\begin{array}{l} H (t; u) = H (0; u) \end{array}$ as long as the solution exists.

In this paper, we consider the nonlinear Kirchhoff equation with Kelvin–Voigt damping. Kelvin–Voigt damping originates from the internal friction of the material of the vibrating structures and in thus called “internal damping”. It is one of the most important physical dampings. The damped equation takes the form: $\begin{array}{l} u_{t t} - (1 + \int_{R^{n}} | \nabla u |^{2} d x) Δ u + λ_{1} Δ^{2} u - div (a (x) \nabla \partial_{t} u) + λ_{2} u_{t} = | u |^{p - 1} u, \\ (1.1) & \forall (t, x) \in R^{+} \times R^{n}, \end{array}$ with the initial data $\begin{array}{l} u (t, x) |_{t = 0} = u_{0} (x), \partial_{t} u (t, x) |_{t = 0} = u_{1} (x), \end{array}$ where the dimension $n ⩾ 1$ , the unknown function $u : R^{+} \times R^{n} \to R$ , this term $div (a (x) \nabla \partial_{t} u)$ is a constructive viscoelastic damping, it models the vibrations of an elastic body which has one part made of viscoelastic material. $λ_{1}$ and $λ_{2}$ are two non-negative parameters. When $λ_{1} = λ_{2} = 0$ , equation (1.1) is reduced into the Kelvin–Voigt damped Kirchhoff equation $\begin{array}{l} (1.2) & u_{t t} - (1 + \int_{R^{n}} | \nabla u |^{2} d x) Δ u - div (a (x) \nabla \partial_{t} u) = | u |^{p - 1} u, \forall (t, x) \in R^{+} \times R^{n} . \end{array}$

In 1940, Bernstein [8] proved the existence of global in time analytic solutions for Kirchhoff equation on an interval of real line. Pohozaev [25] extended Bernstein’s result to more general Hilbert spaces. The global existence for real analytic data has been shown by D’Ancona–Spagnolo [10]. Ghisi–Gobbino [14] showed some counterexamples in order to prove that the regularity required in the existence results is sharp. After that, they [15] obtained a uniqueness result for a general Kirchhoff equation with non-Lipschitz nonlinear term. D’Ancona–Spagnolo [11] considered the following nonlinear Kirchhoff equation $\begin{array}{l} u_{t t} - (1 + \int_{R^{d}} | \nabla u |^{2} d x) △ u = F (u, u_{t}, \nabla u), (t, x) \in R^{+} \times R^{d}, \end{array}$ with small smooth initial data and smooth function $\begin{array}{l} F (x) = O (| x |^{a + 1}), near x = 0, for some integer a, \end{array}$ then they showed that above nonlinear equations admitted a unique $C^{\infty}$ solution for the dimension $d ⩾ 2$ . Autuori–Pucci–Salvatori [7] studied a more general nonlinear equation $\begin{array}{l} u_{t t} - M (F u (t)) △_{p (x)} u + μ | u |^{p (x) - 2} u + Q (t, x, u, u_{t}) = f (t, x, u), (t, x) \in R^{+} \times Ω \subset R^{d}, \end{array}$ where the $p (x)$ -Dirichlet energy integral is denoted by $F u (t) = \int_{Ω} {| D u (t, x) |^{p (x)} / p (x)} d x$ , and $△_{p (x)}$ denotes the vectorial $p (x)$ -Laplacian operator. By some assumption on nonlinear term, they showed above nonlinear equation admited the global non-existence of solution. After that, they gave the lifespan estimate of solution in [4]. We refer the reader’s to [3,5,6,13,22,26,35] for more interesting results on this equation.

Clearly, the existence of ground state (or stationary solutions) for the nonlinear Kirchhoff equation (1.1) and (1.2) is related to the non-local elliptic problem $\begin{array}{l} (1.3) & - (1 + \int_{R^{n}} | \nabla u |^{2} d x) Δ u + λ_{1} Δ^{2} u = | u |^{p - 1} u, \end{array}$ and $\begin{array}{l} (1.4) & - (1 + \int_{R^{n}} | \nabla u |^{2} d x) Δ u = | u |^{p - 1} u, \end{array}$ respectively, the existence of solution for stationary equations (1.3)–(1.4) depend on the nonlinear term strongly. It has been studied widely by using variational method, we refer the readers to [9,12,13,16,21] for more details. By Giorgi–Moser iteration, one can obtain the existence of smooth solution for above problem. Thus a nature question is

Is the smooth stationary solution stable in functional space?

In the present paper, we will answer this question. Meanwhile, our result gives the global well-posedness for the nonlinear Kirchhoff equations with Kelvin–Voigt damping (1.1) and (1.2). Since we deal with the dissipative case, we can consider the general initial data of the dissipative equation. Assume that the stationary equation (1.3) admits a smooth solution $U (x)$ , and $a (x)$ satisfies the following conditions: $\begin{array}{l} a (x) > σ > 0, {‖ \partial^{s} a ‖}_{L^{\infty}} \sim ε, \forall s \in {1, 2, 3, \dots}, \end{array}$ for a positive small constant ε.

Then we have the following result.

Theorem 1.1.
Let constants $λ_{i} ⩾ 0$ with $i = 1, 2$ . The smooth stationary solution of nonlinear Kirchhoff equations with Kelvin–Voigt damping ( 1.1 ) is asymptotic stable in the Sobolev space $H^{s} (R^{n})$ for any integer $s ⩾ 1$ , that is, for a sufficient small $ε > 0$ , if the initial data $\begin{array}{l} {‖ u_{0} (x) - U (x) ‖}_{H^{s + 2} (R^{n})} + {‖ u_{1} (x) ‖}_{H^{s} (R^{n})} ≲ ε, \end{array}$ then the nonlinear Kirchhoff equation ( 1.1 ) admits a global solution $u (t, x)$ such that $\begin{array}{l} lim_{t \to + \infty} ({‖ u (t, x) - U (x) ‖}_{H^{s + 2} (R^{n})} + {‖ u_{t} (t, x) ‖}_{H^{s} (R^{n})}) = 0, \forall (t, x) \in R^{+} \times R^{n} . \end{array}$

Sketch of the proof. Assume that $U (x) \in C^{\infty} (R^{n})$ is a stationary solution of equation (1.1), i.e. it satisfies the stationary Kirchhoff equation $\begin{array}{l} (1.5) & \{\begin{matrix} - (1 + \int_{R^{n}} | \nabla U |^{2} d x) Δ U + λ_{1} Δ^{2} U = | U |^{p - 1} U, \\ {lim}_{| x | \to + \infty} U (x) = 0 . \end{matrix} \end{array}$ Let $\begin{array}{l} u (t, x) = U (x) + w (t, x), \end{array}$ then substituting it into the equation (1.1) to get the following perturbation equation $\begin{array}{l} L (w) : = & w_{t t} - (1 + \int_{R^{n}} | \nabla U + \nabla w |^{2} d x) Δ w - Δ U \int_{R^{n}} (| \nabla w |^{2} + 2 \nabla U \cdot \nabla w) d x \\ + λ_{1} Δ^{2} w + λ_{2} w_{t} - div (a (x) \nabla \partial_{t} w) - U \sum_{k = 1}^{p - 1} (\binom{p - 1}{k}) | w |^{k} | U |^{(p - 1) - k} \\ (1.6) & - w \sum_{k = 0}^{p - 1} (\binom{p - 1}{k}) | w |^{k} | U |^{(p - 1) - k} = 0, \end{array}$ with the initial data $\begin{array}{l} (1.7) & w (0, x) = w_{0} (x), w_{t} (t, x) |_{t = 0} = u_{1} (x), \end{array}$ where the symbol $\begin{array}{l} (\binom{p - 1}{k}) = \frac{(p - 1)!}{k! (p - 1 - k)!} . \end{array}$

From the structure of nonlinear equation (1.6), the loss of derivative phenomenon should appear, thus we have to smooth the linearized equation by using the smooth operator (see [2] for more details about this operator), and then a suitable Nash–Morse iteration scheme will be constructed to show a unique approximation solution. This method has been used in [28–34]. We refer the reader to [17,18,23,24] for more details on the classical iteration scheme.

We introduce the iteration scheme. Suppose that we have chosen a smooth initial function $w^{(0)} (t, x)$ . Since it is not a solution of nonlinear equation (1.6), there must be an error term, we denoted it by $\begin{array}{l} E^{(0)} : = L (w^{(0)}) . \end{array}$ Meanwhile, the choice of $w^{(0)} (t, x)$ should make the error term small enough in Sobolev space, i.e. $\begin{array}{l} E^{(0)} = L (w^{(0)}) \sim ε in H^{s} (R^{n}), \forall s ⩾ 1 . \end{array}$

The first approximation solution takes the following form $\begin{array}{l} w^{(1)} (t, x) : = w^{(0)} (t, x) + h^{(1)} (t, x) \end{array}$ where $h^{(1)}$ is the solution of linear damped non-local wave equation $\begin{array}{l} L_{w^{(0)}}^{(0)} h^{(1)} = E^{(0)}, \end{array}$ where the linearized operator $\begin{array}{l} L_{w^{(0)}}^{(0)} h^{(1)} : = & h_{t t}^{(1)} - (1 + \int_{R^{n}} {| \nabla w^{(0)} |}^{2} d x) Δ h^{(1)} - 2 Δ w^{(0)} \int_{R^{n}} \nabla h^{(1)} \cdot \nabla w^{(0)} d x + λ_{1} Δ^{2} h^{(1)} \\ - div (a (x) \nabla h_{t}^{(1)}) + λ_{2} h_{t}^{(1)} - {| w^{(0)} |}^{p - 2} ((p - 1) w^{(0)} + | w^{(0)} |) h^{(1)} . \end{array}$

According to this idea, we get the m-th approximation step $h^{(m)}$ by solving the linear equation $\begin{array}{l} L_{w^{(m - 1)}}^{(m - 1)} h^{(m)} = E^{(m - 1)}, \forall m \in N, \end{array}$ where the error term $E^{(m - 1)} : = L (w^{(m - 1)})$ . So the m-th approximation solution has the following form $\begin{array}{l} w^{(m)} (t, x) : = w^{(0)} (t, x) + \sum_{i = 1}^{m} h^{(i)} (t, x), \end{array}$ thus we need to prove $\begin{array}{l} lim_{m \to + \infty} w^{(m)} (t, x) = w^{(\infty)} (t, x) < \infty, \end{array}$ and the error term $\begin{array}{l} lim_{m \to + \infty} E^{(m)} (t, x) = lim_{m \to + \infty} L (w^{(m)}) = 0, \end{array}$ where $w^{(\infty)}$ is the solution of (1.6).

Notation. In this paper, we use $a ≲ b$ to represent that there exists a positive constant C such that $a ⩽ C b$ , and $a \sim b$ means that a and b are equivalent. The notation $C_{0}^{\infty} (R^{+} \times R^{n})$ is the space of $u : R^{+} \times R^{n} \to R$ , and u is infinitely differentiable with a compact support. Furthermore, we denote the usual norm of Sobolev space $H^{s} (R^{n})$ by $‖ \cdot ‖_{H^{s}}$ for convenience.

The paper is organize as follows. In Section 2, we prove the global well-posedness of linear damped wave equation with variable coefficients. In Section 3, we construct an approximate solution of nonlinear approximation perturbation problem. The last section is devoted to show the convergence of approximation scheme.
2. Well-posedness of the linearized problem

Assume the fixed function $U (x) \in H^{s} (R^{n})$ . Inspired by (1.6), we now consider the following linear damped non-local wave equation with variable coefficients $\begin{array}{l} w_{t t} - (1 + \int_{R^{n}} | \nabla U |^{2} d x) Δ w - 2 Δ U \int_{R^{n}} \nabla U \cdot \nabla w d x + λ_{1} Δ^{2} w \\ (2.1) & - div (a (x) \nabla w_{t}) + λ_{2} w_{t} - | U |^{p - 2} ((p - 1) U + | U |) w = f (t, x), \end{array}$ with the initial data $\begin{array}{l} (2.2) & w (0, x) = w_{0} (x), \partial_{t} w (0, x) = w_{1} (x), \forall x \in R^{n} . \end{array}$

Assume that coefficients of (2.1) satisfy the following conditions: $\begin{array}{l} (2.3) & a (x) > σ > 0, {‖ \partial^{s} a ‖}_{L^{\infty}} \sim ε, \forall s \in {1, 2, 3, \dots}, \end{array}$ for a positive small constant ε.

We choose two positive, bounded smooth functions $φ (t, x)$ and $\overline{φ} (t, x)$ in $C_{0}^{\infty} (R^{+} \times R^{n})$ , and satisfying $\begin{array}{l} (2.4) & φ_{t} + c_{1} φ ⩽ 0, \partial^{s} φ < ε, \\ (2.5) & {\overline{φ}}_{t} + c_{1} \overline{φ} ⩽ 0, {\overline{φ}}_{t t} ⩾ c_{1}^{2} \overline{φ}, \partial^{s} \overline{φ} < ε, \\ (2.6) & c_{1} \overline{φ} ⩽ φ ⩽ c_{2} \overline{φ}, \end{array}$ with two positive constants $c_{1}$ and $c_{2}$ , $1 ⩽ c_{1} ⩽ c_{2}$ .

Furthermore, there exists a positive constant C depend on $c_{1}$ , $c_{2}$ and σ such that $\begin{array}{l} (2.7) & {\overline{φ}}^{- 1} e^{- C t} ⩽ e^{- ε t}, \forall t > 0 . \end{array}$ And it also need to suppose $\begin{array}{l} (2.8) & {‖ φ^{\frac{1}{2}} \partial^{s} U ‖}_{L^{2}} {‖ φ^{- \frac{1}{2}} | \partial^{s} U | ‖}_{L^{2}} \sim ε, {‖ {\overline{φ}}^{\frac{1}{2}} \partial^{s} U ‖}_{L^{2}} {‖ {\overline{φ}}^{- \frac{1}{2}} | \partial^{s} U | ‖}_{L^{2}} \sim ε, \end{array}$ and $\begin{array}{l} (2.9) & c_{1}^{2} + c_{1} λ_{2} - 1 - p (c_{2} + 2) | U |^{p - 1} - 2 ε ⩾ C, \\ (2.10) & c_{1} (2 λ_{2} + c_{1}) - c_{2} (1 + p | U |^{p - 1} + 2 ε) - 2 ⩾ C . \end{array}$

We have the following weighted $L^{2}$ -estimate of solution for the linear damped wave equation with variable coefficients (2.1).

Lemma 2.1.
Let constants $λ_{i} ⩾ 0$ with $i = 1, 2$ . Assume that ( 2.3 ) hold. Then the solution of equation ( 2.1 ) satisfies $\begin{array}{l} \int_{R^{n}} (w^{2} + {(w_{t})}^{2} + | \nabla w |^{2} + λ_{1} \sum_{i, k = 1}^{n} {(\partial_{x_{i}} \partial_{x_{k}} w)}^{2}) d x \\ (2.11) & ≲ e^{- ε t} [\int_{R^{n}} (w_{0}^{2} + w_{1}^{2} + | \nabla w_{0} |^{2} + λ_{1} \sum_{i, k = 1}^{n} {(\partial_{x_{i}} \partial_{x_{k}} w_{0})}^{2}) d x + \int_{0}^{t} \int_{R^{n}} f^{2} d x d τ] . \end{array}$
Proof.
On one hand, we multiply equation (2.1) with $φ (t, x) w_{t}$ , then integrating it over $R^{n}$ on x, it holds $\begin{array}{l} \int_{R^{n}} φ w_{t t} w_{t} d x - (1 + \int_{R^{n}} | \nabla U |^{2} d x) \int_{R^{n}} φ Δ w w_{t} d x + λ_{1} \int_{R^{n}} φ Δ^{2} w w_{t} d x \\ - 2 \int_{R^{n}} φ Δ U (\int_{R^{n}} \nabla U \cdot \nabla w d x) w_{t} d x - \int_{R^{n}} φ div (a (x) \nabla w_{t}) w_{t} d x + λ_{2} \int_{R^{n}} φ {(w_{t})}^{2} d x \\ - \int_{R^{n}} φ | U |^{p - 2} ((p - 1) U + | U |) w w_{t} d x \\ (2.12) & = \int_{R^{n}} φ f w_{t} d x . \end{array}$ For some terms, integration by part give that $\begin{array}{l} (2.13) & \int_{R^{n}} φ w_{t t} w_{t} d x = \frac{1}{2} \frac{d}{d t} \int_{R^{n}} φ {(w_{t})}^{2} d x - \frac{1}{2} \int_{R^{n}} φ_{t} {(w_{t})}^{2} d x, \\ λ_{1} \int_{R^{n}} φ Δ^{2} w w_{t} d x = - λ_{1} \sum_{i, k = 1}^{n} \int_{R^{n}} \partial_{x_{i}} φ \partial_{x_{i}} (\partial_{x_{k}}^{2} w) w_{t} d x - λ_{1} \sum_{i, k = 1}^{n} \int_{R^{n}} φ \partial_{x_{i}} (\partial_{x_{k}}^{2} w) \partial_{x_{i}} w_{t} d x \\ = λ_{1} \sum_{i, k = 1}^{n} \int_{R^{n}} \partial_{x_{i}} \partial_{x_{k}} φ \partial_{x_{i}} \partial_{x_{k}} w w_{t} d x + 2 λ_{1} \sum_{i, k = 1}^{n} \int_{R^{n}} \partial_{x_{i}} φ \partial_{x_{i}} \partial_{x_{k}} w \partial_{x_{k}} w_{t} d x \\ (2.14) & + \frac{1}{2} λ_{1} \frac{d}{d t} \sum_{i, k = 1}^{n} \int_{R^{n}} φ {(\partial_{x_{i}} \partial_{x_{k}} w)}^{2} d x - \frac{1}{2} λ_{1} \sum_{i, k = 1}^{n} \int_{R^{n}} φ_{t} {(\partial_{x_{i}} \partial_{x_{k}} w)}^{2} d x, \end{array}$ and $\begin{array}{l} - & \int_{R^{n}} φ div (a (x) \nabla \partial_{t} w) w_{t} d x \\ = \sum_{k = 1}^{n} \int_{R^{n}} \partial_{x_{k}} φ a (x) \partial_{x_{k}} w_{t} w_{t} d x + \sum_{k = 1}^{n} \int_{R^{n}} φ a (x) {(\partial_{x_{k}} w_{t})}^{2} d x \\ (2.15) & = - \frac{1}{2} \sum_{k = 1}^{n} \int_{R^{n}} \partial_{x_{k}} (\partial_{x_{k}} φ a (x)) {(w_{t})}^{2} d x + \sum_{k = 1}^{n} \int_{R^{n}} φ a (x) {(\partial_{x_{k}} w_{t})}^{2} d x, \\ - (1 + \int_{R^{n}} | \nabla U |^{2} d x) \int_{R^{n}} φ Δ w w_{t} d x \\ = (1 + \int_{R^{n}} | \nabla U |^{2} d x) [\sum_{k = 1}^{n} \int_{R^{n}} \partial_{x_{k}} φ \partial_{x_{k}} w w_{t} d x + \sum_{k = 1}^{n} \int_{R^{n}} φ \partial_{x_{k}} w \partial_{x_{k}} w_{t} d x] \\ = (1 + \int_{R^{n}} | \nabla U |^{2} d x) [\sum_{k = 1}^{n} \int_{R^{n}} \partial_{x_{k}} φ \partial_{x_{k}} w w_{t} d x + \frac{1}{2} \frac{d}{d t} \sum_{k = 1}^{n} \int_{R^{n}} φ {(\partial_{x_{k}} w)}^{2} d x \\ (2.16) & - \frac{1}{2} \sum_{k = 1}^{n} \int_{R^{n}} φ_{t} {(\partial_{x_{k}} w)}^{2} d x] . \end{array}$ Combining (2.13)–(2.16), we reduce (2.12) into $\begin{array}{l} \frac{1}{2} \frac{d}{d t} \int_{R^{n}} φ [{(w_{t})}^{2} + (1 + \int_{R^{n}} | \nabla U |^{2} d x) | \nabla w |^{2} + λ_{1} \sum_{i, k = 1}^{n} {(\partial_{x_{i}} \partial_{x_{k}} w)}^{2}] d x \\ + \frac{1}{2} \int_{R^{n}} (2 λ_{2} φ - φ_{t} - \sum_{k = 1}^{n} \partial_{x_{k}} (\partial_{x_{k}} φ a (x))) {(w_{t})}^{2} d x \\ - \frac{1}{2} (1 + \int_{R^{n}} | \nabla U |^{2} d x) \int_{R^{n}} φ_{t} | \nabla w |^{2} d x \\ - \frac{1}{2} λ_{1} \sum_{i, k = 1}^{n} \int_{R^{n}} φ_{t} {(\partial_{x_{i}} \partial_{x_{k}} w)}^{2} d x + \int_{R^{n}} φ a (x) | \nabla w_{t} |^{2} d x \\ - 2 \int_{R^{n}} φ Δ U (\int_{R^{n}} \nabla U \cdot \nabla w d x) w_{t} d x \\ + (1 + \int_{R^{n}} | \nabla U |^{2} d x) \sum_{k = 1}^{n} \int_{R^{n}} \partial_{x_{k}} φ \partial_{x_{k}} w w_{t} d x + 2 λ_{1} \sum_{i, k = 1}^{n} \int_{R^{n}} \partial_{x_{i}} φ \partial_{x_{i}} \partial_{x_{k}} w \partial_{x_{k}} w_{t} d x \\ + λ_{1} \sum_{i, k = 1}^{n} \int_{R^{n}} \partial_{x_{i}} \partial_{x_{k}} φ \partial_{x_{i}} \partial_{x_{k}} w w_{t} d x - \int_{R^{n}} φ | U |^{p - 2} ((p - 1) U + | U |) w w_{t} d x \\ (2.17) & = \int_{R^{n}} φ f w_{t} d x . \end{array}$

In addition, we use Cauchy’s inequality and Hölder’s inequality to obtain $\begin{array}{l} - 2 \int_{R^{n}} φ Δ U (\int_{R^{n}} \nabla U \cdot \nabla w d x) w_{t} d x \\ ⩽ 2 {(\int_{R} φ | Δ U |^{2} d x)}^{\frac{1}{2}} {(\int_{R} φ {(w_{t})}^{2} d x)}^{\frac{1}{2}} {(\int_{R} φ^{- 1} | \nabla U |^{2} d x)}^{\frac{1}{2}} {(\int_{R} φ | \nabla w |^{2} d x)}^{\frac{1}{2}} \\ ⩽ {(\int_{R} φ | Δ U |^{2} d x)}^{\frac{1}{2}} {(\int_{R} φ^{- 1} | \nabla U |^{2} d x)}^{\frac{1}{2}} (\int_{R} φ {(w_{t})}^{2} d x + \int_{R} φ | \nabla w |^{2} d x) \\ = {‖ φ^{\frac{1}{2}} Δ U ‖}_{L^{2}} {‖ φ^{- \frac{1}{2}} | \nabla U | ‖}_{L^{2}} (\int_{R} φ {(w_{t})}^{2} d x + \int_{R} φ | \nabla w |^{2} d x), \end{array}$ and $\begin{array}{l} (1 + \int_{R^{n}} | \nabla U |^{2} d x) \sum_{k = 1}^{n} \int_{R^{n}} \partial_{x_{k}} φ \partial_{x_{k}} w w_{t} d x \\ ⩽ \frac{1}{2} (1 + \int_{R^{n}} | \nabla U |^{2} d x) \sum_{k = 1}^{n} \int_{R^{n}} | \partial_{x_{k}} φ | ({(\partial_{x_{k}} w)}^{2} + {(w_{t})}^{2}) d x, \end{array}$ and $\begin{array}{l} λ_{1} \sum_{i, k = 1}^{n} \int_{R^{n}} \partial_{x_{i}} \partial_{x_{k}} φ \partial_{x_{i}} \partial_{x_{k}} w w_{t} d x ⩽ \frac{1}{2} λ_{1} \sum_{i, k = 1}^{n} \int_{R^{n}} | \partial_{x_{i}} \partial_{x_{k}} φ | ({(\partial_{x_{i}} \partial_{x_{k}} w)}^{2} + {(w_{t})}^{2}) d x, \\ 2 λ_{1} \sum_{i, k = 1}^{n} \int_{R^{n}} \partial_{x_{i}} φ \partial_{x_{i}} \partial_{x_{k}} w \partial_{x_{k}} w_{t} d x ⩽ λ_{1} \sum_{i, k = 1}^{n} \int_{R^{n}} | \partial_{x_{i}} φ | ({(\partial_{x_{i}} \partial_{x_{k}} w)}^{2} + {(\partial_{x_{k}} w_{t})}^{2}) d x, \\ - \int_{R^{n}} φ | U |^{p - 2} ((p - 1) U + | U |) w w_{t} d x ⩽ \frac{p}{2} \int_{R^{n}} φ | U |^{p - 1} (w^{2} + {(w_{t})}^{2}) d x, \\ \int_{R^{n}} φ f w_{t} d x ⩽ \frac{1}{2} \int_{R^{n}} φ (f^{2} + {(w_{t})}^{2}) d x . \end{array}$

Based on the above inequality, we can deduce (2.17) into $\begin{array}{l} \frac{1}{2} \frac{d}{d t} \int_{R^{n}} φ [{(w_{t})}^{2} + (1 + \int_{R^{n}} | \nabla U |^{2} d x) | \nabla w |^{2} + λ_{1} \sum_{i, k = 1}^{n} {(\partial_{x_{i}} \partial_{x_{k}} w)}^{2}] d x - \frac{p}{2} \int_{R^{n}} φ | U |^{p - 1} w^{2} d x \\ + \frac{1}{2} \int_{R^{n}} [2 λ_{2} φ - φ_{t} - \sum_{k = 1}^{n} \partial_{x_{k}} (\partial_{x_{k}} φ a (x)) - p φ | U |^{p - 1} - λ_{1} \sum_{i, k = 1}^{n} | \partial_{x_{i}} \partial_{x_{k}} φ | \\ - 2 φ {‖ φ^{\frac{1}{2}} Δ U ‖}_{L^{2}} {‖ φ^{- \frac{1}{2}} | \nabla U | ‖}_{L^{2}} - (1 + \int_{R^{n}} | \nabla U |^{2} d x) \sum_{k = 1}^{n} | \partial_{x_{k}} φ | - φ] {(w_{t})}^{2} d x \\ - \frac{1}{2} \sum_{k = 1}^{n} \int_{R^{n}} [(1 + \int_{R^{n}} | \nabla U |^{2} d x) (φ_{t} + | \partial_{x_{k}} φ |) + 2 φ {‖ φ^{\frac{1}{2}} Δ U ‖}_{L^{2}} {‖ φ^{- \frac{1}{2}} | \nabla U | ‖}_{L^{2}}] {(\partial_{x_{k}} w)}^{2} d x \\ - \frac{1}{2} λ_{1} \sum_{i, k = 1}^{n} \int_{R^{n}} (φ_{t} + | \partial_{x_{i}} \partial x_{k} φ | + 2 | \partial_{x_{i}} φ |) {(\partial_{x_{i}} \partial_{x_{k}} w)}^{2} d x \\ (2.18) & + \sum_{k = 1}^{n} \int_{R^{n}} (φ a (x) - λ_{1} \sum_{i = 1}^{n} | \partial_{x_{i}} φ |) {(\partial_{x_{k}} w_{t})}^{2} d x ⩽ \frac{1}{2} \int_{R^{n}} φ f^{2} d x . \end{array}$

On the other hand, we multiply equation (2.1) with $\overline{φ} (t, x) w$ , then integrating it over $R^{n}$ on x, it holds $\begin{array}{l} \int_{R^{n}} \overline{φ} w_{t t} w d x - (1 + \int_{R^{n}} | \nabla U |^{2} d x) \int_{R^{n}} \overline{φ} Δ w w d x + λ_{1} \int_{R^{n}} \overline{φ} Δ^{2} w w d x \\ - 2 \int_{R^{n}} \overline{φ} Δ U (\int_{R^{n}} \nabla U \cdot \nabla w d x) w d x - \int_{R^{n}} \overline{φ} div (a (x) \nabla \partial_{t} w) w d x + λ_{2} \int_{R^{n}} \overline{φ} w_{t} w d x \\ (2.19) & - \int_{R^{n}} \overline{φ} | U |^{p - 2} ((p - 1) U + | U |) w^{2} d x = \int_{R^{n}} \overline{φ} f w d x . \end{array}$

Integrating by parts to (2.19), we derive $\begin{array}{l} \int_{R^{n}} \overline{φ} w_{t t} w d x \\ = \frac{d}{d t} \int_{R^{n}} \overline{φ} w_{t} w d x - \int_{R^{n}} {\overline{φ}}_{t} w_{t} w d x - \int_{R^{n}} \overline{φ} {(w_{t})}^{2} d x \\ (2.20) & = \frac{1}{2} \frac{d}{d t} \int_{R^{n}} (2 \overline{φ} w_{t} w - {\overline{φ}}_{t} w^{2}) d x + \frac{1}{2} \int_{R^{n}} {\overline{φ}}_{t t} w^{2} d x - \int_{R^{n}} \overline{φ} {(w_{t})}^{2} d x, \\ λ_{1} \int_{R^{n}} \overline{φ} Δ^{2} w w d x \\ = - λ_{1} \sum_{i, k = 1}^{n} \int_{R^{n}} \partial_{x_{i}} \overline{φ} \partial_{x_{i}} (\partial_{x_{k}}^{2} w) w d x - λ_{1} \sum_{i, k = 1}^{n} \int_{R^{n}} \overline{φ} \partial_{x_{i}} (\partial_{x_{k}}^{2} w) \partial_{x_{i}} w d x \\ = \frac{1}{2} λ_{1} \sum_{i, k = 1}^{n} \int_{R^{n}} \partial_{x_{i}}^{2} \partial_{x_{k}}^{2} \overline{φ} w^{2} d x - λ_{1} \sum_{i, k = 1}^{n} \int_{R^{n}} \partial_{x_{i}} \partial_{x_{k}} \overline{φ} \partial_{x_{i}} w \partial_{x_{k}} w d x \\ (2.21) & - λ_{1} \sum_{i, k = 1}^{n} \int_{R^{n}} \partial_{x_{i}}^{2} \overline{φ} {(\partial_{x_{k}} w)}^{2} d x + λ_{1} \sum_{i, k = 1}^{n} \int_{R^{n}} \overline{φ} {(\partial_{x_{i}} \partial_{x_{k}} w)}^{2} d x, \\ - \int_{R^{n}} \overline{φ} div (a (x) \nabla \partial_{t} w) w d x \\ = \sum_{k = 1}^{n} \int_{R^{n}} \partial_{x_{k}} \overline{φ} a (x) \partial_{x_{k}} \partial_{t} w w d x + \sum_{k = 1}^{n} \int_{R^{n}} \overline{φ} a (x) \partial_{x_{k}} \partial_{t} w \partial_{x_{k}} w d x \\ = \sum_{k = 1}^{n} \int_{R^{n}} \partial_{x_{k}} \overline{φ} a (x) \partial_{x_{k}} w_{t} w d x + \frac{1}{2} \frac{d}{d t} \sum_{k = 1}^{n} \int_{R^{n}} \overline{φ} a (x) {(\partial_{x_{k}} w)}^{2} d x \\ (2.22) & - \frac{1}{2} \sum_{k = 1}^{n} \int_{R^{n}} {\overline{φ}}_{t} a (x) {(\partial_{x_{k}} w)}^{2} d x, \\ (2.23) & λ_{2} \int_{R^{n}} \overline{φ} w_{t} w d x = \frac{1}{2} λ_{2} \frac{d}{d t} \int_{R^{n}} \overline{φ} w^{2} d x - \frac{1}{2} λ_{2} \int_{R^{n}} {\overline{φ}}_{t} w^{2} d x, \end{array}$ and $\begin{array}{l} - (1 + \int_{R^{n}} | \nabla U |^{2} d x) \int_{R^{n}} \overline{φ} Δ w w d x \\ = (1 + \int_{R^{n}} | \nabla U |^{2} d x) [\sum_{k = 1}^{n} \int_{R^{n}} \partial_{x_{k}} \overline{φ} \partial_{x_{k}} w w d x + \sum_{k = 1}^{n} \int_{R^{n}} \overline{φ} {(\partial_{x_{k}} w)}^{2} d x] \\ (2.24) & = (1 + \int_{R^{n}} | \nabla U |^{2} d x) [- \frac{1}{2} \sum_{k = 1}^{n} \int_{R^{n}} \partial_{x_{k}}^{2} \overline{φ} w^{2} d x + \sum_{k = 1}^{n} \int_{R^{n}} \overline{φ} {(\partial_{x_{k}} w)}^{2} d x] . \end{array}$

Combining (2.20)–(2.24) with (2.19), it yields $\begin{array}{l} \frac{1}{2} \frac{d}{d t} \int_{R^{n}} [2 \overline{φ} w_{t} w + (λ_{2} \overline{φ} - {\overline{φ}}_{t}) w^{2} + \sum_{k = 1}^{n} \overline{φ} a (x) {(\partial_{x_{k}} w)}^{2}] d x - \int_{R^{n}} \overline{φ} {(w_{t})}^{2} d x \\ + \frac{1}{2} \int_{R^{n}} [{\overline{φ}}_{t t} - (1 + \int_{R^{n}} | \nabla U |^{2} d x) \sum_{k = 1}^{n} \partial_{x_{k}}^{2} \overline{φ} \\ + λ_{1} \sum_{i, k = 1}^{n} \partial_{x_{i}}^{2} \partial_{x_{k}}^{2} \overline{φ} - 2 \overline{φ} | U |^{p - 2} ((p - 1) U + | U |) - λ_{2} {\overline{φ}}_{t}] w^{2} d x \\ + \frac{1}{2} \sum_{k = 1}^{n} \int_{R^{n}} [2 \overline{φ} (1 + \int_{R^{n}} | \nabla U |^{2} d x) - 2 λ_{1} \sum_{i = 1}^{n} \partial_{x_{i}}^{2} \overline{φ} - a (x) {\overline{φ}}_{t}] {(\partial_{x_{k}} w)}^{2} d x \\ + λ_{1} \sum_{i, k = 1}^{n} \int_{R^{n}} \overline{φ} {(\partial_{x_{i}} \partial_{x_{k}} w)}^{2} d x - 2 \int_{R^{n}} \overline{φ} Δ U (\int_{R^{n}} \nabla U \cdot \nabla w d x) w d x \\ + \sum_{k = 1}^{n} \int_{R^{n}} \partial_{x_{k}} \overline{φ} a (x) \partial_{x_{k}} w_{t} w d x \\ (2.25) & - λ_{1} \sum_{i, k = 1}^{n} \int_{R^{n}} \partial_{x_{i}} \partial_{x_{k}} \overline{φ} \partial_{x_{i}} w \partial_{x_{k}} w d x = \int_{R^{n}} \overline{φ} f w d x, \end{array}$ furthermore, we can use Hölder’s inequality and Cauchy’s inequality to get $\begin{array}{l} - 2 \int_{R^{n}} \overline{φ} Δ U (\int_{R^{n}} \nabla U \cdot \nabla w d x) w d x \\ ⩽ 2 {(\int_{R^{n}} \overline{φ} | Δ U |^{2} d x)}^{\frac{1}{2}} {(\int_{R^{n}} \overline{φ} w^{2} d x)}^{\frac{1}{2}} {(\int_{R^{n}} {\overline{φ}}^{- 1} | \nabla U |^{2} d x)}^{\frac{1}{2}} {(\int_{R^{n}} \overline{φ} | \nabla w |^{2} d x)}^{\frac{1}{2}} \\ ⩽ {(\int_{R^{n}} \overline{φ} | Δ U |^{2} d x)}^{\frac{1}{2}} {(\int_{R^{n}} {\overline{φ}}^{- 1} | \nabla U |^{2} d x)}^{\frac{1}{2}} (\int_{R^{n}} \overline{φ} w^{2} d x + \int_{R^{n}} \overline{φ} | \nabla w |^{2} d x) \\ = {‖ {\overline{φ}}^{\frac{1}{2}} Δ U ‖}_{L^{2}} {‖ {\overline{φ}}^{- \frac{1}{2}} | \nabla U | ‖}_{L^{2}} (\int_{R^{n}} \overline{φ} w^{2} d x + \int_{R^{n}} \overline{φ} | \nabla w |^{2} d x), \end{array}$ and $\begin{array}{l} \sum_{k = 1}^{n} \int_{R^{n}} \partial_{x_{k}} \overline{φ} a (x) \partial_{x_{k}} w_{t} w d x ⩽ \frac{1}{2} \sum_{k = 1}^{n} \int_{R^{n}} | \partial_{x_{k}} \overline{φ} a (x) | ({(\partial_{x_{k}} w_{t})}^{2} + w^{2}) d x, \\ - λ_{1} \sum_{i, k = 1}^{n} \int_{R^{n}} \partial_{x_{i}} \partial_{x_{k}} \overline{φ} \partial_{x_{i}} w \partial_{x_{k}} w d x ⩽ λ_{1} \sum_{i, k = 1}^{n} \int_{R^{n}} | \partial_{x_{i}} \partial_{x_{k}} \overline{φ} | {(\partial_{x_{k}} w)}^{2} d x, \\ \int_{R^{n}} \overline{φ} f w d x ⩽ \frac{1}{2} \int_{R^{n}} \overline{φ} (f^{2} + w^{2}) d x . \end{array}$

Thus using above results, we reduce (2.25) to $\begin{array}{l} \frac{1}{2} \frac{d}{d t} \int_{R^{n}} [2 \overline{φ} w_{t} w + (λ_{2} \overline{φ} - {\overline{φ}}_{t}) w^{2} + \sum_{k = 1}^{n} \overline{φ} a (x) {(\partial_{x_{k}} w)}^{2}] d x - \int_{R^{n}} \overline{φ} {(w_{t})}^{2} d x \\ + \frac{1}{2} \int_{R^{n}} [{\overline{φ}}_{t t} - (1 + \int_{R^{n}} | \nabla U |^{2} d x) \sum_{k = 1}^{n} \partial_{x_{k}}^{2} \overline{φ} - 2 \overline{φ} | U |^{p - 2} ((p - 1) U + | U |) \\ + λ_{1} \sum_{i, k = 1}^{n} \partial_{x_{i}}^{2} \partial_{x_{k}}^{2} \overline{φ} - λ_{2} {\overline{φ}}_{t} - \sum_{k = 1}^{n} | \partial_{x_{k}} \overline{φ} a (x) | - \overline{φ} - 2 \overline{φ} {‖ {\overline{φ}}^{\frac{1}{2}} Δ U ‖}_{L^{2}} {‖ {\overline{φ}}^{- \frac{1}{2}} | \nabla U | ‖}_{L^{2}}] w^{2} d x \\ + \frac{1}{2} \sum_{k = 1}^{n} \int_{R^{n}} [2 \overline{φ} (1 + \int_{R^{n}} | \nabla U |^{2} d x) \\ - 2 λ_{1} \sum_{i = 1}^{n} \partial_{x_{i}}^{2} \overline{φ} - a (x) {\overline{φ}}_{t} - 2 \overline{φ} {‖ {\overline{φ}}^{\frac{1}{2}} Δ U ‖}_{L^{2}} {‖ {\overline{φ}}^{- \frac{1}{2}} | \nabla U | ‖}_{L^{2}} \\ - 2 λ_{1} \sum_{i = 1}^{n} | \partial_{x_{i}} \partial_{x_{k}} \overline{φ} |] {(\partial_{x_{k}} w)}^{2} d x + λ_{1} \sum_{i, k = 1}^{n} \int_{R^{n}} \overline{φ} {(\partial_{x_{i}} \partial_{x_{k}} w)}^{2} d x \\ - \frac{1}{2} \sum_{k = 1}^{n} \int_{R^{n}} | \partial_{x_{k}} \overline{φ} a (x) | {(\partial_{x_{k}} w_{t})}^{2} d x \\ (2.26) & ⩽ \frac{1}{2} \int_{R^{n}} \overline{φ} f^{2} d x . \end{array}$

Therefore summing up (2.18) with (2.26), it holds $\begin{array}{l} \frac{d}{d t} \int_{R^{n}} [2 \overline{φ} w_{t} w + φ {(w_{t})}^{2} + (λ_{2} \overline{φ} - {\overline{φ}}_{t}) w^{2} + ((1 + \int_{R^{n}} | \nabla U |^{2} d x) φ + \overline{φ} a (x)) \sum_{k = 1}^{n} {(\partial_{x_{k}} w)}^{2} \\ + λ_{1} φ \sum_{i, k = 1}^{n} {(\partial_{x_{i}} \partial_{x_{k}} w)}^{2}] d x + \int_{R^{n}} A_{1} (t, x) w^{2} d x + \int_{R^{n}} A_{2} (t, x) {(w_{t})}^{2} d x \\ + \sum_{k = 1}^{n} \int_{R^{n}} A_{3 k} (t, x) {(\partial_{x_{k}} w)}^{2} d x \\ + \sum_{k = 1}^{n} \int_{R^{n}} (2 a (x) φ - 2 λ_{1} \sum_{i = 1}^{n} | \partial_{x_{i}} φ | - | \partial_{x_{k}} \overline{φ} a (x) |) {(\partial_{x_{k}} w_{t})}^{2} d x \\ (2.27) & + λ_{1} \sum_{i, k = 1}^{n} \int_{R^{n}} (2 \overline{φ} - φ_{t} - 2 | \partial_{x_{i}} φ | - | \partial_{x_{i}} \partial_{x_{k}} φ |) {(\partial_{x_{i}} \partial_{x_{k}} w)}^{2} d x ⩽ \int_{R^{n}} (φ + \overline{φ}) f^{2} d x, \end{array}$ where $\begin{array}{l} A_{1} (t, x) : = {\overline{φ}}_{t t} - (1 + \int_{R^{n}} | \nabla U |^{2} d x) \sum_{k = 1}^{n} \partial_{x_{k}}^{2} \overline{φ} + λ_{1} \sum_{i, k = 1}^{n} \partial_{x_{i}}^{2} \partial_{x_{k}}^{2} \overline{φ} - λ_{2} {\overline{φ}}_{t} - \sum_{k = 1}^{n} | \partial_{x_{k}} \overline{φ} a (x) | \\ - \overline{φ} - 2 \overline{φ} {‖ {\overline{φ}}^{\frac{1}{2}} Δ U ‖}_{L^{2}} {‖ {\overline{φ}}^{- \frac{1}{2}} | \nabla U | ‖}_{L^{2}} - p φ | U |^{p - 1} - 2 \overline{φ} | U |^{p - 2} ((p - 1) U + | U |), \\ A_{2} (t, x) : = 2 λ_{2} φ - φ_{t} - \sum_{k = 1}^{n} \partial_{x_{k}} (\partial_{x_{k}} φ a (x)) - 2 \overline{φ} - 2 φ {‖ φ^{\frac{1}{2}} Δ U ‖}_{L^{2}} {‖ φ^{- \frac{1}{2}} | \nabla U | ‖}_{L^{2}} - p φ | U |^{p - 1} \\ - λ_{1} \sum_{i, k = 1}^{n} | \partial_{x_{i}} \partial_{x_{k}} φ | - (1 + \int_{R^{n}} | \nabla U |^{2} d x) \sum_{k = 1}^{n} | \partial_{x_{k}} φ | - φ, \\ A_{3 k} (t, x) : = (1 + \int_{R^{n}} | \nabla U |^{2} d x) (2 \overline{φ} - φ_{t} - \sum_{k = 1}^{n} | \partial_{x_{k}} φ |) - 2 λ_{1} \sum_{i = 1}^{n} \partial_{x_{i}}^{2} \overline{φ} - 2 λ_{1} \sum_{i = 1}^{n} | \partial_{x_{i}} \partial_{x_{k}} \overline{φ} | \\ - a (x) {\overline{φ}}_{t} - 2 φ {‖ φ^{\frac{1}{2}} Δ U ‖}_{L^{2}} {‖ φ^{- \frac{1}{2}} {| \nabla U ‖}_{L^{2}} - 2 \overline{φ} {‖ {\overline{φ}}^{\frac{1}{2}} Δ U ‖}_{L^{2}} ‖ {\overline{φ}}^{- \frac{1}{2}} | \nabla U | ‖}_{L^{2}} . \end{array}$

Let us now deal with coefficients of inequality (2.27). Using the assumptions given in (2.3) and (2.5)–(2.6), there exists a positive constant C depending on $c_{1}$ , $c_{2}$ , σ, and ε such that $\begin{array}{l} (2.28) & φ - \overline{φ} ⩾ (c_{1} - 1) \overline{φ} ⩾ C \overline{φ}, λ_{1} φ ⩾ λ_{1} c_{1} \overline{φ} ⩾ C \overline{φ}, \\ (2.29) & λ_{2} \overline{φ} - {\overline{φ}}_{t} - \overline{φ} ⩾ (λ_{2} + c_{1} - 1) \overline{φ} ⩾ C \overline{φ}, \\ (2.30) & (1 + \int_{R^{n}} | \nabla U |^{2} d x) φ + \overline{φ} a (x) ⩾ [c_{1} (1 + \int_{R^{n}} | \nabla U |^{2} d x) + σ] \overline{φ} ⩾ C \overline{φ} . \end{array}$ Similarly, by (2.3)–(2.9), we can get $\begin{array}{l} A_{1} (t, x) = {\overline{φ}}_{t t} - λ_{2} {\overline{φ}}_{t} - \overline{φ} - p | U |^{p - 1} φ - 2 | U |^{p - 2} ((p - 1) U + | U |) \overline{φ} \\ - 2 \overline{φ} {‖ {\overline{φ}}^{\frac{1}{2}} Δ U ‖}_{L^{2}} {‖ {\overline{φ}}^{- \frac{1}{2}} | \nabla U | ‖}_{L^{2}} + O (ε) \\ ⩾ [c_{1}^{2} + c_{1} λ_{2} - 1 - p (c_{2} + 2) | U |^{p - 1} - 2 ε] \overline{φ} + O (ε) ⩾ C \overline{φ} + O (ε), \\ A_{2} (t, x) = 2 λ_{2} φ - φ_{t} - 2 \overline{φ} - φ - p | U |^{p - 1} φ - 2 φ {‖ φ^{\frac{1}{2}} Δ U ‖}_{L^{2}} {‖ φ^{- \frac{1}{2}} | \nabla U | ‖}_{L^{2}} + O (ε) \\ ⩾ [c_{1} (2 λ_{2} + c_{1}) - c_{2} (1 + p | U |^{p - 1} + 2 ε) - 2] \overline{φ} + O (ε) ⩾ C \overline{φ} + O (ε), \\ A_{3 k} (t, x) = (1 + \int_{R^{n}} | \nabla U |^{2} d x) (2 \overline{φ} - φ_{t}) - a (x) {\overline{φ}}_{t} - 2 φ {‖ φ^{\frac{1}{2}} Δ U ‖}_{L^{2}} ‖ φ^{- \frac{1}{2}} {| \nabla U ‖}_{L^{2}} \\ - 2 \overline{φ} {‖ {\overline{φ}}^{\frac{1}{2}} Δ U ‖}_{L^{2}} ‖ {\overline{φ}}^{- \frac{1}{2}} | \nabla U | ‖_{L^{2}} + O (ε) \\ ⩾ [(c_{1}^{2} + 2) (1 + \int_{R^{n}} | \nabla U |^{2} d x) - 2 ε (c_{2} + 1) + c_{1} σ] \overline{φ} + O (ε) \\ ⩾ C \overline{φ} + O (ε), \end{array}$ and $\begin{array}{l} 2 a (x) φ - 2 λ_{1} \sum_{i = 1}^{n} | \partial_{x_{i}} φ | - | \partial_{x_{k}} \overline{φ} a (x) | ⩾ 2 c_{1} σ \overline{φ} + O (ε) ⩾ C \overline{φ} + O (ε), \\ (2.31) & 2 \overline{φ} - φ_{t} - 2 | \partial_{x_{i}} φ | - | \partial_{x_{i}} \partial_{x_{k}} φ | ⩾ (c_{1}^{2} + 2) \overline{φ} + O (ε) ⩾ C \overline{φ} + O (ε) . \end{array}$

Based on above estimation, we deduce (2.27) into $\begin{array}{l} \int_{R^{n}} \overline{φ} (w^{2} + {(w_{t})}^{2} + | \nabla w |^{2} + λ_{1} \sum_{i, k = 1}^{n} {(\partial_{x_{i}} \partial_{x_{k}} w)}^{2}) d x \\ + C \int_{0}^{t} \int_{R^{n}} \overline{φ} (w^{2} + {(w_{t})}^{2} + | \nabla w |^{2} + | \nabla w_{t} |^{2} + λ_{1} \sum_{i, k = 1}^{n} {(\partial_{x_{i}} \partial_{x_{k}} w)}^{2}) d x d τ \\ ≲ \int_{R^{n}} \overline{φ} (0, x) ({(w_{0})}^{2} + {(w_{1})}^{2} + | \nabla w_{0} |^{2} + λ_{1} \sum_{i, k = 1}^{n} {(\partial_{x_{i}} \partial_{x_{k}} w_{0})}^{2}) d x \\ (2.32) & + \int_{0}^{t} \int_{R^{n}} (φ + \overline{φ}) f^{2} d x d τ . \end{array}$ Applying Gronwall’s inequality to (2.32) we can get $\begin{array}{l} \int_{R^{n}} \overline{φ} (w^{2} + {(w_{t})}^{2} + | \nabla w |^{2} + λ_{1} \sum_{i, k = 1}^{n} {(\partial_{x_{i}} \partial_{x_{k}} w)}^{2}) d x \\ ≲ e^{- C t} [\int_{R^{n}} \overline{φ} (0, x) ({(w_{0})}^{2} + {(w_{1})}^{2} + | \nabla w_{1} |^{2} + λ_{1} \sum_{i, k = 1}^{n} {(\partial_{x_{i}} \partial_{x_{k}} w_{0})}^{2}) d x \\ (2.33) & + \int_{0}^{t} \int_{R^{n}} (φ + \overline{φ}) f^{2} d x d τ] . \end{array}$

In particularly, we can take weighted functions $\begin{array}{l} (2.34) & φ (t, x) : = 2 e^{- (C - ε) t}, \overline{φ} (t, x) : = e^{- (C - ε) t} \end{array}$ with a positive constant C, then assumptions (2.4)–(2.8) hold. Then (2.11) can be obtained from (2.33). □

In what follows, we derive $H^{s}$ -estimates for any $s ⩾ 1$ and $s \in N$ . Applying $\partial_{x_{j}}^{s}$ $(\forall j = 1, 2, \dots, n)$ to both side of (2.1), it holds $\begin{array}{l} \partial_{x_{j}}^{s} w_{t t} - (1 + \int_{R^{n}} | \nabla U |^{2} d x) Δ (\partial_{x_{j}}^{s} w) + λ_{1} Δ^{2} (\partial_{x_{j}}^{s} w) - div (a (x) \nabla (\partial_{x_{j}}^{s} w_{t})) \\ (2.35) & + λ_{2} \partial_{x_{j}}^{s} w_{t} - | U |^{p - 2} ((p - 1) U + | U |) \partial_{x_{j}}^{s} w = g_{s} (t, x), \end{array}$ where $\begin{array}{l} g_{s} (t, x) = & \partial_{x_{j}}^{s} f + 2 Δ (\partial_{x_{j}}^{s} U) \int_{R^{n}} \nabla U \cdot \nabla w d x + \sum_{k = 1}^{n} \sum_{\begin{array}{c} s_{1} + s_{2} = s \\ 1 ⩽ s_{1} ⩽ s \end{array}} (\binom{s}{s_{1}}) \partial_{x_{k}} (\partial_{x_{j}}^{s_{1}} a (x) \partial_{x_{k}} \partial_{x_{j}}^{s_{2}} w_{t}) \\ + \sum_{\begin{array}{c} s_{1} + s_{2} = s \\ 1 ⩽ s_{1} ⩽ s \end{array}} (\binom{s}{s_{1}}) \partial_{x_{j}}^{s_{1}} (| U |^{p - 2} ((p - 1) U + | U |)) \partial_{x_{j}}^{s_{2}} w, \end{array}$ with the symbol $\begin{array}{l} (\binom{s}{s_{1}}) = \frac{s!}{s_{1}! (s - s_{1})!} . \end{array}$

Then we have the following priori estimation:
Lemma 2.2.
Let constants $λ_{i} ⩾ 0$ with $i = 1, 2$ . Assume that ( 2.3 ) hold. Then the solution of equation ( 2.1 ) satisfies $\begin{array}{l} \sum_{j = 1}^{n} \int_{R^{n}} ({(\partial_{x_{j}}^{s} w)}^{2} + {(\partial_{x_{j}}^{s} w_{t})}^{2} + {| \nabla \partial_{x_{j}}^{s} w |}^{2} + λ_{1} \sum_{i, k = 1}^{n} {(\partial_{x_{i}} \partial_{x_{k}} \partial_{x_{j}}^{s} w)}^{2}) d x \\ ≲ e^{- ε t} \sum_{θ = 0}^{s} \sum_{j = 1}^{n} [\int_{R^{n}} ({(\partial_{x_{j}}^{θ} w_{0})}^{2} + {(\partial_{x_{j}}^{θ} w_{1})}^{2} + {| \nabla \partial_{x_{j}}^{θ} w_{0} |}^{2} + λ_{1} \sum_{i, k = 1}^{n} {(\partial_{x_{i}} \partial_{x_{k}} \partial_{x_{j}}^{θ} w_{0})}^{2}) d x \\ (2.36) & + \int_{0}^{t} \int_{R^{n}} {(\partial_{x_{j}}^{θ} f)}^{2} d x d τ] . \end{array}$
Proof.
This proof is based on the induction. When $s = 1$ , we rewrite the liner equation as $\begin{array}{l} \partial_{x_{j}} w_{t t} - (1 + \int_{R^{n}} | \nabla U |^{2} d x) Δ (\partial_{x_{j}} w) + λ_{1} Δ^{2} (\partial_{x_{j}} w) - div (a (x) \nabla (\partial_{x_{j}} w_{t})) \\ (2.37) & + λ_{2} \partial_{x_{j}} w_{t} - | U |^{p - 2} ((p - 1) U + | U |) \partial_{x_{j}} w = g_{1} (t, x), \end{array}$ where $\begin{array}{l} g_{1} (t, x) = & \partial_{x_{j}} f + 2 Δ (\partial_{x_{j}} U) \int_{R^{n}} \nabla U \cdot \nabla w d x + \sum_{k = 1}^{n} \partial_{x_{k}} (\partial_{x_{j}} a (x) \partial_{x_{k}} w_{t}) \\ + \partial_{x_{j}} (| U |^{p - 2} ((p - 1) U + | U |)) w . \end{array}$

We can see that the linear equation (2.37) has the same structure with equation (2.1), so we can multiply both sides of equation (2.37) with $φ (t, x) \partial_{x_{j}} w_{t}$ and $\overline{φ} (t, x) \partial_{x_{j}} w$ , respectively, using the same method of getting (2.27), we deduce $\begin{array}{l} \frac{d}{d t} \int_{R^{n}} [2 \overline{φ} \partial_{x_{j}} w_{t} \partial_{x_{j}} w + φ {(\partial_{x_{j}} w_{t})}^{2} + (λ_{2} \overline{φ} - {\overline{φ}}_{t}) {(\partial_{x_{j}} w)}^{2} + λ_{1} φ \sum_{i, k = 1}^{n} {(\partial_{x_{i}} \partial_{x_{k}} \partial_{x_{j}} w)}^{2} \\ + ((1 + \int_{R^{n}} | \nabla U |^{2} d x) φ + \overline{φ} a (x)) \sum_{k = 1}^{n} {(\partial_{x_{k}} \partial_{x_{j}} w)}^{2}] d x \\ + \int_{R^{n}} {\overline{A}}_{1} (t, x) {(\partial_{x_{j}} w)}^{2} d x + \int_{R^{n}} {\overline{A}}_{2} (t, x) {(\partial_{x_{j}} w_{t})}^{2} d x \\ + \sum_{k = 1}^{n} \int_{R^{n}} {\overline{A}}_{3 k} (t, x) {(\partial_{x_{k}} \partial_{x_{j}} w)}^{2} d x \\ + \sum_{k = 1}^{n} \int_{R^{n}} [2 a (x) φ - 2 λ_{1} \sum_{i = 1}^{n} | \partial_{x_{i}} φ | - | \partial_{x_{k}} \overline{φ} a (x) |] {(\partial_{x_{k}} \partial_{x_{j}} w_{t})}^{2} d x \\ + λ_{1} \sum_{i, k = 1}^{n} \int_{R^{n}} [2 \overline{φ} - φ_{t} - 2 | \partial_{x_{i}} φ | - | \partial_{x_{i}} \partial_{x_{k}} φ |] {(\partial_{x_{i}} \partial_{x_{k}} \partial_{x_{j}} w)}^{2} d x \\ (2.38) & ⩽ 2 \int_{R^{n}} (φ \partial_{x_{j}} w_{t} + \overline{φ} \partial_{x_{j}} w) g_{1} d x, \end{array}$ where $\begin{array}{l} {\overline{A}}_{1} (t, x) : = {\overline{φ}}_{t t} - (1 + \int_{R^{n}} | \nabla U |^{2} d x) \sum_{k = 1}^{n} \partial_{x_{k}}^{2} \overline{φ} + λ_{1} \sum_{i, k = 1}^{n} \partial_{x_{i}}^{2} \partial_{x_{k}}^{2} \overline{φ} - λ_{2} {\overline{φ}}_{t} - \sum_{k = 1}^{n} | \partial_{x_{k}} \overline{φ} a (x) | \\ - p φ | U |^{p - 1} - 2 \overline{φ} | U |^{p - 2} ((p - 1) U + | U |), \\ {\overline{A}}_{2} (t, x) : = 2 λ_{2} φ - φ_{t} - \sum_{k = 1}^{n} \partial_{x_{k}} (\partial_{x_{k}} φ a (x)) - 2 \overline{φ} - λ_{1} \sum_{i, k = 1}^{n} | \partial_{x_{i}} \partial_{x_{k}} φ | \\ - p φ | U |^{p - 1} - (1 + \int_{R^{n}} | \nabla U |^{2} d x) \sum_{k = 1}^{n} | \partial_{x_{k}} φ |, \\ {\overline{A}}_{3 k} (t, x) : = (1 + \int_{R^{n}} | \nabla U |^{2} d x) (2 \overline{φ} - φ_{t} - \sum_{k = 1}^{n} | \partial_{x_{k}} φ |) - 2 λ_{1} \sum_{i = 1}^{n} \partial_{x_{i}}^{2} \overline{φ} \\ - 2 λ_{1} \sum_{i = 1}^{n} | \partial_{x_{i}} \partial_{x_{k}} \overline{φ} | - a (x) {\overline{φ}}_{t} . \end{array}$

We next estimate the right hand side term of (2.38). We first use Hölder’s inequality and Cauchy’s inequality to get the following inequalities $\begin{array}{l} 2 \int_{R^{n}} (φ \partial_{x_{j}} w_{t} + \overline{φ} \partial_{x_{j}} w) \partial_{x_{j}} f d x ⩽ & \int_{R^{n}} [φ {(\partial_{x_{j}} w_{t})}^{2} + \overline{φ} {(\partial_{x_{j}} w)}^{2} + (φ + \overline{φ}) {(\partial_{x_{j}} f)}^{2}] d x, \end{array}$ and $\begin{array}{l} 2 \sum_{k = 1}^{n} \int_{R^{n}} (φ \partial_{x_{j}} w_{t} + \overline{φ} \partial_{x_{j}} w) \partial_{x_{k}} (\partial_{x_{j}} a (x) \partial_{x_{k}} w_{t}) d x \\ ⩽ \sum_{k = 1}^{n} \int_{R^{n}} | \partial_{x_{k}} \overline{φ} \partial_{x_{j}} a (x) | {(\partial_{x_{j}} w)}^{2} d x + \sum_{k = 1}^{n} \int_{R^{n}} \overline{φ} | \partial_{x_{j}} a (x) | {(\partial_{x_{k}} \partial_{x_{j}} w)}^{2} d x \\ + \sum_{k = 1}^{n} \int_{R^{n}} | \partial_{x_{k}} φ \partial_{x_{j}} a (x) | {(\partial_{x_{j}} w_{t})}^{2} d x + \sum_{k = 1}^{n} \int_{R^{n}} φ | \partial_{x_{j}} a (x) | {(\partial_{x_{k}} \partial_{x_{j}} w_{t})}^{2} d x \\ + \sum_{k = 1}^{n} \int_{R^{n}} | \partial_{x_{j}} a (x) | (φ + \overline{φ} + | \partial_{x_{k}} φ | + | \partial_{x_{k}} \overline{φ} |) {(\partial_{x_{k}} w_{t})}^{2} d x, \\ 2 \int_{R^{n}} (φ \partial_{x_{j}} w_{t} + \overline{φ} \partial_{x_{j}} w) \partial_{x_{j}} (| U |^{p - 2} ((p - 1) U + | U |)) w d x \\ ⩽ \int_{R^{n}} (φ + \overline{φ}) | \partial_{x_{j}} (| U |^{p - 2} ((p - 1) U + | U |)) | w^{2} d x \\ + \int_{R^{n}} φ | \partial_{x_{j}} (| U |^{p - 2} ((p - 1) U + | U |)) | {(\partial_{x_{j}} w_{t})}^{2} d x \\ + \int_{R^{n}} \overline{φ} | \partial_{x_{j}} (| U |^{p - 2} ((p - 1) U + | U |)) | {(\partial_{x_{j}} w)}^{2} d x, \\ 2 \int_{R^{n}} φ Δ (\partial_{x_{j}} U) (\int_{R^{n}} \nabla U \cdot \nabla w d x) \partial_{x_{j}} w_{t} d x \\ ⩽ 2 {(\int_{R} φ {| Δ (\partial_{x_{j}} U) |}^{2} d x)}^{\frac{1}{2}} {(\int_{R} φ {(\partial_{x_{j}} w_{t})}^{2} d x)}^{\frac{1}{2}} \\ \times {(\int_{R} φ^{- 1} | \nabla U |^{2} d x)}^{\frac{1}{2}} {(\int_{R} φ | \nabla w |^{2} d x)}^{\frac{1}{2}} \\ ⩽ {(\int_{R} φ {| Δ (\partial_{x_{j}} U) |}^{2} d x)}^{\frac{1}{2}} {(\int_{R} φ^{- 1} | \nabla U |^{2} d x)}^{\frac{1}{2}} (\int_{R} φ {(\partial_{x_{j}} w_{t})}^{2} d x + \int_{R} φ | \nabla w |^{2} d x) \\ = {‖ φ^{\frac{1}{2}} Δ (\partial_{x_{j}} U) ‖}_{L^{2}} {‖ φ^{- \frac{1}{2}} | \nabla U | ‖}_{L^{2}} (\int_{R} φ {(\partial_{x_{j}} w_{t})}^{2} d x + \int_{R} φ | \nabla w |^{2} d x), \\ - 2 \int_{R^{n}} \overline{φ} Δ \partial_{x_{j}} U (\int_{R^{n}} \nabla U \cdot \nabla w d x) \partial_{x_{j}} w d x \\ ⩽ 2 {(\int_{R} \overline{φ} | Δ \partial_{x_{j}} U |^{2} d x)}^{\frac{1}{2}} {(\int_{R} φ {(\partial_{x_{j}} w)}^{2} d x)}^{\frac{1}{2}} {(\int_{R} {\overline{φ}}^{- 1} | \nabla U |^{2} d x)}^{\frac{1}{2}} {(\int_{R} \overline{φ} | \nabla w |^{2} d x)}^{\frac{1}{2}} \\ ⩽ {(\int_{R} \overline{φ} | Δ \partial_{x_{j}} U |^{2} d x)}^{\frac{1}{2}} {(\int_{R} {\overline{φ}}^{- 1} | \nabla U |^{2} d x)}^{\frac{1}{2}} (\int_{R} \overline{φ} {(\partial_{x_{j}} w)}^{2} d x + \int_{R} \overline{φ} | \nabla w |^{2} d x) \\ = {‖ {\overline{φ}}^{\frac{1}{2}} Δ \partial_{x_{j}} U ‖}_{L^{2}} {‖ {\overline{φ}}^{- \frac{1}{2}} | \nabla U | ‖}_{L^{2}} (\int_{R} \overline{φ} {(\partial_{x_{j}} w)}^{2} d x + \int_{R} \overline{φ} | \nabla w |^{2} d x) . \end{array}$

Based on above estimation, summing up (2.38) from $j = 1$ to $j = n$ , it holds $\begin{array}{l} \sum_{j = 1}^{n} \frac{d}{d t} \int_{R^{n}} [2 \overline{φ} \partial_{x_{j}} w_{t} \partial_{x_{j}} w + φ {(\partial_{x_{j}} w_{t})}^{2} + (λ_{2} \overline{φ} - {\overline{φ}}_{t}) {(\partial_{x_{j}} w)}^{2} \\ + λ_{1} φ \sum_{i, k = 1}^{n} {(\partial_{x_{i}} \partial_{x_{k}} \partial_{x_{j}} w)}^{2} + ((1 + \int_{R^{n}} | \nabla U |^{2} d x) φ + \overline{φ} a (x)) \sum_{k = 1}^{n} {(\partial_{x_{k}} \partial_{x_{j}} w)}^{2}] d x \\ + \sum_{j = 1}^{n} \int_{R^{n}} B_{1 j} (t, x) {(\partial_{x_{j}} w)}^{2} d x + \sum_{j = 1}^{n} \int_{R^{n}} B_{2 j} (t, x) {(\partial_{x_{j}} w_{t})}^{2} d x \\ + \sum_{k, j = 1}^{n} \int_{R^{n}} B_{3 k j} (t, x) {(\partial_{x_{k}} \partial_{x_{j}} w)}^{2} d x \\ + λ_{1} \sum_{i, j, k = 1}^{n} \int_{R^{n}} [2 \overline{φ} - φ_{t} - 2 | \partial_{x_{i}} φ | - | \partial_{x_{i}} \partial_{x_{k}} φ |] {(\partial_{x_{i}} \partial_{x_{k}} \partial_{x_{j}} w)}^{2} d x \\ + \sum_{k, j = 1}^{n} \int_{R^{n}} [2 a (x) φ - 2 λ_{1} \sum_{i = 1}^{n} | \partial_{x_{i}} φ | - | \partial_{x_{k}} \overline{φ} a (x) | - φ | \partial_{x_{j}} a (x) |] {(\partial_{x_{k}} \partial_{x_{j}} w_{t})}^{2} d x \\ (2.39) & ⩽ \sum_{j = 1}^{n} \int_{R^{n}} (φ + \overline{φ}) {(\partial_{x_{j}} f)}^{2} d x + \sum_{j = 1}^{n} R_{1 j} (t), \end{array}$ where $\begin{array}{l} B_{1 j} (t, x) : = {\overline{A}}_{1} (t, x) - \overline{φ} - \sum_{k = 1}^{n} | \partial_{x_{k}} \overline{φ} \partial_{x_{j}} a | - 2 \overline{φ} {‖ {\overline{φ}}^{\frac{1}{2}} Δ (\partial_{x_{j}} U) ‖}_{L^{2}} {‖ {\overline{φ}}^{- \frac{1}{2}} | \nabla U | ‖}_{L^{2}} \\ - \overline{φ} | \partial_{x_{j}} (| U |^{p - 2} ((p - 1) U + | U |)) |, \\ B_{2 j} (t, x) : = {\overline{A}}_{2} (t, x) - φ - φ | \partial_{x_{j}} (| U |^{p - 2} ((p - 1) U + | U |)) | \\ - 2 φ {‖ φ^{\frac{1}{2}} Δ (\partial_{x_{j}} U) ‖}_{L^{2}} {‖ φ^{- \frac{1}{2}} | \nabla U | ‖}_{L^{2}} - | \partial_{x_{k}} φ \partial_{x_{j}} a (x) |, \\ B_{3 k j} (t, x) : = {\overline{A}}_{3 k} (t, x) - \overline{φ} | \partial_{x_{j}} a (x) |, \end{array}$ and $\begin{array}{l} R_{1 j} (t) : = & 2 \int_{R^{n}} [φ {‖ φ^{\frac{1}{2}} Δ (\partial_{x_{j}} U) ‖}_{L^{2}} {‖ φ^{- \frac{1}{2}} | \nabla U | ‖}_{L^{2}} + \overline{φ} {‖ {\overline{φ}}^{\frac{1}{2}} Δ (\partial_{x_{j}} U) ‖}_{L^{2}} {‖ {\overline{φ}}^{- \frac{1}{2}} | \nabla U | ‖}_{L^{2}}] | \nabla w |^{2} d x \\ + \sum_{k = 1}^{n} \int_{R^{n}} | \partial_{x_{j}} a (x) | [φ + \overline{φ} + | \partial_{x_{k}} φ | + | \partial_{x_{k}} \overline{φ} |] {(\partial_{x_{k}} w_{t})}^{2} d x \\ + \int_{R^{n}} (φ + \overline{φ}) | \partial_{x_{j}} (| U |^{p - 2} ((p - 1) U + | U |)) | w^{2} d x . \end{array}$

We now analyze the coefficients of inequality (2.39). By (2.3)–(2.8), it holds $\begin{array}{l} B_{1 j} (t, x) = {\overline{φ}}_{t t} - λ_{2} {\overline{φ}}_{t} - \overline{φ} - p | U |^{p - 1} φ - 2 \overline{φ} {‖ {\overline{φ}}^{\frac{1}{2}} Δ (\partial_{x_{j}} U) ‖}_{L^{2}} {‖ {\overline{φ}}^{- \frac{1}{2}} | \nabla U | ‖}_{L^{2}} \\ - 2 | U |^{p - 2} ((p - 1) U + | U |) \overline{φ} - \overline{φ} | \partial_{x_{j}} (| U |^{p - 2} ((p - 1) U + | U |)) | + O (ε) \\ ⩾ [c_{1}^{2} + c_{1} λ_{2} - 1 - p (c_{2} + 2) | U |^{p - 1} - 2 ε - | \partial_{x_{j}} (| U |^{p - 2} ((p - 1) U + | U |)) |] \overline{φ} + O (ε) \\ ⩾ C \overline{φ} + O (ε), \\ B_{2 j} (t, x) = 2 λ_{2} φ - φ_{t} - 2 \overline{φ} - φ - p | U |^{p - 1} φ - φ | \partial_{x_{j}} (| U |^{p - 2} ((p - 1) U + | U |)) | \\ - 2 φ {‖ φ^{\frac{1}{2}} Δ (\partial_{x_{j}} U) ‖}_{L^{2}} {‖ φ^{- \frac{1}{2}} | \nabla U | ‖}_{L^{2}} + O (ε) \\ ⩾ [c_{1} (2 λ_{2} + c_{1}) - c_{2} (1 + p | U |^{p - 1} + | \partial_{x_{j}} (| U |^{p - 2} ((p - 1) U + | U |)) | + 2 ε) - 2] \overline{φ} \\ + O (ε) \\ ⩾ C \overline{φ} + O (ε), \\ B_{3 k j} (t, x) = (1 + \int_{R^{n}} | \nabla U |^{2} d x) (2 \overline{φ} - φ_{t}) - a (x) {\overline{φ}}_{t} - \overline{φ} | \partial_{x_{j}} a (x) | + O (ε) \\ ⩾ [(c_{1}^{2} + 2) (1 + \int_{R^{n}} | \nabla U |^{2} d x) + c_{1} σ - ε] \overline{φ} + O (ε) \\ ⩾ C \overline{φ} + O (ε), \end{array}$ and $\begin{array}{l} 2 a (x) φ - 2 λ_{1} \sum_{i = 1}^{n} | \partial_{x_{i}} φ | - | \partial_{x_{k}} \overline{φ} a (x) | - φ | \partial_{x_{j}} a (x) | ⩾ (2 c_{1} σ - c_{2} ε) \overline{φ} + O (ε) ⩾ C \overline{φ} + O (ε) . \end{array}$

According to above estimation and (2.28)–(2.30) and (2.31), (2.39) can be reduced into $\begin{array}{l} \sum_{j = 1}^{n} \int_{R^{n}} \overline{φ} ({(\partial_{x_{j}} w_{t})}^{2} + {(\partial_{x_{j}} w)}^{2} + | \nabla \partial_{x_{j}} w |^{2} + λ_{1} \sum_{i, k = 1}^{n} {(\partial_{x_{i}} \partial_{x_{j}} \partial_{x_{k}} w)}^{2}) d x \\ + C \sum_{j = 1}^{n} \int_{0}^{t} \int_{R^{n}} \overline{φ} ({(\partial_{x_{j}} w)}^{2} + {(\partial_{x_{j}} w_{t})}^{2} + | \nabla \partial_{x_{j}} w |^{2} + | \nabla \partial_{x_{j}} w_{t} |^{2} \\ + λ_{1} \sum_{i, k = 1}^{n} {(\partial_{x_{i}} \partial_{x_{j}} \partial_{x_{k}} w)}^{2}) d x d τ \\ ≲ \sum_{j = 1}^{n} \int_{R^{n}} \overline{φ} (0, x) ({(\partial_{x_{j}} w_{0})}^{2} + {(\partial_{x_{j}} w_{1})}^{2} + | \nabla \partial_{x_{j}} w_{0} |^{2} + λ_{1} \sum_{i, k = 1}^{n} {(\partial_{x_{i}} \partial_{x_{k}} \partial_{x_{j}} w_{0})}^{2}) d x \\ (2.40) & + \sum_{j = 1}^{n} \int_{0}^{t} \int_{R^{n}} (φ + \overline{φ}) {(\partial_{x_{j}} f)}^{2} d x d τ + \sum_{j = 1}^{n} \int_{0}^{t} R_{1 j} (τ) d τ . \end{array}$

We notice that the term $\int_{0}^{t} R_{1 j} (τ) d τ$ can be controlled by (2.32), thus by Gronwall’s inequality, it derives $\begin{array}{l} \sum_{j = 1}^{n} \int_{R^{n}} \overline{φ} ({(\partial_{x_{j}} w)}^{2} + {(\partial_{x_{j}} w_{t})}^{2} + | \nabla \partial_{x_{j}} w |^{2} + λ_{1} \sum_{i, k = 1}^{n} {(\partial_{x_{i}} \partial_{x_{j}} \partial_{x_{k}} w)}^{2}) d x \\ ≲ e^{- C t} \sum_{θ = 0}^{1} \sum_{j = 1}^{n} [\int_{R^{n}} \overline{φ} (0, x) ({(\partial_{x_{j}}^{θ} w_{0})}^{2} + {(\partial_{x_{j}}^{θ} w_{1})}^{2} + {| \nabla \partial_{x_{j}}^{θ} w_{0} |}^{2} + λ_{1} \sum_{i, k = 1}^{n} {(\partial_{x_{i}} \partial_{x_{k}} \partial_{x_{j}}^{θ} w_{0})}^{2}) d x \\ + \int_{0}^{t} \int_{R^{n}} (φ + \overline{φ}) {(\partial_{x_{j}}^{θ} f)}^{2} d x d τ] . \end{array}$

Let $1 ⩽ θ ⩽ s - 1$ . Assume it holds $\begin{array}{l} \sum_{j = 1}^{n} \int_{R^{n}} \overline{φ} ({(\partial_{x_{j}}^{θ} w)}^{2} + {(\partial_{x_{j}}^{θ} w_{t})}^{2} + {| \nabla \partial_{x_{j}}^{θ} w |}^{2} + λ_{1} \sum_{i, k = 1}^{n} {(\partial_{x_{i}} \partial_{x_{j}}^{θ} \partial_{x_{k}} w)}^{2}) d x \\ ≲ e^{- C t} \sum_{θ = 0}^{s - 1} \sum_{j = 1}^{n} [\int_{R^{n}} \overline{φ} (0, x) ({(\partial_{x_{j}}^{θ} w_{0})}^{2} + {(\partial_{x_{j}}^{θ} w_{1})}^{2} + {| \nabla \partial_{x_{j}}^{θ} w_{0} |}^{2} + λ_{1} \sum_{i, k = 1}^{n} {(\partial_{x_{i}} \partial_{x_{k}} \partial_{x_{j}}^{θ} w_{0})}^{2}) d x \\ (2.41) & + \int_{0}^{t} \int_{R^{n}} (φ + \overline{φ}) {(\partial_{x_{j}}^{θ} f)}^{2} d x d τ] . \end{array}$

When $θ = s$ , we multiply both sides of (2.35) with $φ (t, x) \partial_{x_{j}}^{s} w_{t}$ and $\overline{φ} (t, x) \partial_{x_{j}}^{s} w$ , respectively, using the same process with (2.40), we derive $\begin{array}{l} \sum_{j = 1}^{n} \int_{R^{n}} \overline{φ} ({(\partial_{x_{j}}^{s} w_{t})}^{2} + {(\partial_{x_{j}}^{s} w)}^{2} + {| \nabla \partial_{x_{j}}^{s} w |}^{2} + λ_{1} \sum_{i, k = 1}^{n} {(\partial_{x_{i}} \partial_{x_{j}}^{s} \partial_{x_{k}} w)}^{2}) d x \\ + C \sum_{j = 1}^{n} \int_{0}^{t} \int_{R^{n}} \overline{φ} ({(\partial_{x_{j}}^{s} w)}^{2} + {(\partial_{x_{j}}^{s} w_{t})}^{2} + {| \nabla \partial_{x_{j}}^{s} w |}^{2} + {| \nabla \partial_{x_{j}}^{s} w_{t} |}^{2} \\ + λ_{1} \sum_{i, k = 1}^{n} {(\partial_{x_{i}} \partial_{x_{j}}^{s} \partial_{x_{k}} w)}^{2}) d x d τ \\ ≲ \sum_{j = 1}^{n} \int_{R^{n}} \overline{φ} (0, x) ({(\partial_{x_{j}}^{s} w_{0})}^{2} + {(\partial_{x_{j}}^{s} w_{1})}^{2} + {| \nabla \partial_{x_{j}}^{s} w_{0} |}^{2} + λ_{1} \sum_{i, k = 1}^{n} {(\partial_{x_{i}} \partial_{x_{k}} \partial_{x_{j}}^{s} w_{0})}^{2}) d x \\ + \sum_{j = 1}^{n} \int_{0}^{t} \int_{R^{n}} (φ + \overline{φ}) {(\partial_{x_{j}}^{s} f)}^{2} d x d τ + \sum_{j = 1}^{n} \int_{0}^{t} R_{s j} (τ) d τ, \end{array}$ with $\begin{array}{l} R_{s j} (t) : = & 2 \int_{R^{n}} [φ {‖ φ^{\frac{1}{2}} Δ (\partial_{x_{j}}^{s} U) ‖}_{L^{2}} {‖ φ^{- \frac{1}{2}} | \nabla U | ‖}_{L^{2}} + \overline{φ} {‖ {\overline{φ}}^{\frac{1}{2}} Δ (\partial_{x_{j}}^{s} U) ‖}_{L^{2}} {‖ {\overline{φ}}^{- \frac{1}{2}} | \nabla U | ‖}_{L^{2}}] | \nabla w |^{2} d x \\ + \sum_{\begin{array}{c} s_{1} + s_{2} = s \\ 1 ⩽ s_{1} ⩽ s \end{array}} (\binom{s}{s_{1}}) \int_{R^{n}} (φ + \overline{φ}) | \partial_{x_{j}}^{s_{1}} (| U |^{p - 2} ((p - 1) U + | U |)) | {(\partial_{x_{j}}^{s_{2}} w)}^{2} d x \\ + \sum_{k = 1}^{n} \sum_{\begin{array}{c} s_{1} + s_{2} = s \\ 1 ⩽ s_{1} ⩽ s \end{array}} (\binom{s}{s_{1}}) \int_{R^{n}} | \partial_{x_{j}}^{s_{1}} a (x) | [φ + \overline{φ} + | \partial_{x_{k}} φ | + | \partial_{x_{k}} \overline{φ} |] {(\partial_{x_{k}} \partial_{x_{j}}^{s_{2}} w_{t})}^{2} d x, \end{array}$ where C is a positive constant depending on $c_{1}$ , $c_{2}$ and σ.

Since the term $R_{s j}$ can be controlled by (2.41), we can use Gronwall’s inequality to derive $\begin{array}{l} \sum_{j = 1}^{n} \int_{R^{n}} \overline{φ} ({(\partial_{x_{j}}^{s} w)}^{2} + {(\partial_{x_{j}}^{s} w_{t})}^{2} + {| \nabla \partial_{x_{j}}^{s} w |}^{2} + λ_{1} \sum_{i, k = 1}^{n} {(\partial_{x_{i}} \partial_{x_{j}}^{s} \partial_{x_{k}} w)}^{2}) d x \\ ≲ e^{- C t} \sum_{θ = 0}^{s} \sum_{j = 1}^{n} [\int_{R^{n}} \overline{φ} (0, x) ({(\partial_{x_{j}}^{θ} w_{0})}^{2} + {(\partial_{x_{j}}^{θ} w_{1})}^{2} + {| \nabla \partial_{x_{j}}^{θ} w_{0} |}^{2} + λ_{1} \sum_{i, k = 1}^{n} {(\partial_{x_{i}} \partial_{x_{k}} \partial_{x_{j}}^{θ} w_{0})}^{2}) d x \\ + \int_{0}^{t} \int_{R^{n}} (φ + \overline{φ}) {(\partial_{x_{j}}^{θ} f)}^{2} d x d τ], \end{array}$ then according to the special weight function chosen in (2.34), we can get (2.36). □

Based on above results, we are ready to prove the global existence of solution for the linear damped non-local wave equation with variable coefficients (2.1). Proposition 2.1.
Let constants $λ_{i} ⩾ 0$ with $i = 1, 2$ . Assume that ( 2.3 ) hold. The linear problem ( 2.1 ) with the initial date ( 2.2 ) admits a unique global solution $\begin{array}{l} w (t, x) \in C^{1} ((0, \infty); H^{s} (R^{n})) \cap C ((0, \infty); H^{s + 2} (R^{n})) . \end{array}$ Moreover, it satisfies $\begin{array}{l} (2.42) & ‖ w ‖_{H^{s + 2}}^{2} + ‖ w_{t} ‖_{H^{s}}^{2} ≲ e^{- C t} (‖ w_{0} ‖_{H^{s + 2}}^{2} + ‖ w_{1} ‖_{H^{s}}^{2} + \int_{0}^{t} ‖ f ‖_{H^{s}}^{2} d t) . \end{array}$
Proof.
The proof follows theorem 3.2 given in page 18 of [27] by applying the standard fixed point iteration. We only state the outline here. We set $ϕ = {(w, w_{t})}^{T}$ and define $A$ and $G (t, x)$ as follows $\begin{array}{l} A = (\begin{matrix} 0 & 1 \\ L_{1} & L_{2} \end{matrix}) \end{array}$ with $\begin{array}{l} L_{1} : = (1 + \int_{R^{n}} | \nabla U |^{2} d x) Δ - λ_{1} Δ^{2} + | U |^{p - 2} ((p - 1 U + | U |), \\ L_{2} : = div (a \nabla) - λ_{2}, \end{array}$ and $\begin{array}{l} G (t, x) : = (\begin{matrix} 0 \\ f - 2 Δ U \int_{R^{n}} \nabla U \cdot \nabla w d x \end{matrix}) . \end{array}$

The linear equation (2.1) can be written as $\begin{array}{l} (2.43) & \frac{d}{d t} ϕ (t) - A ϕ (t) = G (t, x), t > 0, \end{array}$ with the initial data $\begin{array}{l} ϕ_{0} : = ϕ (0, x) = {(h_{0} (x), h_{1} (x))}^{T} . \end{array}$

We now consider the approximation problem $\begin{array}{l} ϕ (t, x) = ϕ_{0} + \int_{0}^{t} (A ϕ (s, x) + G (s, x)) d s . \end{array}$ In fact, it has a Cauchy sequence ${ϕ_{j}}_{j \in Z^{+}}$ in $H^{s} (R^{n}) \times H^{s + 2} (R^{n})$ that converges to $ϕ (t, x)$ , and it is the solution of linearized system (2.43) in $(0, T]$ . Furthermore, according to Lemma 2.1–2.2, it holds $\begin{array}{l} ‖ w ‖_{H^{s + 2}}^{2} + ‖ w_{t} ‖_{H^{s}}^{2} ≲ e^{- C t} (‖ w_{0} ‖_{H^{s + 2}}^{2} + ‖ w_{1} ‖_{H^{s}}^{2} + \int_{0}^{t} ‖ f ‖_{H^{s}}^{2} d t), \end{array}$ thus the constructed local solution $ϕ (t, x)$ can be extended to the global solution in time.

Suppose that $ϕ_{1}$ and $ϕ_{2}$ are two solutions of system (2.43) with the same initial data, then $ϕ : = ϕ_{1} - ϕ_{2}$ admits zero Cauchy data, and $\begin{array}{l} \frac{d}{d τ} ϕ (t) - A ϕ (t) = 0, ϕ = {(w, w_{t})}^{T}, \end{array}$ through (2.42) we can launch $ϕ \equiv 0$ , so the solution of system (2.43) is unique. This completes the proof. □

3. Construction of approximation solutions

In this section, we will construction an approximation solution of nonlinear approximation problem. We first introduce a family of smooth operators (see the [1,2]). Let $χ \in C_{0}^{\infty} (R)$ such that $χ = 1$ in ${x \in R | | x | ⩽ \frac{1}{2}}$ , otherwise, $χ \equiv 0$ . Following Proposition 1.6 of [2, p.83] or page 72 of [20], we define $Y_{θ}^{(1)}$ by $\begin{array}{l} Y_{θ}^{(1)} U (t, x) = \sum_{n} χ (\frac{x}{θ}) U (t, x), \end{array}$ and $Y_{θ^{'}}^{(2)}$ by $\begin{array}{l} Y_{θ^{'}}^{(1)} U (t, x) = \sum_{n} χ (\frac{t}{θ^{'}}) U (t, x) . \end{array}$

We define $\begin{array}{l} Y_{θ, θ^{'}} : = Y_{θ}^{(1)} Y_{θ^{'}}^{(2)}, \end{array}$ according to the proof given in [1, p.192] or [2], it holds $\begin{array}{l} (3.1) & {‖ Y_{θ, θ^{'}} \partial_{t}^{j} U ‖}_{H^{s_{1}}} ⩽ C θ^{{(s_{1} - s_{2})}_{+}} {(θ^{'})}^{{(j - j^{'})}_{+}} {‖ \partial_{t}^{j^{'}} U ‖}_{H^{s_{2}}}, \forall s_{1}, s_{2} ⩾ 0 \\ (3.2) & {‖ Y_{θ, θ^{'}} \partial_{t}^{j} U - \partial_{t}^{j} U ‖}_{H^{s_{1}}} ⩽ C θ^{s_{1} - s_{2}} {(θ^{'})}^{(j - j^{'})} {‖ \partial_{t}^{j^{'}} U ‖}_{H^{s_{2}}}, 0 ⩽ s_{1} ⩽ s_{2}, \end{array}$ where ${(s_{1} - s_{2})}_{+} : = max (0, s_{1} - s_{2})$ .

In our iteration scheme, we assume that $\begin{array}{l} (3.3) & θ = θ^{'} = N_{m} = N_{0}^{m}, \forall m = 0, 1, 2, \dots, \end{array}$ where $N_{0}$ is a fixed positive constant, we denote it by $Y_{N_{m}}$ for convenience, then by (3.1), it holds $\begin{array}{l} (3.4) & {‖ Y_{N_{m}} \partial_{t}^{s_{1}} U ‖}_{H^{s_{1}}} ≲ N_{m}^{2 (s_{1} - s_{2})} {‖ \partial_{t}^{s_{2}} U ‖}_{H^{s_{2}}}, \forall s_{1} ⩾ s_{2} . \end{array}$

3.1. The first iteration step

We suppose that the initial approximation function $w^{(0)} (t, x)$ satisfies the following conditions $\begin{array}{l} (3.5) & w^{(0)} (t, x) \in C^{\infty} (R^{+} \times R^{n}), {‖ w^{(0)} ‖}_{H^{s}} ⩽ e^{- C t}, \\ (3.6) & w^{(0)} (0, x) = \partial_{t} w^{(0)} (t, x) |_{t = 0} = 0, \\ (3.7) & {‖ \partial^{s} w^{(0)} ‖}_{L^{\infty}} ≲ ε, lim_{| x | \to + \infty} \partial_{x^{i}}^{s} w^{(0)} (t, x) = 0, \forall i = 2, \dots, n, s \in {0, 1, 2, \dots,} . \end{array}$

Then we get the nonhomogeneous linear damped wave equation (3.8) with variable coefficients by linearizing nonlinear equation (1.6) at $w^{(0)} (t, x)$ : $\begin{array}{l} L_{w^{(0)}}^{(0)} h^{(1)} : = & h_{t t}^{(1)} - (1 + \int_{R^{n}} {| \nabla U + \nabla w^{(0)} |}^{2} d x) Δ h^{(1)} - 2 (Δ U + Δ w^{(0)}) \int_{R^{n}} \nabla U \cdot \nabla h^{(1)} d x \\ (3.8) & + λ_{1} Δ^{2} h^{(1)} - div (a (x) \nabla h_{t}^{(1)}) + λ_{2} h_{t}^{(1)} - D_{k} h^{(1)} = E^{(0)} (t, x), \end{array}$ where $\begin{array}{l} D_{k} : = U \sum_{k = 1}^{p - 1} (\binom{p - 1}{k}) k {| w^{(0)} |}^{k - 1} | U |^{p - 1 - k} + \sum_{k = 0}^{p - 1} (\binom{p - 1}{k}) (| w^{(0)} | + k w^{(0)}) {| w^{(0)} |}^{k - 1} | U |^{p - 1 - k} . \end{array}$

We now consider the linear damped non-local wave equation with variable coefficient derive external force as follows $\begin{array}{l} (3.9) & L_{w^{(} 0)}^{(0)} h^{(1)} = Y_{N_{1}} E^{(0)} (t, x), \forall (t, x) \in R^{+} \times R^{n}, \end{array}$ with the initial data $\begin{array}{l} (3.10) & h^{(1)} (0, x) = h_{0}^{(1)}, h_{t}^{(1)} (0, x) = h_{1}^{(1)}, \end{array}$ where the external force $E^{(0)} (t, x)$ is related to the error term at the initial approximation function.

Furthermore, It can be inferred that there is a global solution of linear equation (3.9) by means of assumptions (3.5)–(3.7) and proposition 2.1.

Proposition 3.1.
Let constants $λ_{i} ⩾ 0$ with $i = 1, 2$ . The linear problem ( 3.9 ) with the initial data ( 3.10 ) admits a unique global solution $\begin{array}{l} h^{(1)} (t, x) \in C^{1} ((0, \infty); H^{s} (R^{n})) \cap C ((0, \infty); H^{s + 2} (R^{n})) . \end{array}$ Moreover, it satisfies $\begin{array}{l} (3.11) & {‖ h^{(1)} ‖}_{H^{s + 2}}^{2} + {‖ h_{t}^{(1)} ‖}_{H^{s}}^{2} ≲ e^{- C t} ({‖ h_{0}^{(1)} ‖}_{H^{s + 2}}^{2} + {‖ h_{1}^{(1)} ‖}_{H^{s}}^{2} + \int_{0}^{t} {‖ Y_{N_{1}} E^{(0)} ‖}_{H^{s}}^{2} d τ) . \end{array}$

3.2. The general iteration step

Let $0 < ε ≪ 1$ , we define $\begin{array}{l} B_{ε} : = {w^{(i)} \in H^{s} (R^{n}) : {‖ w^{(i)} ‖}_{H^{s + 2}} ⩽ ε} \end{array}$ with the integer $1 ⩽ i ⩽ m - 1$ .

We denote the m-th approximate step of the nonlinear equation (1.6) with $h^{(m)} (t, x)$ $(m = 2, 3, \dots)$ , where we set $\begin{array}{l} h^{(m)} (t, x) : = w^{(m)} (t, x) - w^{(m - 1)} (t, x), \end{array}$ it holds $\begin{array}{l} w^{(m)} (t, x) : = w^{(0)} (t, x) + h^{(1)} (t, x) + \sum_{i = 2}^{m} h^{(i)} (t, x) . \end{array}$

We consider the following initial value problem (3.12) by the linearization of nonlinear equation (1.6) around $w^{(m - 1)} (t, x)$ : $\begin{array}{l} (3.12) & \{\begin{matrix} L_{w^{(m - 1)}}^{(m - 1)} h^{(m)} = Y_{N_{m}} E^{(m - 1)}, \forall (t, x) \in R^{+} \times R^{n}, \\ h^{(m)} (0, x) = h_{0}^{(m)} (x), h_{t}^{(m)} (0, x) = h_{1}^{(m)} (x), \end{matrix} \end{array}$ where the error term is $\begin{array}{l} (3.13) & E^{(m - 1)} : = L (w^{(m - 1)}) = R (h^{(m)}), \end{array}$ and $\begin{array}{l} (3.14) & R (h^{(m)}) : = L (w^{(m)}) - L (w^{(m - 1)}) - L_{w^{(m - 1)}}^{(m - 1)} (h^{(m)}), \end{array}$ which is also the nonlinear term in the approximation problem (1.6) at $w^{(m - 1)} (t, x)$ .

By the same process of getting proposition 3.1, we can construct the m-th approximation solution.

Proposition 3.2.
Let constants $λ_{i} ⩾ 0$ with $i = 1, 2$ . The linearized problem ( 3.12 ) admits a unique global solution $\begin{array}{l} h^{(m)} (t, x) \in C^{1} ((0, \infty); H^{s} (R^{n})) \cap C ((0, \infty); H^{s + 2} (R^{n})), \end{array}$ which satisfies $\begin{array}{l} (3.15) & {‖ h^{(m)} ‖}_{H^{s + 2}}^{2} + {‖ h_{t}^{(m)} ‖}_{H^{s}}^{2} ≲ e^{- C t} ({‖ h_{0}^{(m)} ‖}_{H^{s + 2}}^{2} + {‖ h_{1}^{(m)} ‖}_{H^{s}}^{2} + \int_{0}^{t} {‖ Y_{N_{m}} E^{(m - 1)} ‖}_{H^{s}}^{2} d τ) . \end{array}$
Proof.
On one hand, we find the m-th $(m ⩾ 2)$ approximation solution $h^{(m)} (t, x)$ , which is equivalent to find $h^{(m)} (t, x)$ such that $\begin{array}{l} (3.16) & w^{(m)} (t, x) = w^{(m - 1)} (t, x) + h^{(m)} (t, x), \end{array}$ substituting (3.16) into (1.6), it holds $\begin{array}{l} (3.17) & L (w^{(m)}) = L (w^{(m - 1)}) + L_{w^{(m - 1)}}^{(m - 1)} (h^{(m)}) + R (h^{(m)}), \end{array}$ then we set $\begin{array}{l} L_{w^{(m - 1)}}^{(m - 1)} (h^{(m)}) = - L (w^{(m - 1)}) = - Y_{N_{m}} E^{(m - 1)}, \end{array}$ which is a linear damping non-local wave equation as in the form of (3.8) by replacing $w^{(0)}$ with $w^{(m - 1)}$ , and the error term $\begin{array}{l} E^{(m - 1)} : = L (w^{(m - 1)}) = R (h^{(m)}) . \end{array}$

On the other hand, we denote the s-th derivatives of time or spacial variables in $\partial^{s}$ , and apply $\partial^{s}$ to both side of (3.16) to get $\begin{array}{l} (3.18) & \partial^{s} w^{(m - 1)} (t, x) = \partial^{s} w^{(0)} (t, x) + \partial^{s} h^{(1)} (t, x) + \sum_{i = 2}^{m - 1} \partial^{s} h^{(i)} (t, x), \forall s \in N, \end{array}$ for a sufficient small positive parameter ε, it holds $\begin{array}{l} {‖ h^{(i)} ‖}_{H^{s}} ≲ ε, \end{array}$ thus we can see that $\begin{array}{l} \partial^{s} w^{(m - 1)} (t, x) ∽ \partial^{s} w^{(0)} (t, x) + O (ε), \end{array}$ it imply that the initial approximation function $w^{(0)} (t, x)$ is the leading term of the m-th approximation solution. We notice that the linear equation (3.9) and the linear equation of m-th approximation solutions have the same structure. By means of the same argument as in the proof of Proposition 3.1, we can verify that the linear problem (3.12) exists a unique global solution $\begin{array}{l} h^{(m)} (t, x) \in C^{1} ((0, \infty); H^{s} (R^{n})) \cap C ((0, \infty); H^{s + 2} (R^{n})) . \end{array}$ Meanwhile, (3.15) can be obtained. The proof is completed. □

4. The nonlinear problem

In this section, the target is to show that $w^{(\infty)} (t, x)$ is indeed a unique global solution of the nonlinear equations (1.6). This is equivalent to prove that the series $\sum_{i = 1}^{m} h^{(i)} (t, x)$ is convergent.

We now give the tame estimation of error term in each iteration scheme.

Lemma 4.1.

Let $s ⩾ 1$ . The error term verifies $\begin{array}{l} (4.1) & {‖ Y_{N_{m}} E^{(m - 1)} ‖}_{H^{s}} = {‖ Y_{N_{m}} R (h^{(m)}) ‖}_{H^{s}} ≲ N_{m}^{4 s} {‖ h^{(m)} ‖}_{H^{s}}^{2} . \end{array}$

Proof.

The error term is $\begin{array}{l} R (h^{(m)}) : = & - 2 Δ w \int_{R^{n}} \nabla U \cdot \nabla w d x - (Δ w + Δ U) \int_{R^{n}} | \nabla w |^{2} d x \\ (4.2) & - U \sum_{k = 2}^{p - 1} (\binom{p - 1}{k}) | w |^{k} | U |^{(p - 1) - k} - w \sum_{k = 1}^{p - 1} (\binom{p - 1}{k}) | w |^{k} | U |^{(p - 1) - k}, \end{array}$ and the highest order for the derivative of x is 4. Since the solution of (3.12) should be constructed in $B_{ε}$ , it holds $\begin{array}{l} {‖ h^{(m)} ‖}_{H^{s}}^{s} ⩽ {‖ h^{(m)} ‖}_{H^{s}}^{2}, for p ⩾ 2 . \end{array}$

Thus we can estimate each term of $R (h^{(m)})$ by applying Cauchy‘s inequality and (4.2) $\begin{array}{l} {‖ Y_{N_{m}} R (h^{(m)}) ‖}_{H^{s}} ≲ N_{m}^{4 s} {‖ h^{(m)} ‖}_{H^{s}}^{2} . \end{array}$ □

We now prove that the iteration scheme is convergent. For any $s > 1$ , let $1 ⩽ \overline{k} < k_{0} ⩽ k ⩽ s$ and $\begin{array}{l} k_{m} : = \overline{k} + \frac{k - \overline{k}}{2^{m}}, α_{m + 1} : = k_{m} - k_{m + 1} = \frac{k - \overline{k}}{2^{m + 1}}, \end{array}$ which gives that $\begin{array}{l} (4.3) & k_{0} > k_{1} > \dots > k_{m} > k_{m + 1} > \dots . \end{array}$

Proposition 4.2.

The nonlinear problem ( 1.6 ) with the initial data ( 1.7 ) admits a unique global Sobolev regularity solution $\begin{array}{l} w^{(\infty)} (t, x) = w^{(0)} (t, x) + \sum_{m = 1}^{\infty} h^{(m)} (t, x) + e^{- t^{2}} w_{0} (x) + t e^{- t} w_{1} (x) . \end{array}$

Proof.

The proof is based on the induction. For convenience, we first consider the case with $w_{0} (x) = w_{1} (x) = 0$ . After that, we discuss the case $w_{0} (x) ≢ 0$ and $w_{1} (x) ≢ 0$ . Note that $N_{m} = N_{0}^{m}$ with $N_{0} > 1$ . For all $m = 1, 2, \dots$ , we claim that there is a positive constant ε small enough to hold $\begin{array}{l} (4.4) & \begin{aligned} {‖ h^{(m)} ‖}_{H^{k_{m - 1}}} < ε^{2^{m - 1}}, \\ {‖ Y_{N_{m}} E^{(m)} ‖}_{H^{k_{m - 1}}} < ε^{2^{m}}, \\ w^{(m)} \in B_{ε} . \end{aligned} \end{array}$ For the case of $m = 1$ , by (3.15), and by setting $0 < ε_{0} < N_{0}^{- (64 + k - \overline{k})} ε^{2} ≪ 1$ , it holds $\begin{array}{l} {‖ h^{(1)} ‖}_{H^{k_{0}}} ≲ N_{1}^{2 (k_{0} - 1)} {‖ Y_{N_{1}} E^{(0)} ‖}_{H^{k_{0}}} < ε_{0} < ε^{2} . \end{array}$

Moreover, according to (4.1) and the above estimation, we derive $\begin{array}{l} {‖ Y_{N_{1}} E^{(1)} ‖}_{H^{k_{0}}} ≲ {‖ Y_{N_{1}} R (h^{(1)}) ‖}_{H^{k_{0}}} < ε^{2}, \end{array}$ and $\begin{array}{l} {‖ w^{(1)} ‖}_{H^{k_{0}}} ≲ {‖ w^{(0)} ‖}_{H^{k_{0}}} + {‖ h^{(1)} ‖}_{H^{k_{0}}} ≲ ε, \end{array}$ which imply that $w^{(1)} \in B_{ε}$ .

Assume that the case of $m - 1$ holds $\begin{array}{l} (4.5) & \begin{aligned} {‖ h^{(m - 1)} ‖}_{H^{k_{m - 2}}} < ε^{2^{m - 2}}, \\ {‖ Y_{N_{m - 1}} E^{(m - 1)} ‖}_{H^{k_{m - 2}}} < ε^{2^{m - 1}}, \\ w^{(m - 1)} \in B_{ε} . \end{aligned} \end{array}$ Then we prove that the case of m holds. Combining with (3.15) and the second inequality of (4.5), we derive $\begin{array}{l} (4.6) & \begin{aligned} {‖ h^{(m)} ‖}_{H^{k_{m - 1}}} ≲ N_{m}^{k_{m - 1}} {‖ Y_{N_{m}} E^{(m - 1)} ‖}_{H^{k_{m - 1}}} < ε^{2^{m - 2}}, \end{aligned} \end{array}$ which combining with (4.1)–(4.3) gives $\begin{array}{l} (4.7) & \begin{aligned} {‖ Y_{N_{m}} E^{(m)} ‖}_{H^{k_{m}}} & ≲ N_{m}^{4} {({‖ Y_{N_{m - 1}} E^{(m - 1)} ‖}_{H^{k_{m - 1}}})}^{2} \\ ≲ N_{0}^{(4 + α_{m + 1}) m + 2 (4 + α_{m + 2}) (m - 1)} {({‖ Y_{N_{m - 2}} E^{(m - 2)} ‖}_{H^{k_{m - 2}}})}^{2^{2}} \\ ≲ \dots, \\ ≲ {(N_{0}^{64 + k - \overline{k}} {‖ Y_{N_{1}} E^{(0)} ‖}_{H^{k_{0}}})}^{2^{m}} . \end{aligned} \end{array}$

We choose a sufficiently small positive constant ε such that $\begin{array}{l} 0 < N_{0}^{64 + k - \overline{k}} {‖ Y_{N_{1}} E^{(0)} ‖}_{H^{k_{0}}} < ε^{2} \end{array}$ thus by (4.7), we have $\begin{array}{l} (4.8) & {‖ Y_{N_{m}} E^{(m)} ‖}_{H^{k_{m}}} < ε^{2^{m}}, \end{array}$ and it holds $\begin{array}{l} lim_{m \to \infty} {‖ Y_{N_{m}} E^{(m)} ‖}_{H^{k_{m}}} = 0 . \end{array}$

On the other hand, note that $N_{m} = N_{0}^{m}$ , by (4.5)–(4.6), it holds $\begin{array}{l} {‖ w^{(m)} ‖}_{H^{k_{m}}} ≲ {‖ w^{(m - 1)} ‖}_{H^{k_{m}}} + {‖ h^{(m)} ‖}_{H^{k_{m}}} ≲ ε . \end{array}$ This means that $w^{(m)} \in B_{ε}$ . Hence we conclude that (4.4) holds.

Therefore, the nonlinear equation (1.6) with the zero initial data exists a unique global Sobolev solution $\begin{array}{l} w^{(\infty)} (t, x) = w^{(0)} (t, x) + \sum_{m = 1}^{\infty} h^{(m)} (t, x), \end{array}$ and we use (3.6) to get $\begin{array}{l} w^{(0)} (0, x) = \partial_{t} w^{(0)} (t, x) |_{t = 0} = 0 . \end{array}$

We now discuss the case of small non-zero initial data. Introduce the auxiliary function $\begin{array}{l} \overline{w} (t, x) = w (t, x) - e^{- t^{2}} w_{0} (x) - t e^{- t} w_{1} (x), \forall x \in R^{n} . \end{array}$ the initial data can be reduced into $\begin{array}{l} \overline{w} (0, x) = \partial_{t} \overline{w} (t, x) |_{t = 0} = 0 . \end{array}$ and nonlinear equation (1.6) is transformed into equations of $\overline{w}$ .

Thus we can construct a unique global Sobolev solution $\overline{w}$ by above iteration scheme. Furthermore, the global Sobolev solution of nonlinear equation (1.6) with a small non-zero initial data takes the form $\begin{array}{l} \overline{w} (t, x) + e^{- t^{2}} w_{0} (x) + t e^{- t} w_{1} (x), \end{array}$ and this solution is unique due to the uniqueness of each iteration step $h^{(m)} (t, x)$ . This completes the proof. □

Footnotes

Acknowledgement

The third author is supported by Guangxi Natural Science Foundation No 2021JJG110002 and NSFC No 12161006.

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